Je Suis

Denis Talay

Yu Alexander

Veretenni- Kov, D'

François Bolley

Arnaud Gloter

Alessandra Lunardi

Stéphane Menozzi

Gilles Pagès

Ta- Lay Denis

Enrico Priola

Oscar Jarrin

Mhamed Gaïgi

Kawther Mayoufi

Florian Rasamoely

Ricardo Romo

Mohamed Seddik De

Halaleh Chiara Amorino

Vincent Runge

Wissal Sabbagh

Elizabeth Zuniga

Yannick Armenti

Quentin Clai- Ron

Virginie Stanislas

Christophe Je N'oublie Pas Bien Sûr

Stéphane Profeta

Monique Crépey

Gilles Jeanblanc

Julia Lacombe

Sergio Matos

Abass Pulido

Vincent Sagna

Alexandre Torri

Vidal

Mamie Lène

Mamie Thé

Basile, Louis, Siegrid, Inaya, Stan Coralie Alaric

Raoul D'

Ma Marraine

Christian, Martin Léa Delphine

Ce manuscrit étant relativement long et dense, je profite de ces premières pages, alors que le lecteur a encore l'esprit frais, pour remercier les personnes sans qui ce travail n'aurait pas été possible.

En premier lieu, je tiens à remercier chaleureusement mon directeur de thèse, Stéphane Menozzi, avec qui j'ai eu la chance de me lancer dans cette aventure doctorale. Ce brillant mathématicien s'est énormément investi dans ma thèse. Depuis l'autre bout du monde, nous avons travaillé d'arrache-pied à des heures indues (quel que soit le fuseau horaire). Lors de mon stage de master sous sa direction, j'ai eu le plaisir de travailler avec Gilles Pagès. Ce fut mon premier contact extérieur avec la recherche en mathématiques. Cette collaboration, très intéressante et fructueuse (cf. Chapitre 3), m'a donné le goût de la constante optimale, cf. Chapitre 4. Ce chapitre assez technique, que j'ai signé seul, a été rendu lisible (non sans souffrance pour la première version) grâce à l'assistance appuyée de Stéphane.

Lors de la thèse, j'ai aussi travaillé avec Arnaud Gloter et Dasha Loukianova, afin d'adapter les techniques établies au Chapitre 3 à des sauts. Cette collaboration de chercheurs a été très enrichissante.

Enfin, ma dernière année de thèse a été marquée par ma collaboration avec Paul-Éric Chaudru de Raynal et Stéphane Menozzi. Ce fut une année intense qui a abouti aux Chapitres 6 et 7. Nous avons reculé (backward ) pour mieux avancer (forward ).

Ces trois belles années de doctorat m'ont ouvert quantité de sujets de recherche que j'ai hâte d'explorer.

Mes années doctorales ont aussi été l'occasion de diriger des TD d'Algèbre. Je remercie Abdelmejid Bayad avec qui j'ai eu le plaisir de m'occuper des L3 pendant trois ans. Je remercie également Charlotte Scribot et Shiqi Song, avec qui j'ai préparé d'autres TD, de m'avoir aidé d'un point de vue pédagogique.

Organisation du mémoire

Afin de faciliter la lecture du document, les principaux symboles sont regroupés à la page 14 avec une brève explication, ainsi que la page où ledit symbole apparaît pour la première fois (dans les deux dernières parties du mémoire).

La première partie de la thèse, rédigée en français, est liminaire et permet de contextualiser les travaux. Dans cette partie, nous présentons également les principaux outils utilisés dans la suite du mémoire. Cette introduction se divise en deux parties comme le reste du manuscrit : -Estimations non-asymptotiques de mesures invariantes -Régularisation par un bruit dégénéré.

La suite de la thèse est rédigée en anglais. Chaque chapitre contient une petite introduction, ainsi qu'une présentation des notations utilisées. La première partie est composée de trois chapitres :

-Le Chapitre 3, A first non-asymptotic concentration result, est un travail effectué en collaboration avec Stéphane Menozzi et Gilles Pagès. Ce chapitre est disponible en prépublication [START_REF] Honoré | Non-Asymptotic Gaussian Estimates for the Recursive Approximation of the Invariant Measure of a Diffusion[END_REF]. Il traite d'inégalités de concentration non-asymptotiques des déviations de la mesure invariante avec une mesure empirique approchant cette dernière.

-Le Chapitre 4, A sharp non-asymptotic concentration, est issu d'un article, écrit seul, disponible en prépublication [START_REF] Honoré | Sharp non-asymptotic Concentration Inequalities for the Approximation of the Invariant Measure of a Diffusion[END_REF]. Il présente des inégalités de concentration du même type que ceux du Chapitre 1 mais avec la constante de concentration optimale.

-Le Chapitre 5, Extension to SDEs driven by a Poisson compound process, a été réalisé en collaboration avec Arnaud Gloter et Dasha Loukianova et a donné lieu à une prépublication [START_REF] Gloter | Non-asymptotic concentration inequality for an approximation of the invariant distribution of a diffusion driven by compound poisson process[END_REF]. Dans ce chapitre, on adapte l'algorithme présenté dans les chapitres précédents à une certaine classe d'EDS à sauts.

La seconde partie de la thèse comporte deux chapitres :

-Le Chapitre 6, Sharp Schauder Estimates for some Degenerate Kolmogorov Equations, en coopération avec Paul Éric Chaudru de Raynal et Stéphane Menozzi. Nous établissons des estimées de Schauder associées au problème de Cauchy pour l'équation de Kolmogorov.

-Le Chapitre 7, Strong uniqueness of degenerate stochastic system, également écrit avec Paul Éric Chaudru de Raynal et Stéphane Menozzi. Nous démontrons l'unicité forte à régularité minimale pour une chaîne d'oscillateurs bruités en la première composante (cadre dégénéré). F k the filtration associated with the scheme, F k :" σppX j q jPrr0,kss q . . . . . . (Σ ) Assumption : Σ i,j pxq " Σ i,j px i^j , ¨¨¨, x d q . . .

Table des matières

Liste des Symboles

R
|| ψ || C k`γ b,d pR nd ,R q || ψ || C k`γ b,d pR nd ,R q " ř n i"1 sup zPR nd sup zPR nd || ψpz, ¨q || C k`γ 2i´1 b pR d ,R q . . . . . . 206 || ¨|| C k`β pR m ,R q the usual Hölder norm, ||ψ|| C k`β pR m ,R q " ř k i"1 sup |α|"i ||D α ψ|| L 8 pR m ,R q `sup

Première partie

Estimations non-asymptotiques de mesures invariantes

Résumé : Dans cette première partie de thèse, nous chercherons à estimer la mesure invariante d'un processus ergodique dirigé par une Équation Différentielle Stochastique. Le Théorème ergodique nous suggère de considérer la mesure empirique associée à un schéma d'approximation du processus sous-jacent qui peut se voir comme le pendant discret de la mesure d'occupation dudit processus. Dans [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF], Lamberton et Pagès ont introduit un algorithme de discrétisation à pas décroissant qui assure la convergence de la mesure empirique du schéma vers la mesure invariante du processus considéré ainsi qu'un théorème central limite (TCL) quantifiant asymptotiquement l'écart entre ces deux mesures. Nous établissons des inégalités de concentration non-asymptotiques pour les déviations de la mesure empirique (cohérentes avec le TCL mentionné ci-avant), ainsi que des contrôles sur la solution de l'équation de Poisson associée, utiles pour ces inégalités.

1 Processus Ergodiques

Mesure invariante

L'étude du comportement de certaines EDS en temps long et l'approximation de leurs états stationnaires interviennent dans de nombreux champs d'applications. Mentionnons par exemple la mécanique Hamiltonienne. On peut penser en particulier à des matériaux soumis à des séismes et des intempéries dont le comportement en temps long doit être estimé, c.f. par exemple le livre de Soize [START_REF] Soize | The Fokker-Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions[END_REF], ou encore à des particules Browniennes interagissant en vitesse/position, voir le Brownien physique de Nelson [START_REF] Nelson | Dynamical theories of Brownian motion[END_REF]. Un autre champ d'application est fourni par la finance, le cadre ergodique apparaît naturellement pour la modélisation de dynamiques de taux d'intérêt ou de volatilités stochastiques, c.f. Fouque et al. [START_REF] Fouque | Derivatives in Financial Markets with Stochastic Volatility[END_REF] Dans une perspective d'approximations numériques concrètes, nous nous concentrerons sur des estimations ergodiques non-asymptotiques. Nous considérons dans un premier temps que la dynamique du processus sous-jacent est donnée par l'équation différentielle stochastique (EDS) suivante : dX t " bpX t qdt `σpX t qdW t , t ą 0,

(1.1) où pW t q tě0 est un mouvement Brownien de dimension r sur un espace de probabilité filtré pΩ, F, pF t q tě0 , Pq, b : R d Ñ R d , σ : R d Ñ R d b R r seront des fonctions Lipschitz. * Tout au long de la Partie II de la thèse, sauf mention explicite, nous supposerons que σ peut éventuellement dégénérer (i.e. être une matrice non-inversible). Nous nous focaliserons sur l'étude de processus pX t q tě0 de dynamique (1.1) qui admettent une unique mesure ν stationnaire et sont ergodiques. Dans la suite de l'introduction, nous réservons la notation en caractères gras pour des objets ayant une dynamique à temps continu. Dans le cas d'un processus stationnaire, on a en particulier pour toute fonction f suffisamment régulière et t ě 0, ż P t f pxqνpdxq " νpf q ": ż f pxqνpdxq, (1.2) où P t est le semi-groupe associé au processus de Markov X t défini par P t f pxq :" Erf pX t q|X 0 " xs.

L'identité (1.2) signifie que si le processus part avec la loi ν alors celle-ci est conservée au cours du temps. On dit d'un processus stationnaire de mesure invariante ν qu'il est ergodique s'il vérifie le théorème ergodique. Introduisons pour tout t ą 0 la mesure d'occupation suivante :

ν t :" 1 t ż t 0 δ Xs ds. (1.3)
Le théorème ergodique est vérifié si pour toute fonction f continue bornée presque sûrement (p.s.) :

lim tÑ`8
ν t pf q " lim tÑ`8 1 t ż t 0 f pX s qds " νpf q " ż f pxqνpdxq.

(1.4)

L'exemple typique vérifiant les propriétés précédentes est fourni par le processus d'Ornstein-Uhlenbeck, associé à bpxq " ´x 2 et σpxq " I d pour la dynamique (1.1), qui s'écrit alors :

dX t " ´1 2 X t dt `dW t , (1.5)
et s'intègre explicitement en X t " e ´t 2 X 0 `e´t 2 ż t 0 e s 2 dW s .

(1.6)

On déduit de (1.6) que pour un X 0 P L 2 pPq indépendant de pW t q tě0 , X t L ÝÑ tÑ`8 N p0, I d q et que si X 0 L " N p0, I d q alors pour tout t ě 0, X t L " N p0, I d q. Les propriétés ergodiques de l'Ornstein-Uhlenbeck setont aisément déduites des conditions garantissant la validité du théorème ergodique données dans la section suivante.

Partie I : Introduction

Théorèmes Ergodiques

Une propriété fondamentale du théorème ergodique est que les moyennes temporelles et les moyennes spatiales sont égales. La quantité aléatoire ν t pf q (intégrale temporelle) converge p.s. vers νpf q (intégrale spatiale par rapport à la mesure invariante). Cette remarquable propriété permet de se focaliser sur l'étude d'une trajectoire du processus pour estimer la mesure invariante à partir d'une seule réalisation observée en temps infini. Cet aspect se révèle très intéressant d'un point de vue numérique lorsque l'on peut observer le processus sur un temps suffisamment long.

Une question naturelle consiste à se demander quels types de critères conduisent au théorème ergodique et à l'unicité de la mesure invariante. Dans le cadre général d'un processus de Markov Fellérien (semi-groupe continu en 0 et contractant), deux grands types d'hypothèses le permettent : l'irréductibilité et la confluence.

Dans ces deux cas, on suppose l'existence d'une fonction V dite de Lyapunov assurant la non-explosion du processus ainsi qu'un retour à la moyenne. Ces propriétés s'expriment en terme du générateur infinitésimal A du processus défini comme suit. Pour tout ϕ P C 2 0 , lim tÑ0 P t ϕ ´ϕ t ": Aϕ.

(1.7)

Dans le cas d'une diffusion de type (1.1), nous avons en particulier par la formule d'Itô :

Aϕ " dP t ϕ dt | t"0 " xb, ∇ϕy `1 2 Tr `σσ ˚D2 ϕ ˘.

(1.8)

Introduisons le Critère d'Has'minskii (voir e.g. [START_REF] Khasminskii | Stochastic Stability of Differential Equations[END_REF]) :

(H) Il existe une fonction de Lyapunov V : R d ÝÑs0, `8r satisfaisant les conditions suivantes : i) Coercivité. lim |x|Ñ8 V pxq " `8.

ii) Non-explosion. Il existe C V ą 0 tel que AV ď C V .

iii) Retour à la moyenne. lim sup |x|Ñ8 AV pxq ă 0.

Présentons maintenant brièvement les deux hypothèses indiquées ci-dessus et renvoyons pour plus d'informations à Ethier et Kurtz [START_REF] Ethier | Markov Processes. Characterization and Convergence[END_REF] ainsi qu'à l'article de synthèse de Pagès [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF].

-Hypothèse d'irréductibilité : on suppose que (H) est vérifiée, et que le processus admet une densité strictement positive. Cette hypothèse assure aussi l'unicité de la mesure invariante.

-Hypothèse de confluence : sous des conditions de type Lyapunov plus fortes que (H) (voir (L V ) ci-après et également [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion: the case of a weakly mean reverting drift[END_REF] et [START_REF] Pagès | A mixed-step algorithm for the approximation of the stationary regime of a diffusion[END_REF] pour des conditions moins fortes), le théorème ergodique est vérifié. Si de plus les coefficients de diffusion satisfont une hypothèse de confluence (on parle aussi de diffusion asymptotiquement plate, i.e. avec les notations usuelles pour les processus de Markov, en un certain sens, on a pour tous points initiaux x, y P R d , |X x t ´Xy t | tÑ`8 0), alors on a également l'unicité de la mesure invariante.

En particulier, le premier jeu d'hypothèses est valable dans le cadre de diffusions uniformément elliptiques verifiant (H) à coefficients réguliers. L'hypothèse de positivité de la densité est clairement liée à la loi du processus et impliquée par une condition de type non-dégenerescence (aspect régularisant du semi-groupe sous-jacent). L'hypothèse de confluence n'est, quant à elle, pas a priori liée à des conditions de non-dégérescence. Elle est de nature beaucoup plus trajectorielle dès que la dimension est supérieure à un † . Notons que le processus d'Ornstein-Uhlenbeck, vérifie ces deux types d'hypothèses et remarquons que Aϕpxq " ´xx 2 , ∇ϕpxqy `1 2 ∆ϕpxq. En effet, en prenant V une forme quadratique, i.e. V pxq " 1 `|x| 2 les hypothèses de (H) sont vraies. On a clairement de (1.6) que la densité de ce processus X x t " N pe ´t 2 x, 1 ´e´t q est séparée de 0 (irréductible), et que la confluence est vérifiée : X x t ´Xy

t " e ´t 2 px ´yq p.s.

Ñ tÑ8 0.
Il est impossible d'observer en temps infini un processus et donc d'avoir exactement la limite de la mesure d'occupation. Nous cherchons donc à l'estimer et à contrôler l'erreur d'approximation. La vitesse de convergence vers la mesure invariante est donnée par le TCL. Pour la diffusion (1.1), Bhattacharya [START_REF] Bhattacharya | On the functional central limit theorem and the law of the iterated logarithm for Markov processes[END_REF], sous des hypothèses de type irréductibilité (diffusions non-dégénérées), a montré que pour toute fonction f à croissance polynomiale on a le théorème central limite :

? t `νt pf ´νpf qq ˘L ÝÑ tÑ`8 N ´0, ż R d |σ ˚∇ϕpxq| 2 νpdxq ¯, (1.9)
avec ϕ solution de l'équation de Poisson

Aϕ " f ´νpf q.

(1.10) La force du TCL est qu'il permet de rendre compte de l'ordre typique des fluctuations de la mesure d'occupation ν t à l'échelle ? t. La variance asymptotique νp|σ ˚∇ϕ| 2 q est l'intégrale de ce que l'on appelle le carré du champ. Pour toutes fonctions ϕ, ψ P C 2 0 , on définit le carré du champ par Γpϕ, ψq :" Apϕ ¨ψq ´Aϕ ¨ψ ´ϕ ¨Aψ. Ce qui conduit à ν pΓ pϕ, ϕqq " ż Γpϕ, ϕqνpdxq " ´2 ż Aϕ ¨ϕ νpdxq " ż |σ ˚∇ϕ| 2 νpdxq.

Nous renvoyons pour plus de précisions, à Bakry et al. [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] et Ledoux [START_REF] Ledoux | Concentration of measure and logarithmic Sobolev inequalities[END_REF]. Nous dirons par la suite que f est un cobord du générateur infinitésimal s'il existe une solution régulière ϕ au problème de Poisson (1.10) (voir la Section 2.2). Cette équation est difficile à résoudre, surtout dans notre approche qui nécessite d'avoir des contrôles ponctuels. Un bon cadre est fourni par les hypothèses précédentes. Notons par ailleurs que ν vérifie l'équation de Fokker-Planck au sens des distributions : A ˚ν " 0, où A ˚est l'opérateur adjoint de A. En fait, ν P I P , où I P est l'ensemble des mesures invariantes pour le processus X t , si et seulement si pour tout ϕ P DpAq (le domaine de définition du générateur A) on a νpAϕq " 0. Si I P est non vide et compact (au sens de la convergence étroite) alors le théorème de Krein-Millman assure qu'il existe au moins une mesure extrémale ν de I P (i.e. qui ne s'écrit pas comme une combinaison convexe de deux autres mesures de †. rappelons en effet, c.f. Has'minskii [START_REF] Khasminskii | Stochastic Stability of Differential Equations[END_REF] et Lemaire et al. [LPP15] qu'en dimension 1 toutes les diffusions ergodiques sont confluentes en un certain sens. I P ). Notons au passage que l'existence de ν ˚implique que pX t q tě0 est ergodique. Pour plus de renseignements, nous renvoyons à [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF] et à [START_REF] Dunford | Linear Operators[END_REF].

Cette caractérisation d'une mesure invariante se comprend bien formellement. En effet, pour tout ϕ P C 2 0 , d'après la définition du générateur infinitésimal dans (1.7) :

νpAϕq " ż Aϕνpdxq " ż lim tÑ0 P t ϕpxq ´ϕpxq t νpdxq " lim tÑ0 ş P t ϕpxqνpdxq ´νpϕq t " 0,

(1.11) l'avant-dernière égalité se justifie par un argument d'interversion limite intégrale, et la dernière identité par la définition (1.2) d'une mesure invariante. Enfin, remarquer que l'identité νpAϕq " 0 se vérifient pour tout ϕ suffisamment régulière telle que l'équation (1.10) est vérifiée.

L'hypothèse de régularité Lipschitz des coefficients b et σ ainsi que l'hypothèse de Lyapunov ad hoc assure que I V ‰ ∅. Notamment, l'Hypothèse de confluence ou l'Hypothèse d'irréductibilité précédement introduis assurent également l'uncité de la mesure invariante. Pour plus d'informations, nous renvoyons à la monographie de Khasminskii [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF] (ainsi qu'à la seconde édition [START_REF] Khasminskii | Stochastic Stability of Differential Equations[END_REF]), ou bien à celle de Villani [Vil09], à [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF] ainsi qu'à [START_REF] Da Prato | Elliptic operators on rd with unbounded coefficients[END_REF] et à [START_REF] Metafune | Feller semigroups on r n[END_REF]. Enfin la compacité de I P se montre à partir des hypothèse de Lyapunov, c.f. [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] 1. 3

Discrétisation du processus

En pratique, nous ne pouvons pas simuler directement le processus continu pX t q tě0 . Il nous faut donc passer par une méthode de discrétisation. Une approche classique est fourni par le schéma d'Euler à pas constant γ ą 0 associé à (1.1) : X γ n`1 " X γ n `γbpX γ n q `?γσpX γ n qU n`1 , (1.12) où pU n q ně2 est une suite de variables aléatoires de R r indépendantes et identiquement distribuées (i.i.d.) avec les trois premiers moments similaires à ceux de la Gaussienne, et indépendantes de X 0 . Ainsi pX γ n q ně1 est une chaîne de Markov homogène, ce qui nous permet de bénéficier d'une riche littérature (voir par exemple Meyn et Tweedie [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF] pour l'aspect temps long et stabilisation). Une abondante littérature existe pour la convergence de schéma d'Euler à pas constant. Dans le cadre de l'erreur faible à temps fini, i.e. pour des quantités du type E :" Erf pX T q ´f pX n qs avec nγ " T ą 0 pour une bonne «classe» de fonctions f , mentionnons le papier fondateur de Talay et Tubaro [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF], qui les premiers établirent un contrôle de l'erreur E à l'ordre γ sous des hypothèses de régularité sur b, σ et f . Indiquons également les extensions proposées par Bally et Talay dans un cas hypoelliptique [START_REF] Bally | The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function[END_REF] et [START_REF] Bally | The law of the Euler scheme for stochastic differential equations, II. Convergence rate of the density[END_REF] ainsi que les approches parametrix considérées par Konakov et Mammen [START_REF] Konakov | Local limit theorems for transition densities of Markov chains converging to diffusions[END_REF] et [START_REF] Konakov | Edgeworth type expansions for euler schemes for stochastic differential equations[END_REF].

Mentionnons par ailleurs qu'un cas ergodique elliptique est déjà considéré dans [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF]. Des cadres dégénérés liés aux applications précédemment évoquées associées au cadre de la mécanique Hamiltonienne ou problèmes cinétiques ont aussi été abordés par Talay [START_REF] Talay | Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme[END_REF] et par Mattingly et al. [START_REF] Mattingly | Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise[END_REF]. Pour simplifier, les dynamiques considérées dans les articles indiqués sont de la forme suivante : dX t " ˆb1 pX t q X 1 t ˙dt `ˆσ 1 pX t q 0 d,d ˙dW t , (1.13) où W t est un mouvement Brownien de R d , X t " pX 1 t , X 2 t q ˚avec X 1 t , X 2 t P R d . Avec les notations de (1.1), la dynamique de (1.13) correspond à bpxq " pb 1 pxq, x 1 q ˚, σpxq " pσ 1 pxq, 0 d,d q. Dans les deux cas, les auteurs obtiennent sous des hypothèses de type irréductibilité, en exhibant une fonction de Lyapunov adéquate et pour un schéma implicite chez [START_REF] Talay | Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme[END_REF], des contrôles sur l'erreur faible νpf q ´νγ pf q en fonction du pas γ.

De plus, sous les conditions considérées, le théorème ergodique donne que pour f à croissance polynomiale, ν γ n pf q p.s.

ÝÑ nÑ8

ν γ pf q la mesure invariante du schéma et non celle de la diffusion (1.1), i.e. la mesure d'occupation converge vers la mesure invariante du schéma et non celle de la diffusion. On pourrait espérer établir un théorème central limite (TCL) pour les déviations de la mesure d'occupation du schéma à pas constant mais cette approche induirait à considérer deux erreurs. Quoiqu'il en soit l'erreur globale se décompose en ce cadre en deux contributions, l'erreur statistique et l'erreur de discrétisation, i.e.

ν γ n pf q ´νpf q " ν γ n pf q ´νγ pf q `νγ pf q ´νpf q ": E S `ED .

Notons que l'étude de l'erreur de discrétisation E D conduit à résoudre deux fois l'équation de Poisson. Celle (1.10) associée à A et celle associée à A γ (générateur du schéma) à savoir A γ ϕ " f ´νpf q, avec A γ ϕ :" γ ´1pP γ ϕ ´ϕq, où P γ est le semi-groupe associé au schéma. L'étude de l'équation de Poisson dans le cadre continu (1.10) est déjà non triviale (voir Section 2.2), ajouter l'étude d'une équation de Poisson discrète rend l'analyse d'autant plus difficile. En outre, d'un point de vue algorithmique, à un pas de temps donné γ, on ne peut espérer atteindre la bonne mesure (sans biais).

La force de l'algorithme à pas décroissant que nous allons utiliser est de supprimer le terme d'erreur de discrétisation E S (la moyenne ergodique du schéma convergera vers la mesure cible ν). Il est également suffisamment robuste pour prendre en compte des dynamiques dégénérées comme celles considérées dans l'équation (1.13) dès lors que le problème associé à la diffusion est, en un certain sens, bien posé.

Les premiers travaux en ce sens remontent à Basak et al. [START_REF] Basak | Weak convergence of recursions[END_REF]. L'algorithme à pas décroissant considéré s'écrit pour n ě 0 :

X n`1 " X n `γn`1 bpX n q `?γ n`1 σpX n qU n`1 , X 0 P L 2 pΩ, F 0 , Pq,

(1.14) où pγ k q kě1 désigne une suite de pas décroissants. Leur résultat principal a été d'établir que la loi de X n converge faiblement (au sens des mesures) vers la mesure invariante ν de la diffusion. Le pas décroissant permet d'une certaine manière d'être de plus en plus précis quand le temps devient grand. Le pas tend vers 0 au fur et à mesure que le temps augmente et, par rapport à un schéma à pas constants comme en (1.12), dans le cadre d'une simulation ergodique, on fait d'un coup tendre n vers l'infini et γ n vers 0.

Mentionnons également [START_REF] Piccioni | An iterative monte carlo scheme for generating lie group-valued random variables[END_REF], qui ont montré un théorème ergodique, et une convergence L 2 associée, pour la mesure d'occupation empirique d'un schéma à pas décroissant de type (1.14) approchant un processus de diffusion à valeurs dans des groupes de Lie compacts. Lamberton et Pagès [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] ont ensuite poussé beaucoup plus loin l'étude de ce type de schémas en établissant dans le cadre de diffusions ergodiques Browniennes générales un théorème ergodique pour la mesure d'occupation empirique du schéma (1.14) ainsi qu'un théorème central limite (TCL) pour les déviations associées.

Pour n ě 0 fixé, la mesure d'occupation empirique se définit précisément de la façon suivante : pour tout A P BpR d q (sous-ensemble Borélien de R d ), ν n pAq :" ν n pω, Aq :"

ř n k"1 γ k δ X k´1 pωq pAq ř n k"1 γ k .
(1.15)

Nous noterons par la suite Γ n :" ř n k"1 γ k qui représente le temps courant associé à (1.14) à l'étape n. Indiquons que Γ n est l'analogue discret du t dans la mesure d'occupation ν t associée à pX s q sPr0,ts introduite en (1.3). Dans une perspective d'étude en temps long, la suite des pas de temps pγ k q kě1 sera choisie telle que Γ n Ñ n `8. Les hypothèses de Lyapunov renforcées (par rapport au critère (H)) considérées par Lamberton et Pagès sont les suivantes :

(L V ) Il existe une fonction de Lyapunov V : R d ÝÑ rv ˚, `8r, avec v ˚ą 0 satisfaisant les conditions suivantes : i) Régularité-Coercivité. V est une fonction de classe C 2 , }D 2 V } 8 ă `8, lim |x|Ñ8 V pxq " `8. ii) Contrôle de la croissance. Il existe C V P p0, `8q tel que pour tout x P R d :

|∇V pxq| 2 `|bpxq| 2 ď C V V pxq.
iii) Stabilité. Il existe α V ą 0, β V P R `tel que pour tout x P R d , AV pxq ď ´αV V pxq `βV .

Ces hypothèses signifient que V est sous-quadratique et b sous-linéaire. Un exemple typique de processus vérifiant ces hypothèses est de nouveau l'Ornstein-Uhlenbeck. Il faut voir la condition iii) comme une contrainte sur la dérive qui doit se comporter comme une force de rappel sous-linéaire afin de faire revenir le processus à la moyenne.

Sous (L V ), Lamberton et Pagès montrent d'abord le théorème ergodique associé à l'algorithme (1.14), i.e. pour toute fonction ν ´p.s. continue f à croissance polynomiale,

ν n pf q p.s. ÝÑ n νpf q " ż R d f pxqνpdxq, (1.16)
qui est le pendant discrétisé du Théorème ergodique (1.4). De ce fait, le principal intérêt de ce schéma vis-à-vis de celui à pas constant est qu'il fait converger la mesure empirique directement vers la mesure invariante de la diffusion. La démonstration de ce théorème ergodique repose sur le caractère tendu de la suite pν n pV qq ně1 où V est une fonction de Lyapunov et sur le théorème d'Echeverria Weiss (identification de la limite en terme de problème de martingale, c.f. [START_REF] Ethier | Markov Processes. Characterization and Convergence[END_REF]). Une fois le résultat ergodique (1.16) à disposition, la question naturelle est de nouveau d'établir un TCL, en regard du résultat de Bhattacharya rappelé en (1.9) dans le cas continu. Notons Γ p q n :" ř n k"1 γ k , ą 0, de sorte qu'en particulier Γ p1q n " Γ n . En se concentrant sur des fonctions f de la forme f ´νpf q " Aϕ où ϕ est une fonction régulière, et pour lesquelles, d'après (1.11), νpAϕq " 0 et ν n pf q ´νpf q " ν n pf ´νpf qq " ν n pAϕq, Lamberton et Pagès ont montré le TCL suivant.

Theorem 1 (TCL de [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF]) Sous les conditions de Lyapunov (L V ), s'il existe une unique mesure invariante à la diffusion (1.1), et si lim n Γ L'algorithme à pas décroissant permet effectivement d'avoir le pendant discret du TCL pour la diffusion énoncé en (1.9). La condition lim n Γ p2q n { ?

Γ n " 0 est une condition sur les pas de temps. Précisément, pour une suite de pas pγ k q kě1 telle que γ k -k ´θ, θ P p0, 1s, où l'on note γ k -k ´θ pour indiquer qu'il existe une constante C ě 1 telle que pour tout k ě 0, C ´1k ´θ ď γ k ď Ck ´θ, le critère lim n Γ p2q n { ?

Γ n " 0 est vérifié pour θ P p 1 3 , 1s. Le TCL précédent est remarquable au sens où, pour la plage de pas considérée, la discrétisation par le schema (1.14) n'est pas visible asymptotiquement. En effet, la loi limite est la même que dans (1.9) et la variance asymptotique correspond bien au carré du champ de la mesure invariante.

Indiquons également que plus le pas est grand plus la convergence sera rapide. Le TCL précédent peut s'étendre au cas θ " 1 3 (meilleure vitesse) au prix de l'apparition d'un biais. L'effet de la discrétisation se fait alors sentir. Le seuil θ " 1 3 est doublement critique au sens où si θ ă 1 3 le TCL n'est plus valable. Il est en quelque sorte caché par l'erreur de discrétisation. Seule subsiste une convergence en probabilité après renormalisation. Nous renvoyons au Chapitre 3 pour de plus amples discussions sur les biais. Indiquons toutefois que celui-ci n'est pas aisé à approcher numériquement dans la mesure où il fait intervenir les dérivées de la fonction ϕ considérée jusqu'à l'ordre 4 ainsi que des intégrales par rapport à la mesure invariante et la loi de l'innovation.

La démonstration du Théorème 1 repose sur le TCL pour des accroissements de martingales de carré intégrable (voir e.g. [START_REF] Hall | Martingale limit theory and its application[END_REF] et [START_REF] Duflo | Méthodes récursives aléatoires. Techniques Stochastiques[END_REF]). Il faut ainsi décomposer ν n pAϕq " M artingale `Reste (comme nous le ferons dans l'équation (2.17) ci-après), puis vérifier que le terme M artingale satisfasse à la condition de Lindeberg (voir e.g. [START_REF] Hall | Martingale limit theory and its application[END_REF]).

En pratique, il est important de connaître les intervalles de confiance associés à l'estimation de la mesure invariante. Or le TCL n'est pas l'outil le plus adapté. A priori notre estimation renormalisée ? Γ n ν n pAϕq ne suit pas une loi normale. Le TCL fournit un résultat asymptotique et ne permet pas, à n ě 0 donné de quantifier l'écart à la loi normale limite.

Le propos de cette première partie de thèse sera d'établir, sous des hypothèses similaires à celles du TCL précédent, une estimation de l'intervalle de confiance nonasymptotique. Nous le ferons grâce à des inégalités de concentration.

Inégalités de concentration

Le but de la première partie de cette thèse est d'établir une inégalité de concentration (sous)-Gaussienne non-asymptotique pour la mesure empirique renormalisée ? Γ n ν n associée au schéma à pas décroissants. Avec des pas constants, Malrieu et Talay dans [START_REF] Malrieu | Concentration inequalities for Euler Schemes[END_REF] ont montré une inégalité de concentration pour la mesure empirique en établissant une inégalité de type Log-Sobolev sur le schéma. On rappelle que l'inégalité de Log-Sobolev implique la concentration Gaussienne (voir par exemple l'argument de Herbst dans [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]). Cette méthode est néanmoins très rigide pour un processus discrétisé.

L'approche développée dans cette thèse, privilégie l'utilisation directe de la propriété de concentration Gaussienne du terme d'innovation pU k q kě1 dans (1.14) qui se transmettra au schéma. Dorénavant, nous supposerons qu'une variable aléatoire U de même loi que les pU k q kě1 dans (1.14) vérifie la propriété de concentration Gaussienne suivante :

(GC) La variable aléatoire U admet les mêmes trois premiers moments que la loi normale N p0, I r q, et vérifie pour toute fonction g : R r Ñ R, 1´Lipschitz et pour tout λ ą 0 :

E " exppλgpU qq ‰ ď exp ˆλE rgpU qs `λ2 2 ˙.
Mentionnons que la propriété (GC) implique directement, via l'inégalité de Bienaymé -Chebyshev -Markov exponentielle et une optimisation en λ, que si l'on note µ la loi de U , alors, pour toute fonction g : R r Ñ R, 1´Lipschitz et pour tout a ą 0 :

µpg ´µpgq ě aq ď expp´a 2 2 q.
La propriété (GC) est entre autres vérifiée par la loi normale et la loi de Rademacher.

1.5 De ν n pAϕq à l'estimation de νpf q pour une source f donnée Nous avons vu que le TCL du Théorème 1 s'énonçait pour des fonctions de la forme f ´νpf q " Aϕ pour ϕ suffisamment régulière. Dans la pratique, la problématique est inverse, on a une source f à disposition dont on souhaite estimer la moyenne νpf q. Ceci pose donc, à source f donnée, la question de trouver ϕ suffisamment régulière telle que Aϕ " f ´νpf q. En d'autres termes, il faut résoudre l'équation de Poisson pour la source f et obtenir une régularité suffisante pour la solution. De façon similaire a ce qui a été évoqué pour le théorème ergodique, deux grands jeux d'hypotèses (la non-dégénerescence et la confluence) permettront d'aboutir au résultat attendu. Nous renvoyons à la Section 2.2 ci-après et au Chapitre 3 pour une présentation plus détaillée. Jusqu'à cette section nous énoncerons nos résultats, de façon similaire au Théorème 1, pour les déviations de ?

Γ n ν n pAϕq.

Premier résultat de concentration

Le premier résultat du Chapitre 3 est le résultat de concentration suivant qui peut se voir comme un premier pendant non-asymptotique du Théorème 1.

Theorem 2 Supposons qu'il existe une unique mesure invariante associée à (1.1), que σ est bornée, que X 0 et pU k q kPrr1,nss vérifient (GC). Si (L V ) est vérifiée alors pour toute fonction ϕ de classe C 3 , telle que ∇ϕ, D 2 ϕ, D 3 ϕ sont bornés et que D 3 ϕ est β-Hölder, β P p0, 1q vérifiant pour une certaine constante C V,ϕ ą 0 |ϕ| ď C V,ϕ p1 `aV qq, et que x Þ Ñ xb, ∇ϕypxq est Lipschitz si Γ p 3`β 2 q n { ?

Γ n Ñ n 0 , alors il existe c n ď 1 ď C n , n ě 1, avec lim n C n " lim n c n " 1, telles que pour tout n ě 1 et a ą 0 (qui peut dépendre de n) : Ainsi, la majoration du Théorème 2 n'est pas optimale vis-à-vis de son pendant asymptotique : le Théorème 1. Nous fournirons plus loin une inégalité de concentration nonasymptotique similaire mais avec le carré du champ dans la borne de droite (voir la Section 3). Contrairement au Théorème 1, le cadre sans biais est pour Γ

P " | a Γ n ν n pAϕq| ě a ‰ ď 2C n exp
p 3`β 2 q n { ? Γ n Ñ n 0 et non Γ p2q n { ?
Γ n Ñ n 0. Pour le choix naturel de pas de temps γ k -k ´θ, θ P p0, 1s, ceci signifie que le théorème précédent est valable pour θ P p 1 2`β , 1s. Il y a donc bien continuité avec le Théorème 1 pour β " 1.

Nous avons réussi à demander moins de régularité à D 3 ϕ dans le Théorème 2 (où D 3 ϕ est β-Hölder) par rapport au Théorème 1 (où D 3 ϕ est Lipschitz). La moindre régularité de ϕ se paye en terme de vitesse de convergence. Mentionnons que l'affaiblissement de la régularité est motivé par la possible prise en considération de sources f Lipschitz pour notre estimation ergodique (cadre usuel des inégalités fonctionnelles). Ce cadre est détaillé en Section 2. 4.

Notons que si ϕ est solution de l'équation de Poisson (1.10), sous les hypothèses du Théorème 2, ∇ϕ et D 2 ϕ sont Lipschitz. Ainsi, pour σ borné et Lipschitz, dès lors que f est aussi Lipschitz, on a clairement de la relation xb, ∇ϕy " f ´1 2 Trpσσ ˚D2 ϕq, que x Þ Ñ xbpxq, ∇ϕpxqy est également Lipschitz.

L'hypothèse |ϕ| ď C V,ϕ p1`?V q est essentiellement technique et utilisée pour contrôler des termes de restes dans la décomposition (2.17) ci-dessous (voir également Chapitre 3). Dans le cas particulier où V pxq -1 `|x| 2 , elle est automatiquement vérifiée de par le caractère uniformément Lipschitzien de ϕ.

Enfin, comme corollaire important du Théorème 2, lorsque ϕ est solution de Aϕ " f ´νpf q, nous déduisons comme annoncé précédemment, l'estimation non-asymptotique suivante pour l'intervalle de confiance associé à la simulation ergodique : P " νpf q P " ν n pf q ´a ? Γ n , ν n pf q `a ?

Γ n ‰ ı ě 1 ´2C n exp `´c n a 2 2}σ} 2 8 }∇ϕ} 2 8 q ˘.

Partie I : Introduction

Soulignons que f n'est pas forcément borné, et qu'à aucun moment de la thèse la source f ne sera supposée bornée, elle sera toutefois au moins Lispschitzienne.

Idée de la preuve. La démonstration s'appuie sur une méthode de martingale (dite approche d'Azuma, voir [START_REF] Azuma | Weighted sums of certain dependent random variables[END_REF]), qui est déjà employée pour prouver la convergence de l'algorithme et le TCL de [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF]. Cette approche a également été utilisée dans l'article [START_REF] Frikha | Concentration bounds for stochastic approximations[END_REF] afin de contrôler la concentration des déviations du schéma d'Euler à horizon de temps fini dans l'algorithme de Monte-Carlo. Nous nous intéresserons aux déviations de ν n pAϕq, l'étude de |ν n pAϕq| se déduit par symétrie. Pour toute fonction ϕ de classe C 2 , on effectue un développement de Taylor entre ϕpX k q et ϕpX k´1 q pour faire apparaître le générateur défini en (1.8). Ce qui nous donne schématiquement :

ϕpX k q ´ϕpX k´1 q " γ k AϕpX k´1 q `∆k pX k´1 , U k q `Rk,k´1 , (2.17) où R k,k´1 est un terme de reste et ∆ k pX k´1 , U k q est un accroissement de martingale. L'écriture explicite de ∆ k pX k´1 , U k q fait apparaître un terme quadratique en U k , mais il se trouve que u Þ Ñ ∆ k pX k´1 , uq est Lipschitz. En effet, dans l'identité (2.17), à droite de l'égalité seul ψ k pX k´1 , U k q dépend de U k alors qu'à gauche de l'égalité ϕpX k q est bien Lipschitz en U k . En particulier, on a conditionnellement à X k´1 :

r∆ k pX k´1 , ¨qs 1 ď ? γ k }σpX k´1 q}}∇ϕ} 8 ď ? γ k }σ} 8 |}∇ϕ} 8 .

(2.18)

Notons qu'à ce stade nous choisissons de majorer le module Lipschitz de ∆ k pX k´1 , ¨q par la dernière inégalité afin d'avoir une borne déterministe, ce qui simplifie l'analyse. Toutefois, nous verrons dans la Section 2.1 que ce contrôle de module de Lipschitz est crucial pour avoir la meilleure constante de concentration possible. On somme le développement de Taylor (2.17) pour aboutir à :

Γ n ν n pAϕq " ´Mn ´Rn , (2.19) avec R n " ϕpX 0 q´ϕpX n q`ř n k"1 R k,k´1 comme terme de reste et M n "

ř n k"1 ∆ k pX k´1 , U k q comme terme martingale qui induit la concentration sous-Gaussienne. Le terme de reste R n se traite de façon assez technique avec des arguments usuels (inégalité de Hölder, Jensen) en exploitant l'intégrabilité exponentielle de la fonction de Lyapunov, i.e. sous les hypothèses du Théorème 2, il existe une constante c V :" c V ppAqq ą 0 telle que : I 1

V :" sup ně0 E rexppc V V pX n qqs ă `8. Pour le terme martingale, l'idée est d'utiliser, en conditionnant itérativement, la propriété (GC) de concentration Gaussienne de l'innovation, couplée au contrôle du module de Lipschitz (2.18). Précisément, l'inégalité de Bienaymé -Chebyshev -Markov exponentielle donne pour tout a ą 0, λ ą 0 :

P " a Γ n ν n pAϕq ě a ‰ ď expp´λ a ?
Γ n qErexppλν n pAϕqqs.

On isole ensuite par l'inégalité d'Hölder le terme martingale et le terme de restes. Pour p, q ą 1, 1 p `1 q " 1, on obtient : (2.20)

P " a Γ n ν n pAϕq ě a ‰ ď
Une optimisation en λ conduit au résultat.

Améliorations du Théorème 2

Pour l'instant, le Théorème 2 nous donne une majoration dudit carré du champ : νp|σ ˚∇ϕ| 2 q ď }σ} 2 8 }∇ϕ} 2 8 . Par la suite, nous réussissons à améliorer la constante de concentration du Théorème 2 et à s'approcher du carré du champ. Nous voyons toujours l'importance d'étudier le problème de Poisson afin de vérifier que ~σ~2 ´νp~σ~2q est cobord. Indiquons que pour cette étape il est appréciable de bien connaître ~¨~, i.e. calculable, en pratique on choisit la norme de Fröbenius. Dans le cas général où ~¨~est une norme d'opérateur, l'étude de l'équation de Poisson Aϑ " ~σ~2 ´νp~σ~2q est plus complexe. On a besoin de savoir que ~σ~est suffisamment régulière. En outre, rappelons que ? Γ n {Γ p2q n Ñ n `8 pour tout θ P p 1 2`β , 1s, la condition sur a est valable pour les intervalles de confiance à a fixé. Ce théorème représente un véritable intérêt d'un point de vue numérique notamment lorsque νp~σ~2q est bien plus petit que }σ} 2 8 (voir la partie numérique du Chapitre 3).

Idée de la preuve La principale idée derrière ce théorème est de constater que l'on peut estimer plus précisément la norme Lipschitz que cela n'a été fait dans l'inégalité (2.18), i.e. on écrit r∆ k pX k´1 , ¨qs 1 ď γ k }∇ϕ} 8 ~σpX k´1 q~. Par souci de simplicité, nous écrirons } ¨} pour la norme matricielle ~¨~. Ainsi, l'idée est de soustraire la somme des carrés de la norme Lipschitz précédente afin de faire une inégalité de Hölder globale avec

Partie I : Introduction Le premier terme de l'inégalité s'apparente à une martingale exponentielle. En fait, par des inégalités de concentration itérées, on voit que ce terme est une sur-martingale majorée par 1. Tandis que pour le second terme, on a par hypothèse que }∇ϕ} 2 8 } " νp}σ} 2 q `Aϑ on réitère ensuite l'analyse du Théorème 2 pour le terme Aϑ. En revanche, le coefficient devant ν n pAϑq est en λ 2 . On obtient in fine une majoration de la déviation qui est en exponentielle d'un polynôme d'ordre 4 en λ :

P " a Γ n ν n pAϕq ě a ‰ ď C n exp ´´aλ ? Γ n `cn `λ2 Γ n
ρrϕs 2 1 νp}σ} 2 q 2 `λ4 ρ}∇ϕ} 4 8 }σ} 2 8 }∇ϑ} 2 8 8Γ 3 n pρ ´1q 3 ˘¯, avec c n , C n ą 0 tels que lim n C n " lim n c n " 1. Pour optimiser en λ, il nous faut donc trouver les racines d'un polynôme de degré 3. On utilise à cet effet la formule de Cardan-Tartaglia selon laquelle il existe une unique solution λ ρ ą 0 dépendante de ρ. Il faut donc encore optimiser suivant ρ. Une courte étude de cette optimisation au Chapitre 3 montre qu'il y a plusieurs régimes possibles. En effet, si a{ ? Γ n Ñ n 0 alors on obtient le résultat, dans le cas contraire on obtient un autre régime strictement sous-Gaussien :

P " | a Γ n ν n pAϕq| ě a ‰ ď 2 C n exp `´c n a 4{3 Γ 1{3 n 2}σ} 2{3 8 rϑs 2{3 1
˘.

(2.22)

Enfin, cette distinction de régimes permet d'obtenir des résultats numériques intéressants. Même si théoriquement dans le cas où a " ? Γ n {Γ p2q n le Théorème 2 est plus fort (une vraie borne Gaussienne), en pratique ce régime n'est jamais atteint. On voit in fine que le Théorème 3 avec la déviation (2.22) rend numériquement mieux compte de la transition du régime a ! ? Γ n {Γ p2q n à celui a -? Γ n {Γ p2q n . Pour plus de détails voir le Chapitre 4. Dans la prochaine section, nous verrons les grandes lignes de notre étude sur l'équation de Poisson. Ainsi, grâce au contrôle des dérivées de ϕ, solution de ladite équation de Poisson, on pourra s'intéresser directement à la déviation de ν n pf ´νpf qq pour tout type de fonction f de classe C 1,β (ou plus régulière).

Sur le problème de Poisson

Formellement, la solution au problème de Poisson s'écrit par la formule de Feynman-Kac :

ϕpxq " ż `8 0 Erf pX 0,x t q ´νpf qsdt.

(2.23)

Nous cherchons à établir un contrôle ponctuel de la solution ainsi que de ses dérivées. Néanmoins, la littérature à disposition se concentre surtout sur des contrôle de type Sobolev, voir par exemple Pardoux et Veretennikov [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF] pour un cadre elliptique. Dans un second papier, les auteurs [START_REF] Pardoux | On poisson equation and diffusion approximation 2[END_REF] établissent également un contrôle ponctuel du gradient mais à dérive bornée et à coefficient de diffusions réguliers. Avec des conditions locales de Doeblin, [START_REF] Pardoux | On the poisson equation and diffusion approximation 3[END_REF] montrent l'unicité au problème de martingale associé à l'équation de Poisson (possiblement dégénérée). Mentionnons par ailleurs, Pagès et Panloup [START_REF] Pagès | A mixed-step algorithm for the approximation of the stationary regime of a diffusion[END_REF], qui contrôlent, sous hypothèses de confluence, le cas d'un processus d'Ornstein-Uhlenbeck dégénéré. Une condition classique est l'hypothèse de Bakry-Emery. Une propriété importante de cette condition est qu'elle implique une relation de commutation entre le semigroupe et le carré du champ associés à la diffusion, pour plus de détails voir e.g. [START_REF] Bakry | Diffusions hypercontractives[END_REF], [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF], [START_REF] Bolley | Phi-entropy inequalities for diffusion semigroups[END_REF]. Cette hypothèse du type courbure-dimension (voir e.g. [ où Db est à la Jacobienne de b et Dσ ¨j est la j-ième colonne de Dσ. Autrement dit, cette condition signifie que la dérive b permet un rappel «suffisamment fort». Nous proposons deux jeux d'hypothèses afin de contrôler la solution au problème de Poisson. Soit on suppose que les coefficients sont assez réguliers pour dériver le flot à la Kunita (voir e.g. [START_REF] Kunita | Stochastic Flows and Stochastic Differential Equations[END_REF]), soit σ est non-dégénéré alors on peut bénéficier du bootstrap elliptique (e.g. [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF], dans le Chapitre 6 nous étudierons le boostrap dans un cadre dégénéré proche des hypothèses d 'Hörmander). dp2p1`βq´pq est vraie ainsi que la structure autonome pσσ ˚qi,j pxq " pσσ ˚qi,j px i^j , ¨¨¨, x d q.

Theorem 4 (Contrôle du gradient) Supposons que (L V ) et (C R ) ou (C U E ) sont vérifiés alors il existe une unique fonction ϕ P C 3`β telle que Aϕ " f ´νpf q, et :

}∇ϕ} 8 ď rf s 1 α .
Idée de la preuve Tout d'abord, on applique la formule d'Itô à la fonction y P R d Þ Ñ |y| p . On exploite ainsi l'hypothèse de confluence (D α p ), que l'on utilise pour contrôler le terme de type |B x i Y 0,x t | p apparaissant dans la formule de Feynman-Kac (2.23) associée à l'équation de Poisson. On conclut ainsi : @i P rr1, dss, @x P R d , |B x i ϕpxq| ď rf s 1 α .

(2.24)

On refait le même type de calcul pour montrer que ∇ϕ est β-Hölder en utilisant la condition }Dσ} 2 8 ď 2α dp2p1`βq´pq ou }Dσ} 2 8 ď 2α dp2p3`βq´pq . Dans le cas où (C U E ) est vérifié, on peut dériver l'équation de Poisson (1.10) afin d'avoir un système d'équations de Poisson avec potentiel et comme source une fonction ∇f de continuité β-Hölder. Le critère de système autonome pσσ ˚qi,j pxq " pσσ ˚qi,j px i^j , ¨¨¨, x d q permet d'avoir un système vérifiant les conditions de [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF] et dont on peut appliquer le bootstrap elliptique terme à terme. Remarquons également que le potentiel Db dans l'équation de Poisson est bien strictement négatif (en termes de valeurs propres) par l'hypothèse (D p α ). Dans le cas où (C R ) est vérifié, on continue de dériver le flot Y 0,x t jusqu'à l'ordre 3 puis on vérifie que la dérivée troisième est bien β-Hölder. À chaque dérivée, la condition de confluence doit être plus restrictive, i.e. }Dσ} 8 doit être de plus en plus petit avec le nombre de dérivées d'où le terme en 3 `β dans la borne supérieure dans la condition (C R ). En d'autre termes, dans les deux jeux d'hypothèses, on obtient le contrôle de la norme }ϕ} C 3,β en fonction de la norme associée à la régularité de f (}f } C 3,β sous (C R ), }f } C 1,β sous (C U E )). Ce contrôle permet en particulier de gérer les termes de reste R n dans (2.19).

Résultat de type Slutsky

Le défaut de l'approche du Théorème 3 est que la constante de concentration, i.e. νp}σ} 2 q}∇ϕ} 2 8 , dépend de la mesure invariante que nous approchons. Pour éviter cette difficulté, nous établissons également un résultat de type Slutsky. En effet, dans une perspective numérique, on cherche à remplacer νp}σ} 2 q par une estimation simulable, ici ν n p}σ} 2 q. Sous les mêmes conditions et notations que le Théorème 3 et le Théorème 4, on a :

P " | a Γ n ν n pf q ´νpf q a ν n p}σ} 2 q | ě a ‰ ď 2 C n exp `´c n a 2 α 2 2rf s 2 1 ˘.
La démonstration est une conséquence directe du Théorème 3. Remarquons que la renormalisation par a ν n p}σ} 2 q ne change pas la vitesse et permet de connaître parfaitement la grandeur de déviation. Dans la section suivante, nous montrerons que l'on peut prendre une source f qui soit seulement de régularité Lipschitz.

Source Lipschitz

Dans la plupart des inégalités fonctionnelles, la source est Lipschitz. On pense notamment à la distance de p-Wasserstein ‡ (ou appelée aussi distance de Kantorovich qui fut le premier à introduire cette distance) entre deux mesures de probabilité µ 1 et µ 2 qui est un fondamental pour le transport optimal. Pour p " 1, on a la formule de dualité de Kantorovich qui associe la distance de Wasserstein au sup sur les fonctions Lipschitz. C'est pourquoi nous avons étendu nos résultats à une source f Lipschitz. En effet, on montre que sous les mêmes hypothèses et notations que les Théorèmes 2 et 4 mais en supposant que la fonction f est seulement Lipschitz, si θ P p1{2, 1s alors P " | a Γ n `νn pf q ´νpf q ˘| ě a ‰ ď 2C n exp ˆ´c n a 2 α 2 2}σ} 2 8 rf s 2 1 ˙, (2.25) ‡. orthographié également Vaserstein où α est la constante issue de l'hypothèse de confluence (D p α ).

Idée de la preuve Il s'agit seulement de se ramener aux cas précédents, en régularisant la source f . Écrivons : f " f δ `rf ´fδ s, où f δ " f ‹ ρ δ correspond à la convolution de f avec une fonction régularisante (ou "mollifier") η δ telle que f δ est de classe C 8 et η δ a pour limite la distribution de Dirac quand δ tend vers 0. Il faut dès lors étudier l'équation de Poisson Aϕ δ " f δ ´νpf δ q. Or le prix à payer de cette régularisation se trouve dans les termes de reste qui explosent en δ. On écrit dès lors :

Pr a Γ n pν n pf q ´νpf qq ě as " P " pν n pf δ q ´νpf δ qq `Rn,δ pf q ě a ? Γ n ı , avec R n,δ pf q un reste tel que |R n,δ pf q| ď 2C η δrf s 1 . Ainsi, il nous faut trouver un équilibre sur δ qui nous conduit à considérer une nouvelle plage de vitesse, à savoir θ P p 1 2 , 1s. Remarquons d'ores et déjà qu'il y a une forme de continuité entre la vitesse de convergence de l'algorithme et la régularité de la source f . C'est-à-dire, pour une source f P C 1,β dans le Théorème 2 il faut que θ P p 1 2`β , 1s. Si β " 1 i.e. ∇f est Lipschitz alors on retrouve bien la condition θ P p 1 3 , 1s du Théorème 1. Enfin si f est juste Lipschitz, cela correspond à la condition θ P p 1 2 , 1s, c'est le même critère que pour le Théorème 2 avec β " 0. En résumé, de (2.25) on obtient :

sup f,rf s 1 ď1 P " | a Γ n `νn pf q ´νpf q ˘| ě a ‰ ď 2C n exp ˆ´c n a 2 α 2 2}σ} 2 8 ˙.
Un futur travail sera de montrer que l'on peut faire « entrer» le sup dans la probabilité et de bénéficier de la formule de dualité de Kantorovich pour avoir une inégalité de concentration non-asymptotique de la distance de Wasserstein. On pourrait imaginer utiliser des méthodes de localisation ou de couplage subtiles pour y parvenir (voir e.g. [START_REF] Boissard | Simple bounds for the convergence of empirical and occupation measures in 1-Wasserstein distance[END_REF], [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF]). Dans la prochaine section, nous présentons l'inégalité de concentration optimale qui correspond asymptotiquement au Théorème 1.

Inégalité de concentration optimale

Comme nous le verrons dans le Chapitre 4, la constante de concentration a une grande importance pour la précision des simulations numériques. D'où l'intérêt d'avoir l'inégalité de concentration non-asymptotique avec comme constante de concentration le carré du champ, c'est-à-dire la variance apparaissant dans le TCL du Théorème 1.

Theorem 5 (Inégalité de concentration non-asymptotique optimale) B Sous les mêmes hypothèses et notations que le Théorème 2, s'il existe une fonction ϑ P C 3,β pR d , Rq de même régularité que ϕ et telle que Aϑ " |σ ˚∇ϕ| 2 ´νp|σ ˚∇ϕ| 2 q, alors, pour tout θ P p 1 2`β , 1s, n ě 1, a ą 0 vérifiant a{ ? Γ n Ñ 0 (déviation Gaussienne) :

P " | a Γ n ν n pAϕq| ě a ‰ ď 2 C n exp `´c n a 2 2νp|σ ˚∇ϕ| 2 q ˘.
L'hypothèse sur le caractère cobord de |σ ˚∇ϕ| 2 ´νp|σ ˚∇ϕ| 2 q est vraie sous les hypothèses du Théorème 4. Le cas où a{ ? Γ n Ñ `8 conduit à l'inégalité (2.22).

Idée de la preuve Il s'agit d'améliorer encore une fois l'estimation de r∆ k pX k´1 , ¨qs 1 faite dans (2.18) et dans l'idée de la preuve du Théorème 3. Nous chercherons à approcher r∆ k pX k´1 , ¨qs 1 par ? γ k σ ˚∇ϕpX k´1 q, à savoir la racine carré du carré du champ. Pour cette estimation du module Lipschitz, par souci de simplification, nous nous placerons dans le cadre unidimensionnel et où l'innovation pU k q kě1 est bornée. Remarquons dans un premier temps que pour d " 1, pour ϕ solution de l'équation de Poisson Aϕ " f ´νpf q, il existe une constante C ϕ ą 0 telle que pour tout x P R : |D 2 ϕpxq| ď C ϕ {p1 `|x|q. On utilise dès lors cette inégalité dans l'estimation de la norme Lipschitz avec l'hypothèse }U k } 8 ă 8 :

|∇ u ∆ k pX k´1 , uq| u"U k | ď | ? γ k σ k´1 ∇ϕpX k´1 q| `Cγ k , où C " Cp}U k }, C ϕ , σ, ∇ϕq ą 0.
Le terme Cγ k est un terme de reste. De (2.21), on a :

Erexpp λM n Γ n qs ď E " exp `λ2 q 2Γ 2 n n ÿ k"1 γ k |σ ˚∇ϕpX k´1 q| 2 `Cγ 3{2 k ˘‰1{q . (3.26)
Or par hypothèse, il existe ϑ de classe C 3 telle que Aϑ " }σ} 2 ´νp}σ} 2 q. Ainsi le terme νp}σ} 2 q est celui qui va donner la constante de concentration, tandis que celui en Aϑ se traite de la même manière que précédemment Aϕ. Dans la section suivante, nous montrons que dans ce contexte on peut encore se ramener à une source f seulement Lipschitz.

Source Lipschitz

Dans le même cadre que la Section 2.4, nous établissons le résultat suivant : sous les mêmes hypothèses et notations que les Théorèmes 5 et 4 mais en supposant que la fonction f est seulement Lipschitz, on a :

P " | a Γ n `νn pf q ´νpf q ˘| ě a ‰ ď 2C n exp `´c n a 2 2νp|σ ˚∇ϕ| 2 q ˘, (3.27)
où ϕ P C 0,1 pR d , Rq X W 2 2,loc pR d , Rq est une solution au sens faible de l'équation de Poisson Aϕ " f ´νpf q. Nous avons le pendant du résultat de la Section 2.4 mais avec la constante de concentration optimale, νp|σ ˚∇ϕ| 2 q.

Idée de la preuve La preuve de ce théorème, comme pour le résultat de la Section 2.4, s'articule aussi autour d'une régularisation. Cependant, ici }ϕ δ } n'est pas forcément borné quand δ Ñ 0. En effet, on a seulement besoin de régularité Sobolev, i.e. que ϕ P W 2 2,loc pR d , Rq ùñ νp|σ ˚∇ϕ| 2 q ă `8. Pour se faire, nous utilisons le Theorem 1 de [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF] qui nous donne cette régularité de la solution de Poisson (1.10).

Il serait tentant d'établir un théorème de type Slutsky comme dans la Section 2.3 avec le carré du champ. Il ne serait pas compliqué d'établir une inégalité de concentration de la déviation de νnpf q´νpf q ?

νnp|σ ˚∇ϕ| 2 q . En revanche, à f donné, nous ne connaissons pas ϕ ni ses dérivées. Donc ce qu'il resterait potentiellement à faire dans cette direction serait de trouver une suite (aléatoire ou non) dépendant de f qui converge presque sûrement vers νp|σ ˚∇ϕ| 2 q. Remarquons aussi que V ar ν pf q " νpf 2 q´νpf q 2 n'a aucune raison d'être égale à celle du TCL νp|σ ˚∇ϕ| 2 q. Dans la partie suivante, nous chercherons à tester la robustesse de notre méthode en considérant des EDS dirigées par des processus à accroissements indépendants et stationnaires plus généraux, des processus de Lévy. Pour utiliser nos techniques, notamment la transformation exponentielle, il nous faut considérer des intensités de sauts sous-Gaussiens, nous nous sommes ainsi concentrés sur des sauts de type Poisson à sauts sous-Gaussiens. Dans un cadre plus général, un schéma de discrétisation à pas décroissant pour cette EDS, (4.28) a été introduit dans [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic differential equation driven by a lévy process[END_REF] ainsi que la preuve du Théorème ergodique (1.16) associé. Il s'agit de l'adaptation du schéma et des arguments pour les EDS Browniennes de Lamberton Pagès [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF]. Cependant, pour la partie saut, le pas décroissant du schéma se retrouve dans le temps et non pas en facteur de l'innovation. Autrement dit, on considère que la probabilité de saut du processus discrétisé de Z t au n-ième pas de temps est linéaire au pas de temps γ n . Cette approche différente liée à la présence de sauts induit de nouvelles difficultés. En effet, un des outils fondamentaux utilisé jusque-là se résume en cette phrase : «plus on fait du développement de Taylor, plus on fait sortir du γ k ». Donc pour pouvoir appliquer nos techniques, on ne peut prendre directement un processus de Poisson composé dans notre schéma, puisqu'il ne vérifie pas a priori l'inégalité de concentration (GC) même avec sauts sous-Gaussien. C'est pourquoi, nous avons choisi un cas particulier du schéma qui permet d'approcher un processus de Poisson composé à sauts sous-Gaussien et qui vérifie (GC) :

X n`1 " X n `γn`1 bpX n q `?γ n`1 σpX n qU n`1 `κpX n qZ n`1 , (4.29) avec U n`1 et γ n´1 définis comme dans (1.14) et pour tout n ě 1 :

Z n :" B n Y n , (4.30)
avec pB n q ně1 est une suite i.i.d. unidimensionnelle et suivant une loi de Bernoulli, i.e. B n L " Bernpγ n q, indépendantes de X 0 , de pU n q ně1 et de pY n q ně1 . Enfin, pY n q ně1 est aussi une suite i.i.d. telle que ErY b2 n s " I r et vérifie (GC). Ce choix d'innovation rend compte du fait que la probabilité d'avoir au moins un saut pour un processus de Poisson composé tend vers 0 avec le temps. En fait, Panloup a établi ces résultat dans un cadre d'hypothèse de Lyapunov plus large que celle présentée en (L V ), qui permet entre autres un retour à la moyenne moins fort, i.e. en particulier il suppose qu'il existe a P p0, 1s tel que xb, ∇V y ď ´αV V a `βV (voir aussi [START_REF] Lemaire | An adaptive scheme for the approximation of dissipative systems[END_REF] pour σ non borné). L'étude d'inégalité de concentration non-asymptotique pour des rappels plus faibles que ceux considérés dans (L V ) sera l'objet de futurs travaux. Notons de plus que l'hypothèse de Lyapunov supposée jusque-là n'a pas besoin d'être changée. En effet, écrivons le générateur infinitésimal associé à la diffusion (4.28) pour toute fonction ϕ P C 2 pR d q et x P R d :

Aϕpxq " bpxq∇ϕpxq `1 2 T r `σσ ˚pxqD 2 ϕpxq ˘`ż R d
pϕpx `κpxqyq ´ϕpxqq πpdyq (4.31) où π est la distribution de Y 1 . On pose r Aϕpxq " bpxq∇ϕpxq `1 2 T r pσσ ˚pxqD 2 ϕpxqq correspondant au générateur infinitésimal de la partie continue de la diffusion. Il se trouve que pour toute fonction ϕ de classe C 2 avec ∇ϕ, D 2 ϕ bornées, on obtient par nos hypothèses que |Aϕ ´r Aϕ| est bornée par 1 2 }D 2 ϕ} 8 }κ} 2 8 πp| ¨|2 q. Sous de bonnes hypothèses de stabilité, dans [START_REF] Panloup | Computation of the invariant measure of a levy driven SDE: Rate of convergence[END_REF] l'auteur établi le TCL correspondant.

Theorem 6 (TCL de [START_REF] Panloup | Computation of the invariant measure of a levy driven SDE: Rate of convergence[END_REF]) Supposons (L V ), s'il existe p ą 0 t.q E|Z t | 2p ă `8 alors pour tout θ P p 1 3 , 1s et ϕ avec la même régularité que pour le Théorème 1 :

a Γ n ν n pAϕq L ÝÑ N ˆ0, ż R d `|σ ˚∇ϕ| 2 pxq `żR d |ϕpx `κpxqyq ´ϕpxq| 2 πpdyq ˘νpdxq ˙.
Remarquons que la variance asymptotique est différente ici par rapport au cas continu. En effet, on y retrouve le carré du champ continu plus un terme dû aux sauts :

ż R d ż R d |ϕpx `κpxqyq ´ϕpxq| 2 πpdyqνpdxq.
La présence de sauts modifie le générateur associé à la diffusion, et donc le carré du champ.

Pour une étude complète et plus pratique, il faudrait étudier le comportement non pas de ν n pAϕq mais de ν n pf q en toute généralité, i.e. étudier l'équation intégro-différentielle de Poisson, ce qui fera l'objet de futurs travaux.

Theorem 7 (Concentration pour une EDS à sauts) Sous les mêmes hypothèses et notations que le Théorème 2, pour @n ě 1 ,0 ă a ! ?

Γ n {Γ p2q n : P " | a Γ n ν n pAϕq| ě a ‰ ď 2 C n exp `´c n a 2 2p2}κ} 2 8 }∇ϕ} 2 8 `}σ} 2 8 }∇ϕ} 2 8 q ˘.
La constante de concentration est «polluée» par une constante multiplicative devant le terme de saut. Contrairement au Théorème 2, nous n'avons pas obtenu une simple borne supérieure du carré du champ mais deux fois la borne supérieure du terme de sauts. Cela est dû à la difficulté de vérifier que pZ n q ně1 satisfasse (GC). Le principal obstacle dans ce travail est la plage de validité de la concentration de pZ n q ně1 . En effet, on établit l'inégalité de concentration de type (GC) mais avec ε P p0, 1q, et tel que 0 ă λ ă ε 6rgs 1 on a

Ee λgpZnq ď e λEgpZnq`λ 2 γnp2`εqrgs 2 1 2 .

(4.32)

Remarquons qu'il y a maintenant une contrainte sur la borne de validité du λ, et qu'on voit apparaître un terme en p2 `εq qui vient «perturber» l'inégalité de concentration. C'est cette perturbation qui permet d'avoir cette inégalité, pour un λ suffisamment petit, en faisant un développement en série entière. Ces contraintes sur la concentration impliquent une plus petite intégrabilité de la fonction de Lyapunov, i.e. p ? V pX n qq ně0 est exponentiellement intégrable. Pour la suite de la preuve, il s'agit dans les grandes lignes de la même chose que pour le Théorème 2. La principale difficulté est de réussir à intégrer les contraintes de la concentration (4.32).

Une ouverture possible pour la suite serait de s'intéresser à des Lévy plus généraux avec des contrôles de type Berry-Esseen.

Chapitre 2

Régularisation par un bruit dégénéré 1 Régularisation par un bruit dégénéré

Résumé : Dans une seconde partie, nous établissons des estimées de Schauder liées à des équations paraboliques associées à un système stochastique dégénéré, où la dérive est un champ de vecteurs vérifiant une condition de type Hörmander (faible) mais en cherchant la régularité Hölder minimale sur les coefficients. Ce travail fait suite à l'article de Delarue et Menozzi [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF]. Enfin, notre approche nous permet de montrer l'unicité forte du système stochastique considéré dans le cadre de coefficients Hölder, étendant ainsi le résultat obtenu en dimension 2 par Chaudru de Raynal [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF].

Équation de Kolmogorov et chaîne dégénérée

L'étude des chaînes dégénérées est prégnante dans différents domaines. Par exemple, en finance, la dynamique d'une option asiatique forme une chaîne dégénérée de dimension 2. Ce sont des options dont le résultat à son échéance (ce que gagne le détenteur, appelé aussi payoff ) dépend du prix moyen de l'actif sous-jacent. En d'autres termes, le payoff d'un call est p 1 T ş S t dt ´Kq `, où T est l'échéance, K le strike, et S t le sous-jacent à l'instant t. Pour plus d'applications en finance, mentionnons par exemple Jeanblanc et al. [START_REF] Jeanblanc | Mathematical Methods for Financial Markets[END_REF]. Ces équations sont dites de type vitesse/position. Cela peut se voir comme un système dont les inconnues sont X t (la position) et 9 X t (la vitesse), où X t ne dépend que de t et de 9 X t . Autrement dit, le bruit n'affecte directement que 9 X t et se propage en X t via sa dépendance en 9 X t . En particulier, on retrouve ce système vitesse/position dans la mécanique Hamiltonienne. Dans un cadre plus général, nous nous intéresserons à des chaînes d'oscillateurs (voir par exemple les oscillateurs élasto-plastiques considérés dans [START_REF] Bensoussan | Degenerate Dirichlet problems related to the invariant measure of elasto-plastic oscillators[END_REF] et [START_REF] Bensoussan | An ultra weak finite element method as an alternative to a Monte Carlo method for an elastoplastic problem with noise[END_REF]) qui sont un modèle typique en sismologie par exemple, où l'on considère la propagation d'un aléa sur plusieurs structures qui se transmettent la secousse. Le modèle souvent utilisé pour représenter ce phénomène est un système de ressorts attachés les uns aux autres.

Enfin mentionnons également les modèles microscopiques associés à la diffusion de la chaleur (voir par exemple Heckmann et Hairer [START_REF] Eckmann | Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators[END_REF]). Dans cette partie, nous nous intéresserons à l'équation de Kolmogorov qui s'écrit pour T ą 0 : # pB t `Lqupt, ¨q " ´f pt, ¨q, t P r0, T q, upT, ¨q " 0, (1.1) où L est un opérateur différentiel qui peut se voir comme le générateur infinitésimal associé à un processus de diffusion pX t q tě0 . Nous étudierons cette équation à temps fini, i.e. pour tout t P r0, T s à T ą 0 donné. Pour se ramener à l'équation de Poisson considérée dans la partie précédente, il faut considérer un potentiel qui permet d'avoir des estimées en temps long apparaissant dans la représentation de Feynman-Kac (2.23). Ce terme de potentiel apparaît notamment quand on dérive l'équation de Kolmogorov et sous la condition de Lyapunov (L V ). L'approche que nous développerons après permet cette procédure.

Dans un premier temps, nous établirons des estimées de Schauder associées à l'équation (1.1) pour des chaînes dégénérées. Ces estimées rendent compte du bootstrap parabolique. Illustrons tout d'abord cette notion de bootstrap parabolique dans un cadre uniformément elliptique à coefficients bornés. Dans ce cas, pour des coefficients suffisamment réguliers en temps et en espace, à l'aide de méthode de contrôles a priori, Friedmann [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] et Krylov [START_REF] Krylov | Lectures on elliptic and parabolic equations in Hölder spaces[END_REF] ont montré le contrôle suivant :

}u} C 2`γ 2 ,2`γ b ď C}f } C γ 2 ,γ b , où } ¨}C 2`γ 2 ,2`γ b , } ¨}C γ 2 ,γ b
sont les normes Hölder classiques. Plus précisément, si f est γ 2 -Hölder en temps et γ-Hölder en espace alors la solution est 2`γ 2 -Hölder en temps et 2 `γ-Hölder en espace. Le caractère borné des coefficients en espace et leur régularité Hölder en temps permet de considérer le gain de régularité sur la variable temporelle.

Néanmoins, sans cette hypothèse de bornitude, à notre connaissance il n'existe pas de contrôles Schauder pour des problèmes paraboliques généraux. Le cas où les coefficients sont bornés en temps et à croissance linéaire en espace

}u} L 8 pC 2`γ b q ď C}f } L 8 pC γ q ,
où } ¨}L 8 pC 2`γq , } ¨}L 8 pC γ q sont les normes uniformes en temps et respectivement 2 `γ et γ Hölder homogène en espace (la non-bornitude de f est gérée par la présence d'un potentiel). Cette estimée, dans un cadre plus proche du nôtre mais avec un potentiel séparé de 0, a été montré par Krylov et Priola [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF]. La méthode usuelle utilisée pour les estimées de Schauder et appliquée dans cet article est également une méthode de contrôle a priori. Nous procéderons par une méthode dite perturbative.

Ce bootstrap parabolique à coefficient non-bornés signifie que si f est bornée et γ-Hölder, γ P p0, 1q, alors la solution u est bornée et 2 `γ-Hölder, c'est-à-dire u est 2 fois dérivable, chaque dérivée est bornée et D 2 u est γ-Hölder.

Mentionnons également, Kruzhkov et al. [START_REF] Kružkov | Schauder type estimates, and theorems on the existence of the solution of fundamental problems for linear and nonlinear parabolic equations[END_REF] qui ont établi des estimés de Schauder dans un cadre elliptique pour des coefficients bornés en temps et en espace mais sans supposer de régularité en temps. Par la suite, Lorenzi [START_REF] Lorenzi | Optimal Hölder regularity for nonautonomous Kolmogorov equations[END_REF] a étendu ces résultats au cadre non-borné.

De plus, cette estimée montre également, en un certain sens, l'unicité de l'équation (1.1). En effet, si u 1 et u 2 sont deux solutions de (1.1) alors on a : pB t `Lqpu 1 ´u2 q " 0, l'estimée de Schauder donne ainsi :

}u 1 ´u2 } C 2`γ ď 0. Ce qui conduit directement à u 1 " u 2 .
L'étude de cette équation de Kolmogorov (1.1) est également centrale dans la transformation de Zvonkin pour établir l'unicité forte de l'EDS associée.

Nous disons que la solution d'une EDS dirigée par un Brownien est forte si elle est l'unique processus vérifiant l'EDS et étant adapté à la filtration Brownienne. Nous montrons l'unicité trajectorielle, et par la propriété de Yamada Watanabe [START_REF] Yamada | On the uniqueness of solutions of stochastic differential equations[END_REF], nous savons que l'unicité forte (ou unicité dans l'espace des réalisations) est déduite de l'unicité trajectorielle et de l'existence faible (ou dans l'espace des probabilités).

Présentation du modèle

Considérons le problème de Cauchy associé à l'équation de Kolmogorov, pour tout T ą 0, pt, xq P r0, T s ˆRnd ,

# B t upt, xq `xFpt, xq, Dupt, xqy `1 2 Tr `D2 x 1 upt, xqapt, xq ˘" ´f pt, xq, upT, xq " gpxq, x " px 1 , ¨¨¨, x n q P R nd , (1.2)
où la dérive a la structure suivante Fpt, xq :" pF 1 pt, xq, ¨¨¨, F n pt, xqq telle que @i P rr2, nss, F i pt, xq :" F i pt, x i´1:n q, x i´1:n :" px i´1 , ¨¨¨, x n q.

(1.3)

Autrement dit, chaque composante F i a une seule variable qui transmet le bruit, à savoir x i´1 . Cette équation de Kolmogorov (1.2) est dégénérée au sens où l'opérateur différentielle d'ordre 2, D 2 x 1 n'est pris que sur la première variable, la variable dite non-dégénérée.

Formellement l'équation de Kolmogorov (1.2), est étroitement liée au processus stochastique suivant :

dX 1 t " F 1 pt, X 1 t , . . . , X n t qdt `σpt, X 1 t , . . . , X n t qdW t , dX 2 t " F 2 pt, X 1 t , . . . , X n t qdt, dX 3
t " F 3 pt, X 2 t , . . . , X n t qdt, . . .

dX n t " F n pt, X n´1 t , X n t qdt, t ě 0, (1.4

Propriété d'échelles

Donnons tout d'abord un exemple classique du système d'équation stochastique (1.33), l'exemple de Kolmogorov (Ann. Math. 1934 [Kol34]) :

X 1 t " x 1 0 `Wt , X 2 t " x 2 0 `ż t 0 X 1 s ds, où l'on a pris n " 2, F 1 pxq " x 1 , F 2 pxq " x 1 , et σpxq " 1 d,d
. Il s'agit d'un processus Gaussien pX 1 t , X 2 t q " N `θt px 0 q, K t ˘tel que sa densité est :

ppt, x 0 , yq " p2 ? 3q d p2πq d t p 1 2 `3 2 qd exp ˆ´1 2 xK ´1 t pθ t px 0 q ´yq, θ t px 0 q ´yy ˙,
avec K t la matrice de covariance qui vérifie la propriété de la «bonne échelle» (voir la Définition 3.2 dans [DM10]) :

K 1{2 t " ˆtI d t 2 2 I d t 2 2 I d t 3 3 I d ˙1{2 - ˆt1{2 I d 0 d 0 d t 3{2 I d ˙.
(1.

L 0 :" B t `xA 0 x, Dy `1 2 ∆ x 1 , avec A 0 " ¨0d,d ¨¨¨¨¨¨¨¨¨0
0 d,d ¨¨¨I d,d 0 d,d 0 d,d ‹ ‹ ‹ ' ,
Considérons dès lors l'opérateur de dilatation adapté à l'échelle du système stochastique, à savoir pour tout λ ą 0,

δ λ : pt, xq P R `ˆR nd Þ Ñ δ λ pt, xq " `λ2 t, λx 1 , λ 3 x 2 , ¨¨¨, λ 2n´1 x n ˘.
Maintenant, remarquons que l'opérateur sus-défini L 0 est invariant par la dilatation, c'està-dire : L 0 v " 0 ùñ L 0 pv ˝δλ q " 0. Nous voyons ainsi l'importance des différentes échelles associées à chaque variable px i q iPrr1,nss . Il est dès lors naturel de considérer la pseudo-distance homogène associée :

d `x, y ˘" d ÿ i"1 |y i ´xi | 1 2i´1 . (1.8)
Une fois cette pseudo-distance en main, nous introduisons les différentes normes Hölder aux bonnes échelles. Pour toute fonction u : r0, T s ˆRnd Þ Ñ R, T ą 0 fixé, et pour tout γ P p0, 1q, on note pour tout i P rr1, nss le module Hölder associé à la variable x i :

ru i s γ d :" sup t,x,x 1 i |upt, xq ´upt, x 1:i´1 , x 1 i , x i`1 q| |x i ´x1 i | γ 2i´1
.

Nous notons le module Hölder homogène complet comme la somme de tous les modules Hölder suivant les variables à l'échelle correspondante,

}u} L 8 pC γ d q :" n ÿ i"1 ru i s γ d -sup pt,x,x 1 q |upt, xq ´upt, x 1 q| d γ px, x 1 q ,
remarquons à ce stade que L 8 signifie la norme uniforme suivant la composante de temps. Notons également la norme Hölder homogène totale quand l'indice est de type 2 `γ :

}u} L 8 pC 2`γ d q :" }D x 1 u} L 8 `}D 2 x 1 u} L 8 `n ÿ i"1 rD 2 x 1 u i s γ d `n ÿ i"2 ru i s 2`γ d .
Pour établir les estimées de Schauder, il conviendra en particulier de montrer que la norme précédente associée à la solution u de (1.2) est finie. C'est-à-dire que suivant la i-ème variable, u est 2`γ 2i´1 -Hölder si i ą 1 (sinon on doit considérer des indices ą 1) et que suivant la i-ème variable, D 2

x 1 u est γ 2i´1 -Hölder . En fait, nous supposerons que la source f et que la condition terminale sont bornées, on écrit ainsi pour tout γ P p0, 3q (en pratique γ P tγ, 2 `γu) les pseudo-normes précédemment introduites avec un «b» pour préciser que la fonction considérée est bornée.

}u} L 8 pC γ b,d q :" }u} L 8 `}u} L 8 pC γ d q .
Nous écrirons de ce fait l'espace de Hölder associé aux normes introduites :

L 8 pC γ d q :" L 8 `r0, T s, C γ d ˘" tu P Bpr0, T s ˆRnd , Rq : }u} L 8 pC γ d q ă 8u
où Bpr0, T s ˆRnd , Rq correspond à l'ensemble des fonctions Boréliennes de r0, T s dans R.

Pareillement pour L 8 pC γ b,d q :" L 8 `r0, T s, C γ b,d ˘" tu P Bpr0, T s ˆRnd , Rq : }u} L 8 pC γ b,d q ă 8u et C γ b,d si la fonction ne dépend pas du temps (la condition terminale g de (1.2) par exemple).

Partie I : Introduction

Résultats existants

Après avoir introduit ces notations, nous pouvons maintenant énoncer un premier contrôle Schauder associé au problème de Cauchy (1.2).

Theorem 1 (Estimées de Schauder de [START_REF] Lunardi | Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in R n[END_REF]) Pour σ P L 8 pC b,d q, Uniformément Elliptique et telle qu'il existe σ 0 P GL nd,d t.q. σpxq Ñ |x|Ñ8 σ 0 , Fpxq " Ax, et si pf, gq P L 8 pC γ b,d q ˆC2`γ b,d alors il existe une unique solution mild et faible u P L 8 pC 2`γ b,d q à (1.2) et il existe une constante C 1 ą 0 telle que

}u} L 8 pC 2`γ b,d q ď C 1 p}f } L 8 pC γ b,d q `}g} C 2`γ b,d q (1.9)
Précisons que nous appelons solution mild de (1.2) au sens de Stroock et Varadhan [START_REF] Stroock | Multidimensional diffusion processes[END_REF] (voir aussi e.g. [START_REF] Kolokoltsov | Markov Processes, Semigroups and Generators[END_REF] ) toute solution u pour laquelle la limite de la régularisation pB t `Lm t qu m " ´fm quand m Ñ `8 existe et est égale à f mais dont les dérivées partielles de u intervenant dans pB t `Lt qu ne sont pas forcément définies. Notons que dans l'article de [START_REF] Lunardi | Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in R n[END_REF], la notion de solution forte correspond à notre version de solution mild. La solution faible correspond à la solution au sens des distributions, i.e. que pour toute fonction ϕ P C 8 0 (fonction de classe C 8 à support compact), on a ż ϕpxqpB t `Lt qϕpxqdx "

ż f pxqϕpxqdx.
Mentionnons l'article de Priola [START_REF] Priola | Global Schauder estimates for a class of degenerate Kolmogorov equations[END_REF] qui établit également des estimées de Schauder comme pour le Théorème 1 avec le même critère asymptotique sur σ et la dérive s'écrivant

Fpxq " Ax `´F 1 pxq 0 pn´1qd,d ¯, et F1 est C γ d .
Intuitivement, on utilise la formule de Girsanov avec une version régularisée de F1 pour se ramener au cas considéré par Lunardi. La régularité de F1 intervient lorsqu'on dérive le flot dans martingales exponentielles. Enfin on injecte l'estimée de Schauder avec comme source la difference la version régularisée et non-régularisée de F1 .

Dans la suite, nous établirons des estimés de Schauder pour des solutions mild et faibles. Nous établirons des contrôles de la solution classique dans le cadre de l'unicité forte. Pour plus de détailles sur la solution au sens classique de l'équation de Kolmogorov, nous renvoyons par exemple à [START_REF] Kunze | Nonautonomous Kolmogorov parabolic equations with unbounded coefficients[END_REF], [START_REF] Lunardi | Analytic Semigroups and Optimal Regularity in Parabolic Problems[END_REF].

Dans la section suivante, nous présentons des estimées de Schauder dans un cadre plus général que celui considéré par [START_REF] Priola | Global Schauder estimates for a class of degenerate Kolmogorov equations[END_REF].

Estimées de Schauder établies

P R `ˆR nd , D x i´1 F i pt, x i´1 , . . . , x n q est une matrice de R d b R d inversible.
(S) Régularité des coefficients. Soit γ P p0, 1q.

-Régularité du coefficient de diffusion. On suppose que a P L 8 pC γ b,d q. -Régularité de la dérive. On suppose que F est mesurable et borné en temps.

' On suppose que F 1 P L 8 pC γ d q. ' Pour tout i P rr2, nss, on suppose que

F i P L 8 pC 2i´3`γ d q. En particulier F i pt, ¨, x i:n q P C 1`γ 2pi´1q´1 .
Quand on fait tendre γ vers 0, on retrouve les exposants de régularité critiques présents dans [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] pour obtenir l'unicité faible.

Résultat principal

Sous ces conditions de régularités minimales, nous établissons des Estimées Schauder.

Theorem 2 (Estimées de Schauder.) Soit γ P p0, 1q donné, on suppose (UE), (H) et (S), si pf, gq P L 8 pC γ b,d q ˆC2`γ b,d alors il existe une unique solution mild et faible u P L 8 pC 2`γ b,d q à l'équation (1.2), et il existe une constante C 2 :" C 2 `pUEq, pHq, pSq, T

telle que }u} L 8 pC 2`γ b,d q ď C 2 `}g} C 2`γ b,d `}f } L 8 pC γ b,d q ˘.
(1.10)

Grandes lignes de l'analyse

Rappelons que nous avons régularisé l'équation (1.2). Il nous faut donc de (1.12), contrôler }u m } L 8 pC 2`γ b,d q uniformément par rapport au coefficient de régularisation m. À cette fin, nous utilisons une méthode de type perturbative en développant la solution u autour d'un proxy Xt de type Ornstein-Uhlenbeck.

Méthode de type perturbative

Soit le générateur infinitésimal L associé au proxy Xt qui a pour densité ppt, s, x, yq. On réécrit le problème de Cauchy (1.2) :

# B t upt
dX m t " dX m,1 t " F m 1 pt, X m,1 t , . . . , X m,n t qdt `σpt, X m,1 t , . . . , X m,n t qdW t , dX m,2 t " F m 2 pt, X m,1 t , . . . , X m,n t qdt, dX m,3 t " F m 3 pt, X m,2 t , . . . , X m,n t qdt, . . . dX m,n t " F m n pt, X m,n´1 t , X m,n t qdt.
(

Linéarisation du proxy

L'idée est de linéariser l'EDS (1.14), pour cela considérons le flot, solution de l'EDO associée à la dérive :

9 θ v,τ pξq " Fpv, θ v,τ pξqq, v P rτ, T s, θ τ,τ pξq " ξ,
(1.16) où pτ, ξq P r0, T s ˆRnd sont des paramètres de gel, respectivement en temps et en espace qui seront explicités plus loin. Cette ODE admet une unique solution, comme nous avons régularisé F " F m est Lipschitz. Nous avons choisi une approche dite forward, dans le sens où le flot (1.16) part de ξ (qui sera choisi égal à x) pour aller vers y (variable d'intégration en (1.15)). Cette technique a notamment été utilisée par Chaudru de Raynald [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF] pour montrer l'unicité forte dans le cas où n " 2. Il existe aussi une approche dite backward introduite par [START_REF] Mckean | Curvature and the eigenvalues of the Laplacian[END_REF], et exploitée par la suite par [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF], [START_REF] Menozzi | Martingale problems for some degenerate Kolmogorov equations[END_REF]. Cette méthode peut sembler naturelle pour écrire formellement si g " 0, u " G ˝pI ´Rq ´1f où R " GpL ´Lq et où l'inverse de l'opérateur pI ´Rq est pris dans l'espace de Hölder approprié. Cette inversion d'opérateur est également la technique employée dans [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] et [START_REF] Stroock | Multidimensional diffusion processes[END_REF]. Cette approche permet d'estimer la densité, néanmoins des difficultés apparaissent lorsqu'il faut contrôler ses dérivées. En effet, dans le cas backward ppt, T, x, yq " py pt, T, x, yq, où py exprime que le paramètre de gel dépend de la variable d'intégration y, n'est pas une densité de probabilité. En particulier, il n'est pas évident que PT,t g conserve la même régularité de g.

Maintenant, écrivons le processus gelé autour du flot défini en (1.16) pour tout v P rt, ss

d Xpτ,ξq v " rFpv, θ v,τ pξqq`DFpv, θ v,τ pξqqp Xpτ,ξq v ´θv,τ pξqqsdv `Bσpv, θ v,τ pξqqdW v , Xpτ,ξq t " x, (1.17) avec B " ´Id 0 pn´1qd,d 0 d,d 0 pn´1qd,d ¯et pour tout z P R nd : DFpv, zq " ¨0d,d ¨¨¨¨¨¨0 d,d D z 1 F 2 pv, zq 0 d,d ¨¨¨0 d,d . . . . . . ¨¨¨. . . 0 d,d ¨¨¨D z n´1 F n pv, z n´1 , z n q 0 d,d ‹ ‹ ‹ ' .
Notons que DF est du même type que A 0 , avec pour tout i P rr1, nss, D x i´1 F i est une matrice d ˆd inversible, il faut donc s'attendre à obtenir la même propriété de «bonne échelle».

Propriété du proxy

Nous pouvons écrire explicitement la dynamique du proxy.

: } Gpτ,ξq f } L 8 et }D x 1 Gpτ,ξq f } L 8 , }D 2 x 1
Gpτ,ξq f } L 8 est établi par des techniques de type cancellation et en utilisant la régularité des paramètres intervenant dans le proxy (1.18). Pour illustrer le principe de cancellation, établissons un contrôle en norme uniforme du noyau de Green :

}D 2 x 1 Gpτ,ξq f } L 8 " sup pt,xqPr0
x β expp´x 2 q ď C ζ expp´C ´1 ζ x 2 q.
Nous contrôlons de manière similaire les premiers termes de reste ş T t ds ş R nd ppt, s, x, yqpLĹ qups, yqdy intervenant dans (1.15), c'est-à-dire les termes faisant intervenir les dérivées suivant les composantes non-dégénérées :

ż T t ds ż R nd D 2
x 1 ppτ,ξq pt, s, x, yq ´xF 1 ps, yq ´F1 ps, θ s,τ pξqq, D y 1 ups, yqy `1 2 ´paps, yq ´aps, θ s,τ pξqqqD 2 y 1 ups, yq ¯dy, (1.20) se contrôle en norme uniforme.

Contrôle de la norme uniforme des derniers termes

Néanmoins, pour les derniers coefficients du terme de reste dans (1.15) faisant intervenir les dérivées suivant les variables dégénérées, i.e. pD y i ups, yqq iPrr2,nss , le contrôle en norme uniforme est plus délicat. En effet, la version non régularisée de u n'est a priori pas dérivable suivant les variables dégénérées px i q iPrr2,nss , mais on s'attend des résultats précédents à ce qu'elle soit seulement p2 `γq{p2i ´1q-Hölder uniformément en m. Pour cela, définissons : ∆ i,F pτ, s, θ s,t pξq, yq :" ´Fi ps, yq ´Fi ps, θ s,τ pξqq ´Dx i´1 F i ps, θ s,τ pξqqpy ´θs,τ pξqq i´1 ¯, (1.21) et par intégrations par parties on a :

n ÿ i"2 ż T t ds ż R nd dyD 2 x 1 ppτ,ξq pt, s, x, yqx∆ i,F pτ, s, θ s,t pξq, yq, D y i ups, yqy " ´n ÿ i"2 ż T t ds ż R nd dyD y i ¨"D 2 x 1
ppτ,ξq pt, s, x, yq∆ i,F pτ, s, θ s,t pξq, yq ‰ ups, yq.

(1.22) L'idée, ensuite, sera de contrôler D y i " D 2

x 1 ppτ,ξq pt, s, x, yq∆ i,F pτ, s, θ s,t pξq, yq ‰ suivant la norme adaptée en dualité à la norme L 8 pC 2`γ b,d q de ups, yq. C'est a priori un contrôle non direct étant donné que le terme D 2 x 1 ppτ,ξq pt, s, x, yq∆ i,F pτ, s, θ s,t pξq, yq dont on dérive en D y i n'est pas dérivable en y i à cause de l'irrégularité des coefficients.

Pour se faire, remarquons que pour α P R, C α b pR d , Rq " B α 8,8 pR d , Rq, où pour tout pp, q, sq P r1, `8s 2 ˆR, B s p,q est un espace de Besov. Il existe plusieurs façons de définir les espaces de Besov (module de continuité, décomposition de Littlewood-Paley, représentation thermique). La représentation thermique est la formulation la plus naturelle dans notre contexte Gaussien. On définit ainsi B s p,q pR d q :" tf P S 1 pR d q : }f } H s p,q ă `8u pour tout m P N 0 tel que m ą s 2 :

}f } H α p,q :" }ϕpDqf } L p pR d q `´ż 1 0 v pm´α 2 qq }B m v h v ‹ f } q L p pR d q dv v ¯1 q ,
(1.23) où ϕ P C 8 0 pR d q telle que ϕp0q ‰ 0, ϕpDqf :" pϕ f q _ où on définit la transformée de Fourier f et la transformée de Fourier inverse pϕ f q _ , et pour tout v ą 0, z P R d ,

h v pzq :" 1 p2πvq d 2 exp ˆ´|z| 2 2v
ėst le noyau de la chaleur usuel. Pour plus d'informations nous pouvons nous référer à Triebel [START_REF] Triebel | Theory of function spaces[END_REF].

La propriété qui nous intéresse dans les espaces de Besov est en fait la relation de dualité entre B αi 8,8 pR d , Rq et B ´α i 1,1 pR d , Rq (voir par exemple [START_REF] Lemarie-Rieusset | Recent developments in the Navier-Stokes problem[END_REF]). Ainsi par des méthodes de cancellation, nous établissons la norme Besov de chaque terme de (1.22). Autrement dit, pour tout i P rr2, nss, on montre que la norme Besov prise pour la i-ème variable :

› › y i Þ Ñ D y i `D2 x 1 ppτ,ξq pt, s, x, yq∆ i,F pτ, s, θ s,t pξq, yq ˘› › B ´2`γ 2i´1 1,1 pR d q ď Cps ´tq γ 2 ´1 ż R d ppτ,ξq pt, s, x, yqdy i .
En effet, la caractérisation thermique permet de faire passer la dérivée Þ Ñ D y i sur le noyau de la chaleur h v .

Contrôle des modules Hölder

Pour étudier les différents modules Hölder, pour px, x 1 q P pR nd q 2 donné et 0 ă c 0 ă 1 supposé «petit», nous procédons classiquement à une distinction entre le régime horsdiagonal : c 0 d 2 px, x 1 q ą ps ´tq et le régime diagonal : c 0 d 2 px, x 1 q ď ps ´tq.

' hors-diagonal, dans ce cas, les points sont loin par rapport au temps courant. Donc il n'y a pas d'intérêt à utiliser des développements des termes à contrôler entre ces deux points x, x 1 . Nous utilisons ainsi le contrôle en norme sup précédemment établi avec comme choix de points de gel pτ, ξ, ξ 1 q " pt, x, x 1 q. Par exemple, nous écrivons pour le contrôle Hölder du noyau de Green :

ˇˇG m,ξ f pt, xq ´G m,ξ f pt, x 1 q ˇˇˇˇˇp τ,ξ,ξ 1 q"pt,x,x 1 q " ˇˇż t`c 0 d 2 px,x 1 q t ds ż R nd rD 2
x 1 ppτ,ξq pt, s, x, yq ´D2

x 1 p pτ,ξ 1 q pt, s, x 1 , yqsf ps, yqdy ˇˇˇˇˇp τ,ξ,ξ 1 q"pt,x,x 1 q ď 2 sup pτ,ξ,ξ 1 q"pt,x,x 1 q

ˇˇż t`c 0 d 2 px,x 1 q t ds ż R nd dyD 2
x 1 ppτ,ξq pt, s, x, yqf ps, yq ˇď Cd γ px, x 1 q.

(1.24)

La première inégalité est une conséquence de (1.19).

' diagonal, dans le cas où x, x 1 sont proches vis-à-vis du temps courant, nous développons la densité gelée entre x et x 1 . Pour se faire, nous choisissons les points de gel pτ, ξ, ξ 1 q " pt, x, xq, de sorte que les densités à contrôler soient les mêmes.

Autrement dit on écrit :

ˇˇż T t`c 0 d 2 px,x 1 q ds ż R nd dyrD 2 x 1 ppτ,ξq pt, s, x, yq ´D2 x 1 p pτ,ξ 1 q pt, s, x 1 , yqsf ps, yq ˇď ˇˇż T t`c 0 d 2 px,x 1 q ds ż R nd dy ż 1 0 dλpx ´x1 qD x D 2 x 1 ppτ,ξq pt, s, x 1 `λpx ´x1 q, yqf ps, yq ˇď C}f } L 8 pC γ b,d q n ÿ i"1 |x i ´x1 i | ż T t`c 0 d 2 px,x 1 q ds ż R nd dyps ´tq ´1´pi´1 2 q ppτ,ξq C ´1 pt, s, x, yq ď C}f } L 8 pC γ b,d q d γ px, x 1 q.
(1.25) L'avant dernière inégalité est une conséquence du contrôle des dérivées de la densité gelée (1.18) et de la cancellation ş R nd dyD x D 2

x 1 ppτ,ξq pt, s, x 1 `λpx ´x1 q, yqf ps, θ s,t pxqq " 0 d,d,d , la dernière identité vient du fait que

|x i ´x1 i | ď d 2i´1 px, x 1 q.
Cependant, pour un tel choix de paramètres de gel pτ, ξ, ξ 1 q dépendant de la variable d'intégration s, il nous faut modifier par la formule de Duhamel l'équation (1.15). Autrement dit pour tout pt, x 1 q P r0, T s ˆRnd et r P pt, T s, ξ 1 P R nd on a :

u m pt, x 1 q " P m,pτ,ξ 1 q r,t u m pr, x 1 q `G m,pτ,ξ 1 q r,t f m pt, x 1 q `ż r t ds ż R nd dyp m,pτ,ξ 1 q pt, s, x 1 , yqpL m s ´L m,pτ,ξ 1 q s qu m ps, yq, avec @0 ď v ă r ď T, Gm,pτ,ξ 1 q r,v f m pt, xq " ż r v ds ż R nd
dyp m,pτ,ξ 1 q pt, s, x 1 , yqf m ps, yq.

(1.26)

Dériver cette expression suivant r, engendre pour tout ξ 1 P R nd :

0 " B r P m,pτ,ξ 1 q r,t u m pr, x 1 q `żR nd dyp m,pτ,ξ 1 q pt, r, x 1 , yqf m pr, yq `żR nd dyp m,pτ,ξ 1 q pt, r, x 1 , yqpL m r ´L m,pτ,ξ 1 q r qu m pr, yqdy.

(1.27) Ensuite, nous intégrons cette équation entre t et t 0 P pt, T s pour un premier ξ 1 et entre t 0 et T avec un ξ1 éventuellement différent :

0 " P m,pτ,ξ 1 q t 0 ,t u m pt 0 , x 1 q ´um pt, x 1 q `ż t 0 t ds ż R nd dyp m,pτ,ξ 1 q pt, s, x 1 , yqf m ps, yq `ż t 0 t ds ż R nd
dyp m,pτ,ξ 1 q pt, s, x 1 , yqpL m s ´L m,pτ,ξ 1 q s qu m ps, yq

P m,pτ, ξ1 q T,t u m pT, x 1 q ´P m,pτ, ξ1 q t 0 ,t u m pt 0 , x 1 q `ż T t 0 ds ż R nd dyp m,pτ, ξ1 q pt, s, x 1 , yqf m ps, yq `ż T t 0 ds ż R nd dyp m,pτ, ξ1 q pt, s, x 1 , yqpL m s ´L m,pτ, ξ1
q s qu m ps, yq.

Rappelons que u m pT, x 1 q " g m px 1 q (condition terminale), et avec les notations de (1.26) l'équation précédente se réécrit :

u m pt, x 1 q " P m,pτ, ξ1 q T,t g m px 1 q `G m,pτ,ξ 1 q t 0 ,t f m pt, x 1 q `G m,pτ, ξ1 q T,t 0 f m pt, x 1 q `P m,pτ,ξ 1 q t 0 ,t u m pt 0 , x 1 q ´P m,pτ, ξ1 q t 0 ,t u m pt 0 , x 1 q `ż T t ds ż R nd dy ´Isďt 0 pm,pτ,ξ 1 q pt, s, x 1 , yqpL m s ´L m,pτ,ξ 1 q s q `Isąt 0 pm,pτ, ξ1 q pt, s, x 1 , yqpL m s ´L m,pτ, ξ1
q s q ¯um ps, yq.

(1.28)

Nous voyons dès lors que pour ξ 1 ‰ ξ1 apparaît un nouveau terme dû à la discontinuité du changement de point de gel. Notons toutefois, que le rôle de t 0 est semblable à celui de pξ, ξ 1 q, dans le sens où après de potentielles dérivations en espace nous posons t 0 " t `c0 d 2 px, x 1 q.

(1.29) Autrement dit, le temps t 0 corresponds exactement au temps critique de changement de régime. Le terme de discontinuité P pτ,ξ 1 q t 0 ,t upt 0 , x 1 q ´P pτ, ξ1 q t 0 ,t u m pt 0 , x 1 q se traite identiquement à ceux précédemment contrôlés.

Conclusion

A ce stade, nous pouvons énoncer le contrôle de la solution de (1.15) uniformément à l'indice de régularisation m :

}u} L 8 pC 2`γ b,d q ď C ´}g} C 2`γ b,d `}f } L 8 pC γ b,d q `}u} L 8 pC 2`γ b,d q " 1 `c γ 2 0 `T γ 2 ‰ ¯.
(1.30)

La constante C ą 0 dépend des normes Hölder de a, et de F. Pour des normes Hölder de a et de F, pour c 0 et pour T ą 0 suffisamment petits, on conclut facilement l'estimée de Schauder. Pour des coefficients a et F quelconques, nous procédons par une méthode de scaling. C'est-à-dire pour tout λ ą 0 petit, on écrit le problème de Cauchy équivalent à

(1.2) :

$ ' & ' % B t u λ pt, xq `xFpt, λ ´1{2 T λ xq, λ 1{2 T ´1 λ Du λ pt, xqy `λ´1 2 Tr `D2 x 1 u λ pt, xqapt, λ ´1{2 T λ xq ˘" ´f pt, λ ´1{2 T λ xq, u λ pT, xq " gpλ ´1{2 T λ xq, x P R nd .
(1.31) En reprenant les contrôles précédents on obtient dès lors :

}u} L 8 pC 2`γ b,d q ď C ´}g} C 2`γ b,d `}f } L 8 pC γ b,d q `}u} L 8 pC 2`γ b,d q " λ γ 2 `cγ 2 0 `T γ 2 ‰ ¯. (1.32)
Pour T , c 0 et λ suffisament petits devant les normes Hölder de a et de F, on obtient

C " λ γ 2 `cγ 2 0 `T γ 2 ‰ ă 1.
Nous avons donc le contrôle Schauder voulu en passant le terme

C}u} L 8 pC 2`γ b,d q " λ γ 2 `cγ 2 0 `T γ 2 ‰
à gauche de l'inégalité (1.32). Pour passer à temps quelconque, nous itérons le raisonnement précédent en remplaçant gpxq par upT n , xq pour T n " N ´k`1 N T , avec N suffisamment grand, k P rr0, N ss. Autrement dit, on écrit pour t P r0, p1 ´k´1

N qT q :

# B t u k pt, xq `xFpt, xq, Du k pt, xqy `1 2 Tr `D2 x 1 u k pt, xqapt, xq ˘" ´f pt, xq, u k pp1 ´k´1 N qT, xq " u k´1 pp1 ´k´1 N qT, xq.
Enfin, par un argument de compacité (le Théorème d'Ascoli) et l'unicité au problème de la martingale associé établie par [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF], on obtient qu'il existe une unique solution mild vérifiant l'estimée de Schauder (1.10).

Pour passer de la solution mild à la solution faible, nous exploitons de nouveau la dualité des espaces de Besov.

Unicité forte

Présentation du problème

Pour un système d'Équations Différentielles Ordinaires (EDO), le caractère bien posé s'avère être un véritable enjeu quand on quitte le cadre Lipschitz, voir par exemple [START_REF] Di Perna | Ordinary Differential Equations, transport theory and Sobolev Spaces[END_REF]. En effet l'exemple de Peano montre que l'unicité de la solution peut ne pas être garantie quand la dérive du système n'est que Hölder. Néanmoins, la présence d'un bruit permet de restaurer ladite unicité. Ce phénomène s'appelle la régularisation par le bruit, voir e.g. Flandoli [START_REF] Flandoli | Random perturbation of PDEs and fluid dynamic models[END_REF]. Zvonkin [START_REF] Zvonkin | A transformation of the phase space of a diffusion process that will remove the drift[END_REF] a été le premier à considérer ce cadre en dimension 1. Puis une généralisation multidimensionnelle et elliptique a été établie par Veretennikov [START_REF] Veretennikov | Strong solutions and explicit formulas for solutions of stochastic integral equations[END_REF]. Mentionnons également Krylov et Röckner [START_REF] Krylov | Strong solutions of stochastic equations with singular time dependent drift[END_REF] qui ont vérifié l'unicité forte pour un système non-dégénéré avec dérive L q ´Lp , ainsi que Zhang [START_REF] Zhang | Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients[END_REF] pour une matrice de diffusion faiblement Lipschitz. Enfin Fedrizzi et Flandoli [START_REF] Fedrizzi | Pathwise Uniqueness and Continuous Dependence for SDEs with Nonregular Drift[END_REF] ont établi l'unicité forte dans un cadre elliptique et avec une dérive L q ´Lp et Hölder.

Le cadre, dégénéré avec n " 2 a été étudié par Chaudru de Raynal [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF], Wang et Zhang [START_REF] Wang | Degenerate SDE with Holder-Dini drift and non-Lipschitz noise coefficient[END_REF], Fedrizzi, Flandoli, Priola et Vovelle [START_REF] Fedrizzi | Regularity of stochastic kinetic equations[END_REF], Zhang [START_REF] Zhang | Stochastic hamiltonian flows with singular coefficients[END_REF].

Nous considérerons comme précédemment dans l'étude des estimées de Schauder, le système suivant :

dX 1 t " F 1 pt, X 1 t , . . . , X n t qdt `σpt, X 1 t , . . . , X n t qdW t , dX 2 t " F 2 pt, X 1 t , . . . , X n t qdt, dX 3
t " F 3 pt, X 2 t , . . . , X n t qdt, . . .

dX n t " F n pt, X n´1 t , X n t qdt, t ě 0.
(1.33) L'unicité forte est triviale lorsque les coefficients sont Lipschitz. En effet, on note :

X t " x `ż t 0 Fps, X s qds `ż t 0 σps, X s qdW s , X 1 t " x `ż t 0 Fps, X 1 s qds `ż t 0 σps, X 1 s qdW s .
(1.34)

Si σ et F sont Lipschitz, on déduit qu'il existe C " Cpσ, Fq ą 0 tel que :

E ˇˇX t ´X1 t ˇˇ2 ď C ż t 0 E|X s ´X1 s | 2 ds. (1.35)
Par l'inégalité de Gronwall on a directement que X t " X 1 t pour tout t ě 0 presque sûrement.

Dans le cas où la dérive n'est pas Lipschitz, l'étude de l'unicité forte s'avère nettement plus difficile. Nous supposerons que :

-Régularité du coefficient de diffusion. σ est Lipschitz (une régularité plus faible reste un problème ouvert), -Régularité de la dérive. pour tout (i P rr1, nss, on a rpF i qs

β j d ă 8 avec β j P p 2j´2 2j´1 , 1s et rpD x i´1 F i qs η d ă 8, pour un «petit» η.
La régularité de la dérive est optimale pour notre approche, ces seuils de régularité sont cohérents à ceux pour l'estimée de Schauder. La différence vient du fait que les composants de la dérive ont tous la même régularité, c'est-à-dire que la régularité de F i ne dépend pas de i.

E " sup sďt |X s ´X1 s | 2  ď 5 ˆE " sup sďt |U m ps, X s q ´Um ps, X 1 s q| 2  `E "ż t 0 |rDU m Bs ps, X s q ´rDU m Bs ps, X 1 s q| 2 }σ} 2 8 ds  `E "ż t 0 p}DU m B} 8 `1q |rσps, X s q ´σps, X 1 s qs| 2 ds  `2}R m ¨p¨}| 8 ˙. L'identité (1.39) implique ainsi qu'il existe T ą 0 tel que pour tout 0 ă T ă T et une constante C T " C T pT q ÝÑ T Ñ0 0 telle que E " sup tďT |X t ´X1 t | 2  ď C T E " sup tďT |X t ´X1 t | 2  `10}R m ¨p¨q} 8 . (1.38)
Pour T suffisamment petit, on conclut donc en faisant tendre m vers `8 et par argument circulaire on a que E rsup tďT |X t ´X1 t | 2 s " 0. Autrement dit, on vient de montrer l'unicité forte de la dynamique (1.33). Pour un temps quelconque T , on itère le raisonnement sur r k N T, k`1 N T s avec k P rr0, N ´1ss et N suffisamment grand.

Partie I : Introduction

Contrôles de gradient

Le coeur de l'analyse est de montrer les contrôles de gradients suivant.

Theorem 4 Il existe T ą 0 tel que pour tout T ď T il existe une constante C T :" CppAq, T q ą 0 dépendant seulement des coefficients de diffusion et tel que C T Ñ 0 quand T Ñ 0 tel que pour tout m ě 0 la solution U m satisfait à l'inégalité suivante :

||DU m || 8 `||DpDU m Bq|| 8 ď C T . ( 1
› › ›DD x 1 ż T t ds ż R nd F m k ps, yqp pτ,ξq pt, s, ¨, yqdy › › › 8 ď C T .
Pour les autres termes de l'équation (1.41) faisant intervenir la dérivé de u i suivant la composante non-dégénérée x 1 , on écrit aussi par cancellation (partielle) :

n ÿ l"1 › › ›D l D x 1 ż T t " P pτ,ξq s,t
ppF 1 ps, ¨q ´F1 ps, θ s,t pξqqqD 1 u i ps, ¨qq

ı p¨q ds › › › 8 `n ÿ l"1 › › ›D l D x 1 ż T t " P pτ,
py k q  ϕ ps,y 1:k´1 ,y k`1:n q,t,ξ,λ k,l,j py k q +ˇˇˇˇˇ. ď ż T t ds › › › › " Ψ ps,y 1:k´1 ,y k`1:n q,pt,xq k,pl,1q,j p¨q › › › › B 1´α j k 1,1 sup y j ,jPrr1,nss,j‰k, λPp0,1q › › › › ϕ ps,y 1:k´1 ,y k`1:n q,t,ξ,λ k,l,j p¨q › › › › B α j k ´1 8,8 ď CT γ p}Du i } 8 `}DD 1 u i } 8 q,
(1.43) avec Ψ ps,y 1:k´1 ,y k`1:n q,pt,xq k,pl,1q,j

:

y k P R d Þ Ñ Ψ ps,y 1:k´1 ,y k`1:n q,pt,xq k,pl,1q,j py k q " " D x l D x 1
pξ pt, s, x, yq ´pF k py 1:l´1 , θ l:n s,t pξqq ´Fk pθ s,t pξqq ´Dx k´1 F k pθ s,t pξqq `y ´θs,t pξq ˘k´1 q ¯py ´θs,t pξqq j  ϕ ps,y 1:k´1 ,y k`1:n q,t,ξ,λ k,l,j

:

y k P R d Þ Ñ ϕ
ps,y 1:k´1 ,y k`1:n q,t,ξ,λ k,l,j py k q " D y k D y j u i ps, y 1:l´1 , y l:n `λpy l:n ´θl:n s,t pξqqq.

(1.44)

Notons que la norme sup y j ,jPrr1,nss,j‰k, λPp0,1q › › ϕ ps,y 1:k´1 ,y k`1:n q,t,ξ,λ k,l,j p¨q

› › B α j k ´1 8,8
est équivalent à la norme α j k -Hölder suivant la k-ème variable de D j u i que l'on note }D j pu i q k } α j k . Pour établir le contrôle de cette norme, nous procédons par argument circulaire, c'est-à-dire qu'on montre une inégalité du type

}D j pu i q k } α j k ď `1 `T γ p}D 1 u i } 8 `}D j pu i q k } α j k q ˘,
en injectant la formule de Duhamel (1.41). Le principal intérêt de cette formulation est que contrairement à la méthode adoptée pour les estimées de Schauder, il n'y a plus besoin de méthode de scalling. En effet, l'inégalité (1.43) associée aux autres contrôles implique

}DD 1 u i } 8 ď C `1 `T γ p}D 1 u i } 8 `}DD 1 u i } 8 q ˘,
on déduit dès lors pour T suffisamment petit que

}DD 1 u i } 8 `}DD 1 u i } 8 q ď C.

Part II

Non-asymptotic concentration inequalities

Chapter 3

A first non-asymptotic concentration result

Abstract : We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant distribution ν of an ergodic Brownian diffusion process and the empirical distribution of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions f such that f ´νpf q is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when some suitable squared-norms of the diffusion coefficient also lie in this class. We apply these estimates to design computable non-asymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem.

Introduction

Setting

The aim of this chapter is to approach the invariant distribution of the solution of the diffusion equation:

dY t " bpY t qdt `σpY t qdW t , (1.1)
where pW t q tě0 is a Wiener process of dimension r on a given filtered probability space pΩ, G, pG t q tě0 , Pq, b : R d Ñ R d , and σ : R d Ñ R d b R r are assumed to be Lipschitz continuous functions and to satisfy a mean-reverting assumption in the following sense. If A denotes the infinitesimal generator of the diffusion (1.1), there exists a twice continuously differentiable Lyapunov function V : R d Ñ p0, `8q such that lim |x|Ñ`8 V pxq " `8 and AV ď β ´αV where β P R and α ą 0 . Such a condition ensures the existence of an invariant distribution. We will also assume uniqueness of the invariant distribution, denoted from now by ν. We refer to the monographs by Khasminskii [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF] (see also its augmented second edition [START_REF] Khasminskii | Stochastic Stability of Differential Equations[END_REF]), or Villani [Vil09] and to the survey paper [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF], for in-depth discussions on the conditions yielding such existence and uniqueness results.

We introduce an approximation algorithm based on an Euler like discretization with decreasing time step, which may use more general innovations than the Brownian increments. Namely, for the step sequence pγ k q kě1 and n ě 0, we define:

X n`1 " X n `γn`1 bpX n q `?γ n`1 σpX n qU n`1 , (1.2)
where X 0 P L 2 pΩ, F 0 , Pq and pU n q ně1 is an i.i.d. sequence of centered random variables matching the moments of the Gaussian law on R r up to order three, independent of X 0 . We define the empirical (random) occupation measure of the scheme in the following way. For all A P BpR d q (where BpR d q denotes the Borel σ-field on R d ):

ν n pAq :" ν n pω, Aq :"

ř n k"1 γ k δ X k´1 pωq pAq ř n k"1 γ k . (1.3)
The measure ν n is here defined accordingly to the intrinsic time scale of the scheme. Namely, Γ n " ř n k"1 γ k represents the current time associated with the Euler scheme (1.2) after n iterations. Since we are interested in long time approximation, we consider steps pγ k q kě1 such that Γ n :"

ř n k"1 γ k Ñ n `8. We also assume γ k Ó k 0.
Observe that, for a bounded ν-a.s. continuous function f , it is proved in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] (see e.g. Theorem 1), that:

ν n pf q " 1 Γ n n ÿ k"1 γ k f pX k´1 q a.s. ÝÑ n νpf q " ż R d f pxqνpdxq, (1.4)
or equivalently that ν n pω, ¨q w ÝÑ n ν, Ppdωq ´a.s. The above result can be seen as an inhomogeneous counterpart of stability results discussed for homogeneous Markov chains in Duflo [START_REF] Duflo | Méthodes récursives aléatoires. Techniques Stochastiques[END_REF]. Intuitively, the decreasing steps make the approximation more and more accurate in long time and, therefore, the ergodic empirical mean of the scheme converges to the quantity of interest. Put it differently, there is no bias. This is a significant advantage w.r.t. a more naive discretization method that would rely on a constant step scheme. Indeed, even if this latter approach gains in simplicity, taking γ k " h ą 0 in (1.2) would lead to replace the r.h.s. of (1.4) by the quantity ν h pf q :" ş R d f pxqν h pdxq, with ν h denoting the invariant distribution of the Euler scheme with step h. In such a case, for the analysis to be complete, one needs to investigate the difference ν ´νh through the corresponding continuous and discrete Poisson problems. We refer to Talay et al. [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF], [START_REF] Talay | Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme[END_REF] for a precise presentation of this approach. Now, once (1.4) is available, the next question naturally concerns the rate of that convergence. This was originally investigated by Lamberton and Pagès [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] for functions f of the form f ´νpf q " Aϕ, i.e. f ´νpf q is a coboundary for A. The specific reason for focusing on such a class of functions is that an invariant distribution ν is characterized as a solution in the distribution sense of the stationary Fokker-Planck equation A ˚ν " 0 (where A ˚stands for the adjoint of A). Thus, for smooth enough functions ϕ (at least C 2 pR d , Rq), one has νpAϕq " ş R d Aϕpxqνpdxq " 0. The authors then investigate the weak convergence of ν n pf q ´νpf q once suitably renormalized. However, in these results, the assumptions are made on the function ϕ itself rather than on f . To overcome this limitation and exploit directly some assumptions on the function f requires to solve the Poisson equation Aϕ " f ´νpf q. This is precisely for this step that some structure conditions are needed, namely (hypo)ellipticity or confluence conditions. We refer for instance to the work of Pardoux and Veretennikov [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF], Rothschield and Stein [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF] or Villani [Vil09] who discuss the solvability of the Poisson problem under some ellipticity or hypoellipticity assumptions. We also mention the work of Pagès and Panloup [START_REF] Pagès | A mixed-step algorithm for the approximation of the stationary regime of a diffusion[END_REF] who exploit some confluence conditions allowing to handle for instance the case of an Ornstein-Uhlenbeck process with degenerate covariance matrix. We refer to Sections 2.2 and 2.3 for precise assumptions giving the uniqueness of the invariant distribution of (1.1) and the expected smoothness properties for the associated Poisson problem.

In the current chapter, our goal is to establish for this recursive procedure a nonasymptotic Gaussian control for the deviations of the quantity ν n pf q ´νpf q for possibly unbounded Lipschitz continuous functions f . Such non-asymptotic bounds are crucial in many applicative fields. Indeed, for specific practical simulations, it is not always possible to run ergodic means for very large values of n. It will be direct to derive, as a byproduct of our deviations estimates, some computable non-asymptotic confidence intervals. A specific feature of such non-asymptotic deviation inequalities is that their accuracy depends again on the status of the diffusion coefficient σ with respect to the Poisson equation. Thus, if }σ} 2 ´νp}σ} 2 q " Aϑ is a coboundary (where } ¨} denotes a matrix norm), we manage to improve our analysis, to derive better concentration bounds in a certain deviation range as well as some additional deviation regimes. Also, this additional study seems rather efficient to capture the numerical behavior of the empirical deviations. We refer to Section 4 and 6.2 for details about these points. Eventually, our main deviation results allow to provide deviation inequalities for plain Lipschitz continuous sources f in the ergodic approximation, by using a suitable regularization procedure, as established in Theorem 7. As expected, dealing with this general class of functions requires more stringent constraints on the time steps, that must be small enough, and prevents from obtaining the fastest convergence rates (see again Theorem 7 and Section 5.3).

The main feature of the sequence (1.4) of weighted empirical measures is that it targets the true invariant distribution ν of the continuous time diffusion. The price to pay is the use of an Euler scheme with decreasing step which is a non-homogeneous Markov chain. This induces new difficulties compared to the extensive literature on deviation inequalities for ergodic homogeneous Markov chains. In particular, our approximation procedure produces some remainder terms that need to be controlled accurately enough in a non-asymptotic way to produce tractable deviation inequalities asymptotically close to their counterparts for the diffusion itself. This a major difficulty compared to a CLT where these remainder terms are simply requested to go to 0 fast enough.

As mentioned above and like for the CLT (see [START_REF] Bhattacharya | On the functional central limit theorem and the law of the iterated logarithm for Markov processes[END_REF] for the diffusion or [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] for the weighted empirical measures ν n ), these deviation inequalities are naturally established for coboundaries f ´νpf q " Apϕq, the assumptions being made on ϕ. Our second objective in this chapter is to state our results so that all assumptions could be read on the source function f itself. This first requires to solve the Poisson equation in that spirit, that means deriving pointwise regularity results on ϕ from those made on f . Again, for Lipschitz sources, this step will require an appropriate regularization procedure.

In particular, we will not rely on the Sobolev regularity (see e.g. Pardoux and Veretennikov [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF]) but rather on some Schauder estimates in line with the works by Krylov and Priola [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF], which allow to benefit from the elliptic regularity for operators with unbounded coefficients. For more details, we refer to the introduction of Section 5.

Assumptions and Related Asymptotic Results

From now on, we will extensively use the following notations. For a given step sequence pγ n q ně1 , we denote:

@ P R, Γ p q n :" n ÿ k"1 γ k , Γ n :" n ÿ k"1 γ k " Γ p1q n .
In practice, we will consider time step sequences: γ n -1 n θ with θ P p0, 1s, where for two sequences pu n q nPN , pv n q nPN the notation u n -v n means that Dn 0 P N, DC ě 1 such that @n ě n 0 , C ´1v n ď u n ď Cv n .

For a vector v P R k , k P td, ru, we denote by |v| :" p ř k j"1 v 2 j q 1 2 its (canonical) Euclidean norm. Also, for a function ψ : R q Ñ R d , we set }ψ} 8 :" sup xPR q |ψpxq|. Hypotheses (C1) The random variable X 0 is supposed to be sub-Gaussian, i.e. its square is exponentially integrable up to some threshold. Namely, there exists λ 0 P R ˚such that:

@λ ă λ 0 , E rexppλ|X 0 | 2 qs ă `8.
(GC) The µ-distributed i.i.d. innovation sequence pU n q ně1 is such that E rU 1 s " 0 and for all pi, j, kq P t1, ¨¨¨,

ru 3 , E rU i 1 U j 1 s " δ ij , E rU i 1 U j 1 U k 1 s " 0.
Also, pU n q ně1 and X 0 are independent. Eventually, U 1 satisfies the following Gaussian concentration property, i.e. for every 1´Lipschitz continuous function g : R r Ñ R and every λ ą 0:

E " exppλgpU 1 qq ‰ ď exp ˆλE rgpU 1 qs `λ2 2 ˙.
Observe that if U 1 plawq " N p0, I r q or U 1 plawq " p 1 2 pδ 1 `δ´1 qq br , i.e. for Gaussian or symmetric Bernoulli increments which are the most commonly used sequences for the innovations, the above identity holds. On the other hand, what follows can be adapted almost straightforwardly for a wider class of sub-Gaussian distributions satisfying that for some ą 0 and for all λ ą 0:

E " exppλgpU 1 qq ‰ ď exp ˆλE rgpU 1 qs ` λ 2 4 ˙, (1.5)
which yields that for all r ą 0, Pr|U 1 | ě rs ď 2 expp´r 2 q (sub-Gaussian concentration of the innovation). The case " 2 corresponds to the standard Gaussian concentration. This is also the constant in the logarithmic Sobolev inequality fulfilled by the standard Gaussian measure.

(C2) There exists a positive constant κ such that,

sup xPR d }σpxq} 2 ď κ,
where }σpxq} stands for the operator norm of σpxq, i.e. }σpxq} " sup zPR r ,|z|ď1 |σpxqz| (keep in mind that }σpxq} " }σ ˚pxq} " }σσ ˚pxq} 1 2 ). We then set }σ} 8 :" sup xPR d }σpxq}.

(L V ) There exists a Lyapunov function V : R d ÝÑ rv ˚, `8r, with v ˚ą 0, satisfying the following conditions: i) Regularity-Coercivity. V is a C 2 function, }D 2 V } 8 ă `8, and lim |x|Ñ8 V pxq " `8.

ii) Growth control. There exists C V P p0, `8q such that for all x P R d :

|∇V pxq| 2 `|bpxq| 2 ď C V V pxq.
iii) Stability. Let A be the infinitesimal generator associated with the diffusion Equation (1.1), defined for all ϕ P C 2 0 pR d , Rq and for all x P R d by:

Aϕpxq " xbpxq, ∇ϕpxqy `1 2 Tr `ΣpxqD 2 ϕpxq ˘, Σpxq :" σσ ˚pxq,

where, for two vectors v 1 , v 2 P R d , the symbol xv 1 , v 2 y stands for the canonical inner product of v 1 and v 2 and for M P R d b R d , TrpM q denotes the trace of the matrix M .

There exist α V ą 0, β V P R `such that for all x P R d , AV pxq ď ´αV V pxq `βV .

As a consequence of (L V ) i), there exist constants K and c such that for |x| ě K, |V pxq| ď c|x| 2 , which in turn implies, from (L V ) ii), that |bpxq| ď a C V c|x|.

(U) There exists a unique invariant distribution, denoted from now on by ν, for Equation (1.1).

(S) For a Lyapunov function V satisfying (L V ), we assume that the step sequence pγ k q kě1 satisfies for all k ě 1, γ k ď 1 2 minp 1 ?

C V c , α V C V }D 2 V }8 q.
Condition (S) means that we assume the time steps are sufficiently small w.r.t. the upper bounds of the coefficients and the Lyapunov function.

Remark 1 We have assumed (U) without imposing some specific non-degeneracy conditions. Observe that (L V ) yields existence of an invariant distribution (see e.g. Chapter 4.9 in [START_REF] Ethier | Markov Processes. Characterization and Convergence[END_REF]). Additional structure conditions on the coefficients ((hypo-)ellipticity [START_REF] Khasminskii | Stochastic Stability of Differential Equations[END_REF], [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF], [START_REF] Pardoux | On poisson equation and diffusion approximation 2[END_REF], [START_REF] Pardoux | On the poisson equation and diffusion approximation 3[END_REF], [Vil09] or confluence [START_REF] Pagès | A mixed-step algorithm for the approximation of the stationary regime of a diffusion[END_REF]) then yield uniqueness.

Assumption (S) is a technical condition which is exploited in order to derive the nonasymptotic controls of Theorem 2 (see especially the proof of Proposition 1 below).

Observe that, as soon as conditions (C2), (L V ), (U) are satisfied and E rU 1 s " 0, ErU b3 1 s " 0, the following Central Limit Theorem (CLT) holds (see Theorems 9, 10 in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF]).

Theorem 1 (CLT) Under (C2), (L V ), (U), if E rU 1 s " 0, E rU b3 1 s " 0 and ErV pX 0 qs ă `8, we have the following results. and µ denotes the distribution of the innovations pU k q kě1 . In the above definition of Φ 4 , the term D 3 ϕ stands for the order 3 tensor pB 3 x i ,x j ,x k ϕq pi,j,kqPrr1,dss 3 and we denote, for all x P R d , by D 3 ϕpxqbpxq the R d b R d matrix with entries `D3 ϕpxqbpxq ˘ij " ř d k"1 pD 3 ϕpxqq ijk b k pxq, pi, jq P rr1, dss 2 .

Remark 2 Let us specify that for a step sequence pγ n q nPN such that γ n -n ´θ, θ P p0, 1s, it is easily checked that case (a) occurs for θ P p ă `8 (sublinear diffusion) in case (b). We refer again to Theorems 9 and 10 in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] for further considerations.

Remark 3

The reader should have in mind that an ergodic result similar to the one stated in the fast decreasing step setting holds for the diffusion itself under the same structure assumptions, i.e. (C2), (L V ), (U) (see Bhattacharya [START_REF] Bhattacharya | On the functional central limit theorem and the law of the iterated logarithm for Markov processes[END_REF]). In fact (C2) can be partially relaxed as well, as mentioned above. Precisely,

1 ? t ż t 0 AϕpY s qds L ÝÑ N ´0, ż R d |σ ˚∇ϕ| 2 dν ¯as t Ñ `8.
Note that the asymptotic variance corresponds to the usual integral of the "carré du champ" w.r.t. to the invariant distribution (see again Bhattacharya [Bha82] or the monograph by Bakry et al. [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]), i.e.:

ż

R d |σ ˚∇ϕpxq| 2 νpdxq " ´2 ż R d xAϕ, ϕypxqνpdxq.
In both settings, the normalization is the same: ? t for the diffusion and ? Γ n for the scheme. Except that, as emphasized by Theorem 1, for slowly decreasing step -when θ ă 1{3 -the time discretization effect becomes prominent and "hides" the CLT so that θ " 1{3 (critical value between "fast" and "slow" settings) yields the fastest rate with a biased CLT.

Remark 4

We would like to mention that, in the biased case pbq, for steps of the form γ k " γ 0 k ´1{3 , k ě 1, it is important for a practical implementation to choose γ 0 in an appropriate way, namely by minimizing the function

γ 0 Þ Ñ c 1 γ 0 `c2 γ ´1{2 0 , c 1 " lim n ř n k"1 k ´2{3 p ř n k"1 k ´1{3 q 1{2 , c 2 " ş R d |σ ˚∇ϕ| 2 dν
, which corresponds to the mean-variance contribution deriving from the biased limit Theorem. Of course, c 2 is usually unknown, and the concrete optimization has to be performed replacing c 2 by a computable estimate, like for instance upper bounds, i.e. c 2 ď }σ} 8 }∇ϕ} 8 .

The purpose of this work is to obtain non-asymptotic deviation results which match with the above CLT. In the current ergodic framework, the very first non-asymptotic results were established for the Euler scheme with constant time step by Malrieu and Talay in [START_REF] Malrieu | Concentration inequalities for Euler Schemes[END_REF] when the diffusion coefficient σ in (1.1) is constant. The key tool in their approach consists in establishing a Log Sobolev inequality, which implies Gaussian concentration, for the Euler scheme. This approach allows to easily control the invariant distribution associated with the diffusion process (1.1), see e.g. Ledoux [Led99] or Bakry et al. [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] in a general framework. However Log Sobolev, and even Poincaré, inequalities turn out to be rather rigid tools and are not very well adapted for discretization schemes like (1.2) with or without decreasing steps.

Our approach relies on martingale techniques, which were already a crucial tool to establish the asymptotic results of [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] and have been successfully used in Frikha and Menozzi [START_REF] Frikha | Concentration bounds for stochastic approximations[END_REF] as well to establish non-asymptotic bounds for the regular Monte Carlo error associated with the Euler discretization of a diffusion over a finite time interval r0, T s and a class of stochastic algorithms of Robbins-Monro type. Roughly speaking, for a given n, we decompose the quantity ? Γ n ν n pAϕq as M n `Rn where pM k q kě0 is a martingale which has Gaussian concentration and R n is a remainder term to be controlled in a non-asymptotic way.

We can as well refer to the recent work by Dedecker and Gouëzel [START_REF] Dedecker | Subgaussian concentration inequalities for geometrically ergodic markov chains[END_REF] who also use a martingale approach to derive non-asymptotic deviation bounds for separately bounded functionals of geometrically ergodic Markov chains on a general state space.

Let us also mention that many non-asymptotic results have been obtained based on functional inequalities. Bolley, Guillin and Villani [START_REF] Bolley | Quantitative concentration inequalities for empirical measures on non-compact spaces[END_REF] derived non-asymptotic controls for the deviations of the Wasserstein distance between a reference measure and its empirical counterpart, establishing a non-asymptotic version of the Sanov theorem. Deviation estimates for sums of weakly dependent random variables (with sub exponential mixing rates) have been considered in Merlevède et al. [START_REF] Merlevède | A Bernstein type inequality and moderate deviations for weakly dependent sequences[END_REF]. From a more dynamical viewpoint, let us mention the work of Joulin and Ollivier [START_REF] Joulin | Curvature, concentration and error estimates for Markov chain Monte Carlo[END_REF], who introduced for rather general homogeneous Markov chains a kind of curvature condition to derive a spectral gap for the chain, and therefore an exponential convergence of the marginal laws towards the stationary distribution. We also mention a work of Blower and Bolley [START_REF] Blower | Concentration inequalities on product spaces with applications to Markov processes[END_REF], who obtain Gaussian concentration properties for deviations of functional of the path for metric space valued homogeneous Markov chains or Boissard [START_REF] Boissard | Simple bounds for the convergence of empirical and occupation measures in 1-Wasserstein distance[END_REF] who established non-asymptotic deviation bounds for the Wasserstein distance between the marginal distributions and the stationary law, still in the homogeneous case. The common idea of these works is to prove some contraction properties of the transition kernel of the Markov chain in Wasserstein metric. However, this usually requires to have some continuity in Wasserstein metric for the transition law involved, see e.g. condition (ii) in Theorems 1.2 and 2.1 of [START_REF] Blower | Concentration inequalities on product spaces with applications to Markov processes[END_REF]. Checking such continuity conditions can be difficult in practice. Sufficient conditions, which require absolute continuity and smoothness of the transition laws are given in Proposition 2.2 of [START_REF] Blower | Concentration inequalities on product spaces with applications to Markov processes[END_REF].

Though potentially less sharp for the derivation of constants, the adopted martingalebased approach in this work turns out to be rather simple, robust and can be very naturally adapted to both discrete innovations and inhomogeneous time steps dynamics like the one we currently consider.

It should as well allow to control deviations for functionals of the path, in the spirit of those considered in [START_REF] Pagès | Ergodic approximation of the distribution of a stationary diffusion: rate of convergence[END_REF], [START_REF] Pagès | A mixed-step algorithm for the approximation of the stationary regime of a diffusion[END_REF]. Also, the approach could possibly extend to diffusions with less stringent Lyapunov conditions, like the weakly mean reverting drifts considered in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion: the case of a weakly mean reverting drift[END_REF], or even to more general ergodic Markov processes, see e.g. Pagès and Rey [START_REF] Pagès | Recursive computation of the invariant distribution of Markov and Feller processes[END_REF]. These aspects will concern further research.

As an application of our non-asymptotic concentration results, we will discuss two important topics:

-The first one is of numerical interest and deals with non-asymptotic confidence intervals associated with the estimation of the ergodic mean. Such results can be very useful in practice when the computational resources are constrained (by time, by the model itself,. . . ). If we assume that ϕ P C 3 pR d , Rq, Lipschitz continuous with pD i ϕq iPt2,3u bounded, such that the mapping x P R d Þ Ñ xbpxq, ∇ϕpxqy and D 3 ϕ are Lipschitz continuous, we then establish that there are explicit sequences c n ď 1 ď C n converging to 1 such that for all n P N, for all a ą 0 and γ k -k ´θ, θ P p 1 3 , 1s,

Pr a Γ n ν n pAϕq ě as ď C n exp ˆ´c n a 2 2}σ} 2 8 }∇ϕ} 2 8 ˙. (1.6)
When the diffusion coefficient σ is such that }σ} 2 ´νp}σ} 2 q is itself a coboundary (or its counterpart for any other norm dominating } ¨}), the previous bound improves in a certain deviation range for a. Namely, we are able to replace }σ} 2 8 by νp}σ} 2 q in (1.6), going thus closer to the theoretical limit variance involving the "carré du champ". Moreover, a mixed regime appears in the non-asymptotic deviation bounds which dramatically improves, from the numerical viewpoint, the general case for a certain deviation range. In particular, the corresponding variance is closer to the asymptotic one given by the "carré du champ" (see Theorem 8 below). In accordance with the limit results of Theorem 1, the drifts associated with the fastest convergence rates can be handled as well. We obtain in full generality, results of type (1.6) under slightly weaker smoothness assumptions, considering e.g. D 3 ϕ being β P p0, 1s-Hölder continuous. Eventually, under suitable ellipticity conditions on σ, we are able to give non-asymptotic deviation bounds for a Lipschitz source f as well as explicit gradient bounds for the solution ϕ of the corresponding Poisson problem.

-The second one is mainly theoretical and concerns non-asymptotic deviation bounds for the celebrated almost-sure CLT first established by Brosamler and Schatte (see [START_REF] Brosamler | An almost everywhere central limit theorem[END_REF] and [START_REF] Schatte | On strong versions of the Central Limit Theorem[END_REF]) and revisited through the ergodic discretization schemes viewpoint in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF].

Both applications require a careful investigation of the corresponding Poisson equation Aϕ " f ´νpf q. We will in particular prove that some pointwise regularity properties can be transferred from f to ϕ.

The chapter is organized as follows. We conclude this section by introducing some notations. Our main results are presented in Section 2. We first state therein the specific concentration results for functions f writing f " Aϕ `νpf q (see Section 2.1). We then proceed with some suitable controls on the Poisson problem associated with A and f in a confluent framework under the two main cases considered: namely a possibly degenerate setting, which requires a strong confluence condition and smooth source and coefficients, and a non degenerate setting, which allows to weaken the confluence condition as well as the smoothness assumptions on the source and the coefficients since in that case we manage to benefit from an elliptic bootstrap property (see Section 2.2). We eventually give in Section 2.3 some practical and tractable deviation bounds and non-asymptotic confidence intervals, including a Slutsky like result, for a given specific source f under the afore mentioned conditions on the coefficients of (1.1).

We prove our main concentration result in Section 3. Section 4 is devoted to the case where }σ} 2 ´νp}σ} 2 q is a coboundary. We then prove in Section 5 the required controls on the Poisson equation for our deviation result to hold as well as the practical controls of Section 2.3. Section 6.1 is dedicated to the non-asymptotic deviation bounds for the almost-sure CLT and Section 6.2 to the numerical illustration of our non-asymptotic confidence intervals.

Notations

In the following, we will denote by C a constant that may change from line to line and depend, uniformly in time, on known parameters appearing in (C1), (GC), (C2), (L V ), (S). Other possible dependencies will be explicitly specified. We will also denote by R n and e n deterministic remainder terms that respectively converge to 1 and 0 with n. The explicit dependencies of those sequences again appear in the proofs.

For a function f P C β pR d , Rq, β P p0, 1s, we denote rf s β :" sup

x‰x 1 |f pxq ´f px 1 q| |x ´x1 | β ă `8
its Hölder modulus of continuity. Observe carefully that, when f is additionally bounded, we have that for all 0 ă β 1 ă β:

rf s β 1 ď rf s β β 1 β p2}f } 8 q 1´β β 1 . (1.7)
Additionally, for f P C p pR d , Rq, p P N, we set for β P p0, 1s:

rf ppq s β :" sup x‰x 1 ,|α|"p |D α f pxq ´Dα f px 1 q| |x ´x1 | β ď `8,
where α (viewed as an element of N d 0 zt0u with N 0 :" N Y t0u) is a multi-index of length p, i.e. |α| :" ř d i"1 α i " p. For notational convenience, we also introduce for k P N 0 , β P p0, 1s and m P t1, d, dˆru the Hölder space

C k,β pR d , R m q :" ! f P C k pR d , R m q : @α, |α| P rr1, kss, sup xPR d |D α f pxq| ă `8, rf pkq s β ă `8) .
We also denote by C k,β b the subset of C k,β for which the functions themselves ares bounded. In particular, C 0,1 pR d , R m q is the space of Lipshitz continuous functions from R d to R m and C 0,β b pR d , R m q denotes the space of bounded β-Hölder continuous functions. Observe as well that, if f P C k,β , k ě 1 then f is Lipschitz continuous.

We will as well use the notation rrn, pss, pn, pq P pN 0 q 2 , n ď p, for the set of integers being between n and p. Also, for a given Borel function f : R d Ñ E, where E can be R, R d , R d b R q , q P tr, du, we set for k P N 0 :

f k :" f pX k q.
Eventually, for k P N 0 , we denote by F k :" σ `pX j q jPrr0,kss ˘.

2 Main results

Result of non-asymptotic Gaussian concentration

Our main concentration result is given by the following theorem. In this theorem, we consider a slightly more general situation than for the CLT recalled in Theorem 1. We only assume ϕ P C 3,β pR d , Rq, β P p0, 1s instead of ϕ P C 4 pR d , Rq with existing bounded partial derivatives up to order four (which in particular implies that in Theorem 1 ϕ P C 3,1 pR d , Rq).

Theorem 2 Assume (C1), (GC), (C2), (L V ), (U), (S) hold. Consider a Lipschitz continuous (possibly unbounded) function ϕ P C 3,β pR d , Rq for some β P p0, 1s. Let us furthermore suppose that:

DC V,ϕ ą 0, @x P R d , |ϕpxq| ď C V,ϕ p1 `aV pxqq. (G V )
Let θ P r1{p2 `βq, 1s and assume the step sequence pγ k q kě1 is of the form γ k -k ´θ.

paq Unbiased Case (sub-optimal convergence rate): Let θ P p 1 2`β , 1s. (i) Assume that the mapping x Þ Ñ x∇ϕpxq, bpxqy is Lipschitz continuous.

Then, there exist two explicit monotonic sequences c n ď 1 ď C n , n ě 1, with lim n C n " lim n c n " 1, such that for all n ě 1 and for every a ą 0: keeping in mind that, since ϕ P C 3,β pR d , Rq, rD 3 ϕs β ă `8. We define subsequently:

P " | a Γ n ν n pAϕq| ě a ‰ ď 2C n exp
E β n :" 1 ? Γ n n ÿ k"1 γ 3{2 k ż 1 0 dt p1 ´tqt ż 1 0 du Λ β k´1 pt, u, X k´1 q.
(2.2)

Set now B n,β :"E β n , if β P p0, 1q, B n,β :"E β n `1 ? Γ n n ÿ k"1 γ 2 k ż 1 0 p1 ´tqTr ´D2 ϕpX k´1 `tγ k b k´1 qb k´1 b b k´1 ¯dt `1 2 ? Γ n n ÿ k"1 γ k Tr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σk´1 ¯, if β " 1.
(

2.3)

There exist two explicit monotonic sequences c n ď 1 ď C n , n ě 1, with lim n C n " lim n c n " 1 such that for all n ě 1 and for every a ą 0:

P " | a Γ n ν n pAϕq `Bn,β | ě a ‰ ď 2C n exp ˆ´c n a 2 2}σ} 2 8 }∇ϕ} 2 8 ˙.
For β P p0, 1q, the random variables

|B n,β | " |E β n | ď rϕ p3q s β }σ} p3`βq 8 E " |U 1 | 3`β ‰ p1`βqp2`βqp3`βq Γ p 3`β 2 q n ?
Γn ÝÑ n a β,8 ą 0 a.s. Also, for β " 1, the pB n,1 q ně1 are exponentially integrable and if, furthermore, D 3 ϕ is C 1 , B n,1 Ñ n ´r γm a.s. where r γm is as in Theorem 1. In any case, a bias appears in our deviation controls when we consider, for a given smoothness of order β P p0, 1s for D 3 ϕ, the fastest associated time steps γ k -k ´θ, θ " 1 2`β .

Remark 5 Observe that, when β " 1, the above result provides a non-asymptotic counterpart of the limit Theorem 1. In particular, the concentration constants appearing in Theorem 2 asymptotically match those of the centered CLT recalled in Theorem 1, up to a substitution of the asymptotic variance ş R d |σ ˚∇ϕpxq| 2 νpdxq by its natural upper bound }σ} 2 8 }∇ϕ} 2 8 . Importantly, these bounds do not require "a priori" non-degeneracy conditions and only depend on the diffusion coefficient through the sup-norm of the diffusion matrix Σ, assumption (C2). It will anyhow be very natural to consider a non-degeneracy condition ([PV01], [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF], [Vil09]), or a confluence condition ([PP14]), when investigating the deviations for a given function f , in order to ensure the solvability of the corresponding Poisson equation Aϕ " f ´νpf q and to derive explicit upper bounds for }∇ϕ} 8 in terms of the coefficients b, σ and the source f which turn out to be crucial to design computable non-asymptotic confidence intervals. These aspects are discussed in Section 2.2 below.

The alternative form of the asymptotic variance (see Remark 3) ş R d |σ ˚∇ϕpxq| 2 νpdxq " ´2 ş R d f pxqϕpxqνpdxq suggests that for bounded source terms f , an associated natural variance bound would be 2}f } 8 }ϕ} 8 . Such a control would a priori require less regularity on ϕ than assumed in Theorem 2. One could for instance try to exploit suitable regularization procedures, like for instance the one proposed in Section 5.3 for the proof of Theorem 7 below, to establish non-asymptotic deviation results under weaker assumptions. Our main objective being to capture unbounded Lipschitz functions f , these aspects will concern further research.

Remark 6 (Smoothness and Convergence Rate) Observe that, in coherence with the asymptotic setting of the CLT recalled in Theorem 1, for a given ϕ P C 3,β pR d , Rq, β P p0, 1s, the fastest convergence rate for the deviations is attained for θ " 1 2`β . A bias appears, which can be difficult to estimate in practice since ϕ is usually unknown.

Remark 7 (On the smoothness property of x Þ Ñ xbpxq, ∇ϕpxqy) The Lipschitz continuity assumption on the above mapping appearing in case (i) might seem awkward at first sight. It is non-intrinsic in the sense that it involves both the drift b of the model and the test function ϕ. However, this condition naturally appears when ϕ is a smooth solution to the Poisson equation Aϕ " f ´νpf q. Indeed, recalling the definition of A in (L V ), iii), we can rewrite: x∇ϕpxq, bpxqy " f pxq ´νpf q ´1 2 Tr ´ΣpxqD 2 ϕpxq ¯.

Hence, the Lipschitz continuity of the function in the above left hand side readily follows as soon as the source f is Lipschitz and if D 2 ϕ is bounded and Lipschitz continuous (since σ is also bounded and Lipschitz). Note that with the previous notations for function spaces the previous conditions are implied if

f P C 1,β pR d , Rq Ă C 0,1 pR d , Rq, ϕ P C 3,β pR d , Rq ñ D 2 ϕ P C 1,β b pR d , R d b R d q Ă C 0,1 b pR d , R d b R d q.
We refer to Section 5.1 for details. We now state an improvement of the previous concentration bound when }σ} 2 ´νp}σ} 2 q is itself a coboundary, i.e. when the Poisson problem Aϑ " }σ} 2 ´νp}σ} 2 q can be solved with ϑ satisfying the assumptions required for ϕ in Theorem 2. Precisely, we have the following result.

Theorem 3 paq Under the assumptions of Theorem 2 and with the notations introduced therein, provided that ϑ solution to the Poisson equation Aϑ " }σ} 2 ´νp}σ} 2 q satisfies the same smoothness and growth conditions as ϕ, for β P p0, 1s and θ P p 1 2`β , 1s (unbiased case), there exist two explicit monotonic sequences cn ď 1 ď Cn , n ě 1, with lim n Cn " lim n cn " 1 such that for all n ě 1 for all 0 ă a ď χ n ?

Γn Γ p2q n for a positive sequence χ n Ñ n 0 arbitrarily slowly, so that χ n

? Γn Γ p2q n Ñ n `8: P " | a Γ n ν n pAϕq| ě a ‰ ď 2 r C n exp ˆ´r c n a 2 2νp}σ} 2 q}∇ϕ} 2 8 ˙.
(2.4)

pbq If ϑ solve the Poisson equation Aϑ " ~σ~2 ´νp~σ~2q mutatis mutandis for a matrix norm dominating the operator norm p~σpxq~ě }σpxq}), then the above bound (2.4) still holds with νp~σ~2q instead of νp}σ} 2 q.

Importantly, the above result allows to improve the natural variance bound }∇ϕ} 2 8 }σ} 2 8 of Theorem 2 by a more refined, namely }∇ϕ} 2 8 νp}σ}q 2 . Such a bound can be particularly interesting when the supremum norm of σ is high but its average w.r.t. the invariant distribution ν significantly lower. We refer to Section 4, Theorem 8 (general form of Theorem 3) and 6.2 (numerical results) for further discussions on that topic.

Of course Claim pbq is less sharp than paq stated with the operator norm } ¨} but solving the Poisson equation for }σpxq} seems highly non trivial. By contrast, if ~σpxq~" }σpxq} F :" " Tr `σσ ˚pxqq ‰ 1{2 stands for the Fröbenius norm, Theorem 4 below yields the expected smoothness properties on }σ} 2 F ´νp}σ} 2 F q that ensure the existence of a solution to Aϑ " }σ} 2 F ´νp}σ} 2 F q meeting the required smoothness conditions. The price to pay with such computable norms being that they usually induce some dependence on the dimension d on the estimates (observe e.g. for the identity matrix

I d of R d b R d , }I d } F " d 1{2 ).

Uniqueness of the invariant distribution and Regularity issues for the Poisson problem

For our deviation analysis to work, we need to have the uniqueness of the invariant distribution ν and to establish some pointwise controls on the solution of the associated Poisson equation. Namely, we need to have quantitative bounds on its derivatives and the associated Hölder continuity modulus up to order 3.

To do so, additionally to our main assumptions introduced for Theorem 2, we will work in the confluent setting. In dimension one, any ergodic diffusion is in some sense confluent (see [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF], Appendix of the English translation, Theorem 2.2 p. 308 and its alternative proof in [START_REF] Lemaire | Invariant measure of duplicated diffusions and application to richardson-romberg extrapolation[END_REF] Theorem 2). Here, we will suppose that the following condition holds:

' Confluence Conditions (D p α )
We assume that there exists α ą 0 and p P p1, 2s such that for all

x P R d , ξ P R d B Dbpxq `Dbpxq 2 ξ, ξ F `1 2 r ÿ j"1 ´pp ´2q |xDσ ¨j pxqξ, ξy| 2 |ξ| 2 `|Dσ ¨j ξ| 2 ¯ď ´α|ξ| 2 ,
where Db stands here for the Jacobian of b, σ ¨j stands for the j th column of the diffusion matrix σ and Dσ ¨j for its Jacobian matrix.

Within the confluent framework, we will consider from now on two kinds of assumptions which first give the uniqueness of ν and that can lead to the required smoothness and to computable gradient bounds, which are crucial since they are precisely the quantities appearing in the non-asymptotic Gaussian deviation controls as emphasized in the statement of Theorem 2.

-Strong Confluence condition and regularity of the coefficients, which means that the drift is sufficiently dominant in the dynamics and the coefficients are smooth (see assumption (C R ) below). Note that these conditions may hold for degenerate diffusion coefficients.

-Non-degeneracy of the diffusion coefficient and mild confluence condition and smoothness on the coefficients (see assumption (C UE ) below).

Under a sufficiently strong confluence condition, i.e. when α is large enough in (D p α ), and provided that the coefficients b, σ, f are sufficiently smooth, it is quite direct to derive, through stochastic flow techniques à la Kunita, the required pointwise bounds for the derivatives of the Feynman-Kac representation of the solution to the Poisson equation (see [START_REF] Pagès | A mixed-step algorithm for the approximation of the stationary regime of a diffusion[END_REF] and Section 5.1).

In the non-degenerate case, the main advantage is that we can alleviate some restrictions on α and the smoothness assumptions on b, σ, f to benefit from an elliptic regularity bootstrap deriving from suitable Schauder estimates available in the current setting from the work by Krylov and Priola [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF].

We now introduce a smoothness assumption on b, σ, f that will be useful in both the considered cases.

' Smoothness of the coefficients and the source. For k P t1, 3u and β P p0, 1q define

(R k,β ) The coefficients in equation (1.1) are s.t. b P C k,β pR d , R d q, σ P C k,β b pR d , R d q.
Also, the source f for which we want to estimate νpf q belong to C k,β pR d , Rq.

With these assumptions at hand, we now introduce the first setting we consider.

˛The confluent and regular assumption (C R ), holds if (D p α ), (R 3,β ), for some β P p0, 1s, are in force and }Dσ} 2 8 ď 2α 2p3`βq´p where }Dσ} 8 :" sup xPR d ´řd j"1 }Dσ ¨j pxq} 2 ¯1 2 recalling that, for every j P rr1, dss, }Dσ ¨j pxq} stands for the operator norm of Dσ ¨j pxq.

In particular, we do not impose in this case any additional structure condition on σ which can degenerate.

In our second main framework, we will assume some uniform ellipticity conditions. ' Non-degeneracy Conditions. (UE) Uniform ellipticity. We assume that w.l.o.g. that r " d (r ě d could also be considered) in (1.1) and that the diffusion coefficient σ is such that D σ ą 0, @ξ P R d , xσσ ˚pxqξ, ξy ě σ|ξ| 2 .

We now introduce our second main setting:

˛The confluent and non-degenerate assumption (C UE ), holds if (D p α ), (R 1,β ), for some β P p0, 1s, are in force. If d ą 1, we also assume that }Dσ} 2 8 ď 2α 2p1`βq´p and that the diffusion matrix Σ is such that, for all pi, jq P rr1, dss 2 , Σ i,j pxq " Σ i,j px i^j , ¨¨¨, x d q.

Theorem 4 Assume that (L V ) and either (C R ) or (C UE ) are in force. Then there exists a unique invariant distribution for the solution of (1.1), i.e. assumption (U) holds.

The associated Poisson equation

@x P R d , Aϕpxq " f pxq ´νpf q, (2.5)
admits a unique solution ϕ P C

About the regularity of the coefficients

Under (C R ), the derivatives can be expressed using iterated tangent processes and we cannot hope, without a priori any non-degeneracy condition, for a smoothing effect to hold. To have ϕ P C 3,β pR d , Rq, we need to consider a source f P C 3,β pR d , Rq and the same smoothness on b, σ (Assumption (R 3,β )). We refer to Section 5.1 for the proof of Theorem 4 under (C R ).

In the non-degenerate case, the solvability of the Poisson problem is usually studied in a Sobolev setting, see e.g. [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF]. Let us also indicate that pointwise gradient bounds have been obtained by the same authors in [START_REF] Pardoux | On poisson equation and diffusion approximation 2[END_REF] for bounded drifts and diffusion coefficients which are additionally supposed to be smooth, i.e. at least C 2,γ b with the notations introduced in paragraph 1.3. We point out that these estimates do not apply in our current setting in which the drift has typically linear growth.

We eventually mention the last paper by these authors, namely [START_REF] Pardoux | On the poisson equation and diffusion approximation 3[END_REF]. They derive therein the uniqueness of the martingale solution to the Poisson equation in a potentially degenerate setting under suitable local Doeblin conditions. In that framework, pointwise controls are obtained as well for the solution itself but not for its derivatives.

To obtain the required smoothness, we use here in the non-degenerate framework of (C UE ) some Schauder estimates, deriving from the work of Krylov and Priola [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF], which allow to benefit from the elliptic regularity. Namely, to obtain the mentioned smoothness on ϕ solving Aϕ " f ´νpf q, that we expect to be in C 3,β pR d , Rq, β P p0, 1q, we can take a source f P C 1,β pR d , Rq and b P C 1,β pR d , R d q, σ P C 1,β b pR d , R d q. We would eventually like to emphasize that the structure condition on Σ might seem weird at first sight. It is actually needed to decouple the PDEs formally satisfied by pB x i ϕq iPrr1,dss in order to exploit the a priori estimates of [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF] established for scalar valued PDEs. We refer to Section 5.1 for a proof and details.

About the confluence condition and the restrictions on σ

We work here in the confluent setting of (D p α ). This assumption will allow, through a pathwise analysis associated with the tangent flow, to derive a pointwise gradient bound. Another possibility to obtain such a bound is to assume a so-called Bakry and Émery curvature criterion, see [START_REF] Bakry | Diffusions hypercontractives[END_REF][START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]. Under this condition, the gradient and semi-group commute up to an exponential multiplicative factor (see equation (2.7) below).

Bakry and Émery curvature criterion. First, we recall that the "carré du champ" operator Γ of a Markov process with generator A reads, for every f, g in its domain DpAq Γpf, gq :" 1 2 ´Apf gq ´f Ag ´gAf ¯and Γpf q :" Γpf, f q.

We also need to define the Γ 2 operator

Γ 2 pf q " 1 2 ´AΓpf q ´2Γpf, Af q ¯.
In our Brownian diffusion setting, we have

@x P R d , Γpf qpxq " |σ ˚∇f pxq| 2 .
whereas the computation of Γ 2 is significantly more involved. However, if the diffusion matrix Σ " σσ ˚is constant then:

Γ 2 pf qpxq :" Tr `pD 2 f pxqΣq 2 ˘´x∇f, DbΣ∇f ypxq.

With these notations at hand, we say that the semi-group pP t q tě0 of A satisfies the Bakry and Émery curvature criterion with parameter ρ ą 0 if

(BE ρ ) @ f P DpAq, Γ 2 pf q ě ρ Γpf q.
Observe that for Σ " I d the condition (BE ρ ) is actually equivalent to (D p α ) with α " ρ (and any p P p1, 2s since Dσ " 0) and reads

B Dbpxq `Dbpxq 2 ξ, ξ F ď ´ρ|ξ| 2 .
The computation of the Γ 2 for a general non-degenerate diffusion of the form (1.1) is not easy and is discussed in [START_REF] Arnold | Large-time behavior of non-symmetric Fokker-Planck type equations[END_REF]. In particular, in whole generality, the computation of the Γ 2 requires the coefficients of the operator itself to be smooth (i.e. at least C 2 ).

We also refer to [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF] An important property when (BE ρ ) holds, see again [START_REF] Bakry | Diffusions hypercontractives[END_REF], [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF], is that the following commutation inequality holds: @t ě 0, @x P R d , ΓpP t f qpxq ď expp´2ρtqP t Γpf q.

(2.7)

To conclude, let us say that the Bakry-Emery curvature condition is a very powerful tool to derive pointwise gradient bounds. In our framework, this is unfortunately not enough as soon as d ą 1, because additionally to this kind of bounds we also need, to enter in the framework of Schauder estimates under (C UE ), a control of the β-Hölder modulus of the gradient (see Section 5.1). It does not seem that the condition (BE ρ ) helps to get such controls. The restrictions on the variations of Dσ appearing in both assumptions (C UE ) and (C R ) are precisely needed to derive in the first case the bounds on rDϕs β and in the second one to prove that the derivatives exist up to order 3 and that rD 3 ϕs β is controlled as well. This explains why the conditions on Dσ are more stringent in the potentially degenerate setting (C R ). In each case, those bounds are obtained through pathwise analysis and the restrictions on Dσ ensure the time integrability of the iterated tangent flows, see again Section 5.1 and Appendix A in [START_REF] Pagès | A mixed-step algorithm for the approximation of the stationary regime of a diffusion[END_REF] for details.

Practical Deviation Bounds

A first Non-asymptotic confidence interval result.

Theorem 5 (Non-asymptotic confidence intervals without bias) Let the assumptions of Theorem 4 be in force. Then, there exists a unique invariant distribution ν for (1.1), i.e. (U) holds. Also, ϕ satisfies

(G V ) introduced in Theorem 2 for V pxq - 1 `|x| 2 .
Assume that (C1) (sub-gaussian tails of the innovation) holds and that the step sequence pγ k q kě1 is such that γ k -k ´θ, θ P p 1 2`β , 1s. Then, for pc n q ně1 , pC n q ně1 like in Theorem 2 with lim n c n " lim n C n " 1, we have that for all n ě 1 and a ą 0 and for any matrix norm ~¨~dominating } ¨}:

P " a Γ n |ν n pf q ´νpf q| ą a ‰ ď 2 C n exp ´´c n a 2 α 2 2~σ~2 8 rf s 2 1 ¯with ~σ~8 :" sup xPR r ~σpxq~,
(2.8)

P " νpf q P " ν n pf q ´a~σ~8rf s 1 α ? Γ n , ν n pf q `a~σ~8rf s 1 α ? Γ n ı  ě 1 ´2C n exp ˆ´c n a 2 2 ˙, (2.9)
where the parameter α is the same as in the pointwise gradient bound of Theorem 4.

Proof: Equation (2.8) is a direct consequence of Theorem 2 and the gradient bound in Theorem 4. Indeed, the mean-value Theorem readily yields that (G V ) holds. It then suffice to observe that ν n pf q ´νpf q " ν n pAϕq. To prove (2.9), setting a σ,f,α :" a~σ~8 rf s 1 α , it suffices to write: P " νpf q P " ν n pf q ´aσ,f,α ? Γ n , ν n pf q `aσ,f,α ? Γ n ı  " 1 ´Pr a Γ n ˇˇν n pf q ´νpf q ˇˇě a σ,f,α s and conclude by (2.8). A more refined non-asymptotic confidence interval when ~σ~2 ´νp~σ~2q is a coboundary.

We provide in Theorem 6 below a kind of Slutsky's Lemma when, for a matrix norm ~¨~dominating }σpxq} ď ~σpxq~, s.t. ~σ~2 ´νp~σ~2q is a coboundary.

Theorem 6 (Slutsky type concentration result for the coboundary case) Under the assumptions of Theorem 5, for β P p0, 1s and θ P p 1 2`β , 1s (unbiased case), assuming as well that there is a unique solution ϑ to Aϑ " ~σ~2 ´νp~σ~2q satisfying the same assumptions as ϕ in Theorem 5, there exist two explicit monotonic sequences c n ď 1 ď C n , n ě 1, with lim n C n " lim n c n " 1 such that for all n ě 1, for all a ą 0, the following bounds hold: if a ? Γn Ñ 0 (Gaussian deviations) then,

P " | a Γ n ν n pf q ´νpf q a ν n p~σ~2q | ě a ‰ ď 2 C n exp ˆ´c n a 2 α 2 2rf s 2 1 ˙, (2.10)
P " νpf q P " ν n pf q ´aa ν n p~σ~2qrf s 1 α ? Γ n , ν n pf q `aa ν n p~σ~2qrf s 1 α ? Γ n ‰ ı ě 1 ´2C n exp `´c n a 2 2 ˘.
(2.11)

Again, the non-asymptotic confidence interval is explicitly computable in function of the given source f , the coefficients in the dynamics and the chosen (computable) matrix norm ~¨~. It is also sharper than the one in (2.9).

Towards Lipschitz sources in the non-degenerate case

We conclude this section stating a non-asymptotic deviation result for Lipschitz sources under some non-degeneracy conditions (assumption (C UE ) of Theorem 4 replacing the condition stated there for f by a Lipschitz condition).

Theorem 7 (Non-asymptotic concentration bounds for Lipschitz functions) Bhououo Let the assumptions of Theorem 4 with (C UE ) hold except that f is here solely a Lipschitz continuous function. For a time step sequence pγ k q kě1 of the form γ k -k ´θ, θ P p1{2, 1s, we have that, there exist two explicit monotonic sequences c n ď 1 ď C n , n ě 1, with lim n C n " lim n c n " 1 such that for all n ě 1 and for every a ą 0:

P " | a Γ n `νn pf q ´νpf q ˘| ě a ‰ ď 2C n exp ˆ´c n a 2 α 2 2}σ} 2 8 rf s 2 1 ˙(2.12)
where α is as in Theorem 4.

Such estimates are important since they allow to get rather close to the natural framework which appear in functional inequalities (that mainly deal with Wasserstein distances and their possible deviations). Indeed, through the Monge-Kantorovich formulation, the Wasserstein distance involves Lipschitz functions, since it is precisely achieved taking the minimum over Lipschitz functions for all possible coupling with marginal corresponding to the arguments of the distance (see [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]).

In the literature, some non-asymptotic bounds can be found for the deviations from its mean for the Wasserstein distance between the empirical measure of a homogeneous Markov chain and its stationary distribution (see Boissard [START_REF] Boissard | Simple bounds for the convergence of empirical and occupation measures in 1-Wasserstein distance[END_REF]). Here, we manage to get directly the non-asymptotic deviation bounds over all possible Lipschitz functions for the empirical measure of the scheme aiming directly to approximate the target stationary distribution of the diffusion. Handling the Wasserstein distance in our framework would amount to consider the supremum over the Lipschitz functions in the probability in (2.12). This will concern further research.

We eventually point out that Theorem 7 is obtained through regularization arguments of the source f exploiting the previous results of Theorems 2 and 4 (see Section 5.3 for details). This leads to a constraint on the steps, i.e. γ n -n ´θ, θ P p 1 2 , 1s. This is the price to pay, indeed a bigger θ yields a lower convergence rate, to handle less regular Lipschitz sources. Also, to perform the approximation procedure we precisely need a kind of elliptic bootstrap (like in Theorem 4 under (C UE )). This is why we impose the non-degeneracy assumptions.

Proof of the concentration results (Theorem 2)

For notational convenience, we say that assumption (A) holds whenever (C1), (GC), (C2), (L V ), (U) and (S) are fulfilled. We assume throughout this section that (A) is in force and that the function ϕ appearing in the lemmas satisfies the smoothness assumptions of Theorem 2.

Strategy

To control the deviations of ν n pAϕq we first give a decomposition lemma, obtained by a standard Taylor expansion. The idea is to perform a kind of splitting between the deterministic contributions in the transitions and the random innovations. Doing so, we manage to prove that the contributions involving the innovations can be gathered into conditionally Lipschitz continuous functions of the noise, with small Lipschitz constant (functions pψ k pX k´1 , ¨qq kPrr1,nss below). These functions precisely give the Gaussian concentration, see Lemma 2. The other terms, that we will call from now on "remainders", will be shown to be uniformly controlled w.r.t. n and do not give any asymptotic contribution in the "fast decreasing" case θ ą 1{p2 `βq (with the terminology of Theorem 2), see Lemmas 3, 4 and 5.

Lemma 1 (Local Decomposition of the empirical measure ) For all n ě 1 and k P rr0, n ´1ss:

ϕpX k q ´ϕpX k´1 q " γ k AϕpX k´1 q `"γ k ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt `1 2 γ k Tr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σ2 k´1 ¯`ψ k pX k´1 , U k q  ": γ k ApX k´1 q `´ψ k pX k´1 , U k q `R1 n,k pX k´1 q ¯, (3.13)
where for all k P rr1, nss, conditionally to F k´1 , the mapping u Þ Ñ ψ k pX k´1 , uq is Lipschitz continuous in u with constant ? γ k }σ k´1 }}∇ϕ} 8 .

Introducing for a given k, the mapping

u Þ Ñ ∆ k pX k´1 , uq :" ψ k pX k´1 , uq ´E rψ k pX k´1 , U k q|F k´1 s,
we then rewrite:

ϕpX k q ´ϕpX k´1 q " γ k AϕpX k´1 q `∆k pX k´1 , U k q `Rn,k pX k´1 q, with R n,k pX k´1 q :" R 1 n,k pX k´1 q `E rψ k pX k´1 , U k q|F k´1 s. The contribution ∆ k pX k´1 , U k q can be viewed as a martingale increment. Introduce now the associated (true) martingale

M n :" n ÿ k"1 ∆ k pX k´1 , U k q.
(3.14)

Summing over k yields:

ϕpX n q ´ϕpX 0 q " Γ n ν n pAϕq `Mn `n ÿ k"1 R n,k pX k´1 q. (3.15)
Defining R n :" ř n k"1 R n,k pX k´1 q `ϕpX 0 q ´ϕpX n q we obtain the following decomposition of the empirical measure:

ν n pAϕq " ´1 Γ n pM n `Rn q. (3.16)
-Unbiased Case (Sub-Optimal Convergence Rate). This case corresponds to fast decreasing steps of the form γ k -k ´θ, θ ą 1{p2 `βq. To investigate the non-asymptotic deviations of the empirical measure, the idea is now to write for a, λ ą 0:

P " a Γ n ν n pAϕq ě a ‰ ď exp ´´aλ ? Γ n ¯E " exp ´´λ Γ n pM n `Rn q ¯ ď exp ´´aλ ? Γ n ¯E " exp ´´qλ Γ n M n ¯1{q E " exp ´pλ Γ n |R n | ¯1{p , (3.17)
where 1 p `1 q " 1, p, q ą 1. We actually aim to choose q :" qpnq Ñ n 1. For a suitable choice of q satisfying the previous condition, we manage, in the fast decreasing case, to show that R n :" E rexpp pλ Γn |R n |qs 1{p Ñ n 1. For the term involving the martingale M n we actually use the Gaussian concentration property (GC) of the innovation on its increments p∆ k pX k´1 , U k qq kPrr1,nss . Namely, using the control of the Lipschitz constant of ∆ k pX k´1 , ¨q stated in Lemma 1, we derive:

E " exp ´´qλ Γ n M n ¯ " E " exp ´´qλ Γ n M n´1 ¯E " exp ´´qλ Γ n ∆ n´1 pX n´1 , U n q ¯ˇF n´1  ď E " exp ´´qλ Γ n M n´1 ¯ exp ˆλ2 q 2 2Γ 2 n γ n }σ} 2 8 }∇ϕ} 2 8 ď exp ˆλ2 q 2 2Γ n }σ} 2 8 }∇ϕ} 2 8 ˙, (3.18)
iterating the procedure to derive the last identity. From (3.17), we thus get:

P " a Γ n ν n pAϕq ě a ‰ ď R n exp ´´aλ ? Γ n `λ2 q 2Γ n }σ} 2 8 }∇ϕ} 2 8 ¯.
Keeping in mind that we manage to find q :" qpnq Ó n 1 such that the remainder R n Ó n 1, the result of Theorem 2 in the considered case then follows from a quadratic optimization over the parameter λ.

-Biased Case (Optimal Convergence Rate). This case corresponds to slow decreasing steps of the form γ k -k ´θ, θ " 1{p2 `βq. In this setting, some terms of the remainder R n in (3.16) give a non trivial asymptotic contribution. We choose to substract them before studying the deviation (term B n,β in (2.3)).

Explicit controls on the remainders

Summing the increments appearing in (3.13), we now choose for the analysis to write for a given n P N the remainder R n defined after (3.15) as

R n " n ÿ k"1
R n,k pX k´1 q `ϕpX 0 q ´ϕpX n q " pD 2,b,n `D2,Σ,n q `Ḡ n ´Ln , where:

D 2,b,n :" n ÿ k"1 γ k ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt D 2,Σ,n :" 1 2 n ÿ k"1 γ k Tr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σ2 k´1 ¯, Ḡn :" n ÿ k"1 E rψ k pX k´1 , U k q|F k´1 s, L n :" ϕpX n q ´ϕpX 0 q. (3.19)
We refer to the proof of Lemma 1 to check that the above definition of Ḡn actually matches the term ?

Γ n E β n introduced in equation (2.2) of Theorem 2. We rewrite from (3.16)

ν n pAϕq " ´1 Γ n pM n `Rn q " ´1 Γ n `Mn `pD 2,b,n `D2,Σ,n q `Ḡ n ´Ln ˘. (3.20)
We now split the analysis according to the cases (a) and (b) introduced in Theorem 2.

(a) θ P p1{p2 `βq, 1s, β P p0, 1s. From (3.20), the exponential Tchebychev and Hölder inequalities yield that, for all λ P R `and all p, q P p1, `8q, 1 p `1 q " 1,

P " a Γ n ν n pAϕq ě a ‰ ď exp ´´aλ ? Γ n ¯ˆE exp ´´qλ Γ n M n ¯˙1 q ˆˆE exp ´2pλ Γ n `ˇL n ˇˇ`ˇˇḠ n ˇˇ˘¯˙1 2p ˆE exp ´4pλ Γ n ˇˇD 2,b,n ˇˇ¯˙1 4p ˆE exp ´4pλ Γ n ˇˇD 2,Σ,n ˇˇ¯˙1 4p . ( 3
P " a Γ n ν n pAϕq `Bn,β ě a ‰ " P " ν n pAϕq `Ḡ n Γ n ě a ? Γ n ı ď exp ´´aλ ? Γ n ¯ˆE exp ´´qλ Γ n M n ¯˙1 q ˆˆE exp ´2pλ Γ n ˇˇL n ˇˇ¯˙1 2p ˆE exp ´4pλ Γ n ˇˇD 2,b,n ˇˇ¯˙1 4p ˆE exp ´4pλ Γ n ˇˇD 2,Σ,n ˇˇ¯˙1 4p .
(3.23)

Remark 8 Observe that in case (a), the "small steps" and the corresponding sufficient smoothness of ϕ prevent from the appearance of a bias. As a result, the concentration bound is, at the non-asymptotic level, the same as in Theorem 1, up to the additional upper-bound for the variance. In case (b), we subtract the terms B n,β that asymptotically give a bias. When β " 1, this is the case for both terms Ḡn Γn , D The lemma below provides the Gaussian contribution to be exploited in inequalities (3.21) -(3.23).

Lemma 2 (Gaussian concentration) For a ą 0, q P p1, `8q, setting

λ n :" a q}σ} 2 8 }∇ϕ} 2 8 a Γ n , (3.24)
we derive:

exp ˆ´λ n a ? Γ n ˙ˆE exp ´´qλ n Γ n M n ¯˙1 q ď exp ˆ´a 2 2q}σ} 2 8 }∇ϕ} 2 8 ˙.
Lemma 3 (Bounds for the Conditional Expectations) With the above notations, we have that for β P p0, 1s, θ P r 1 2`β , 1s:

|E β n | " | Ḡn | ? Γ n ď a n :" rϕ p3q s β › › σ › › p3`βq 8 E " |U 1 | 3`β ‰ p1 `βqp2 `βqp3 `βq Γ p 3`β 2 q n ?
Γ n , a.s.

Moreover, a n Ñ n a 8 , with a 8 " 0 if θ P p 1 2`β , 1s and a 8 ą 0 if θ " 1 2`β . Also, for β P p0, 1s, θ P p 1 2`β , 1s:

˜E exp ´2pλ n Γ n | Ḡn | ¯¸1 2p ď exp ´λn ? Γ n a n ¯ď exp ´λ2 n 2Γ n p `a2 n p 2 ¯, @p ą 1. (3.25)
As indicated before, we now aim at controlling the remainders. In particular, from (3.17) and (3.19), we are led to handle terms of the form

E exp ´c n ÿ k"1 γ 2 k |bpX k´1 q| 2 ¯ď pL V q E exp ´c C V n ÿ k"1 γ 2 k |V pX k´1 q|
for small enough real constants c ą 0.

To this end, we will thoroughly rely on the following important integrability result for the Lyapunov function.

Proposition 1 Under (A) there is a constant c V :" c V ppAqq ą 0 such that for all λ P r0, c V s, ξ P r0; 1s:

I ξ V :" sup ně0 E rexppλV ξ n qs ă `8.
We now have the following results for the terms appearing in (3.19).

Lemma 4 (Initial term) Let q P p1, `8q be fixed and λ n be as in (3.24) in Lemma 2. For functions ϕ satisfying (G V ), i.e. there exists C V,ϕ ą 0 such that for all x P R d , |ϕpxq| ď C V,ϕ p1 `aV pxqq, for p :" q q´1 and j P t1, 2u:

ˆE exp ´jpλ n |L n | Γ n ¯˙1 jp ď pI 1 V q 1 jp exp ˆpj `1qpC 2 V,ϕ λ 2 n c V Γ 2 n `cV p " pI 1 V q 1 jp exp ˆpj `1qpC 2 V,ϕ a 2 c V q 2 }σ} 4 8 }∇ϕ} 4 8 Γ n `cV p ˙, with c V , I 1 V like in Proposition 1.
Lemma 5 (Remainders) Let q P p1, `8q be fixed and λ n be as in Lemma 2. Then, there exists C 3.26 :" C 3.26 ppAq, ϕq such that for p " q q´1 :

ˆE exp ´4pλ n Γ n ˇˇD 2,Σ,n ˇˇ¯˙1 4p ď exp ˜C3.26 pλ 2 n pΓ p2q n q 2 Γ 2 n ¸pI 1 V q 1 4p .
(3.26)

We also have:

-If the mapping x Þ Ñ x∇ϕpxq, bpxqy is Lipschitz continuous, then there exists C 3.27 :" CppAq, ϕq ą 0 such that

ˆE exp ´4pλ n Γ n ˇˇD 2,b,n ˇˇ¯˙1 4p ď exp ´C3.27 pλ 2 n pΓ p2q n q 2 Γ 2 n ¯pI 1 V q 1 4p . (3.27) -For a ď cvq 4C V p }σ} 2 8 }∇ϕ} 2 8 }D 2 ϕ} 2 8 ? Γn Γ p2q n
, there exists an R `-valued sequence pv n q ně1 such that ˇˇv n ˇˇď C 3.28 :" C 3.28 ppAq, ϕq and

ˆE exp ´4pλ n Γ n ˇˇD 2,b,n ˇˇ¯˙1 4p ď pI 1 V q vn . ( 3
P " ν n pAϕq ě a ? Γ n  ď exp ˆ´a 2 2q}σ} 2 8 }∇ϕ} 2 8 ˙exp ´λ2 n 2Γ n p `pa 2 n 2 ¯exp ˆ3pC 2 V,ϕ λ 2 n c V Γ 2 n `cV p ṗI 1 V q 1 2p exp ˜C3.26 pλ 2 n pΓ p2q n q 2 Γ 2 n ¸pI 1 V q 1 4p ˆexp ˜C3.27 pλ 2 n pΓ p2q n q 2 Γ 2 n ¸pI 1 V q 1 4p ď exp ˆ´a 2 2q}σ} 2 8 }∇ϕ} 2 8 ´1 ´1 q}σ} 2 8 }∇ϕ} 2 8 ! p Γ n ´6C 2 V,ϕ c V `2" C 3.26 `C3.27 spΓ p2q n q 2 ¯`1 p )¯ṗ I 1 V q 1 p exp ´cV p `pa 2 n 2 ¯. (3.30) Recall now that for θ ą 1 2`β ě 1{3, Γ p 3`β 2 q n { ? Γ n Ñ n 0, Γ p2q n { ? Γ n Ñ n 0 (see Lemma 3
and Remark 2). We now take p :" p n Ñ n `8, and therefore q :" q n Ñ n 1, such that

p 1{2 n Γ p 3`β 2 q n ? Γn Ñ n 0 so that from Lemma 3, p n a 2 n Ñ n 0. Since Γ p 3`β 2 q n ? Γn ě Γ p2q n ?
Γn this in turn implies:

d n :" 1 q n }σ} 2 8 }∇ϕ} 2 8 ! p n Γ n ´6C 2 V,ϕ c V `"2C 3.26 `3C 3.27 ‰ pΓ p2q n q 2 ¯`1 p n ) Ñ n 0. (3.31)
We conclude from (3.30) setting c n " q ´1 n p1´d n q, C n :"

pI 1 V q 1 pn expp 1 pn rc V `C3.27 2 s`p na 2 n 2 q Ñ n 1.
Observe that taking an increasing sequence pp n q ně1 readily yields C n Ó n 1, and q n Ó n 1. Also, the sequence pp n q ně1 can be chosen in order to have, for n large enough, d n Ó n 0 so that c n Ò n 1.

(ii) Assume a ď c V q 4C V p }σ} 2 8 }∇ϕ} 2 8 }D 2 ϕ} 2 8 ? Γn Γ p2q n
. Plugging in (3.21) the controls from (3.29), Lemma 3, equation (3.25) , Lemmas 4 (with j " 2), 5 (equations (3.26), (3.28)) we then derive:

P " ν n pAϕq ě a ? Γ n ı ď exp ˆ´a 2 2q}σ} 2 8 }∇ϕ} 2 8 ˙exp ´λ2 n 2Γ n p `pa 2 n 2 ¯exp ˆ3pC 2 V,ϕ λ 2 n c V Γ 2 n `cV p ṗI 1 V q 1 2p exp ˜C3.26 pλ 2 n pΓ p2q n q 2 Γ 2 n ¸pI 1 V q 1 4p pI 1 V q vn ď pI 1 V q vn`3 4p exp ˆcV p `pa 2 n 2 ėxp ˆ´a 2 2q}σ} 2 8 }∇ϕ} 2 8 ´1 ´1 q}σ} 2 8 }∇ϕ} 2 8 ! p Γ n ´6C 2 V,ϕ c V `2C 3.26 pΓ p2q n q 2 ¯`1 p )¯˙.
(3.32)

Since θ ą 1 2`β ě 1{3 (see Remark 2), we again take p :" p n Ò n `8 so that p 1{2 n a n Ñ n 0 which also guarantees:

d n :" 1 q n }σ} 2 8 }∇ϕ} 2 8 ! p n ´6C 2 V,ϕ c V Γ n `2C 3.26 pΓ p2q n q 2 Γ n ¯`1 p n ) Ñ n 0. (3.33)
In this case, we derive the result by setting c n :" q ´1 n p1 ´dn q Ñ n 1, and C n :"

pI 1 V q vn`3 4pn expp c V pn `pna 2 n 2 q Ñ n
1 (see the limits of v n following equation (3.28) and (3.46)).

Again, pp n q ně1 can be chosen in order to have the stated monotonicity for n large enough. Set now

χ n :" c V }σ} 2 8 }∇ϕ} 2 8 4C V }D 2 ϕ} 2 8 q n p n , (3.34) so that a ď χ n ? Γn Γ p2q n
. Thus, the slower p n goes to infinity, the wider the domain of validity for the estimate in the parameter a.

(b) It remains to analyze the case β P p0, 1s, θ " 1 2`β . Let us deal with β " 1. From (3.22), the controls of (3.29) and Lemma 4 (with j " 1) we get:

P " ν n pAϕq `Ḡ n `D2,n Γ n ě a ? Γ n  ď exp ˆ´a 2 2q}σ} 2 8 }∇ϕ} 2 8 ˙exp ˆ2pC 2 V,ϕ λ 2 n c V Γ 2 n `cV p ˙pI 1 V q 1 p .
Recalling the definition of λ n in (3.24), we conclude as previously with obvious modifications of pc n q ně1 , pC n q ně1 . The case β P p0, 1q is handled similarly starting from (3.23).

Also, when D 3 ϕ P C 1 , we derive similarly to the proof of Theorem 10 in

[LP02] that B n,1 Ñ n ´r γm.
Eventually, the final control involving the two sided deviation is derived by symmetry.

Proof of the Technical Lemmas

This section is devoted to the proof of the previously used Lemmas 1-5 and Proposition 1 which were the key ingredients to derive Theorem 2.

Proof of Lemma 1.

For k P rr1, nss, we first write: ϕpX k q ´ϕpX k´1 q " pϕpX k q ´ϕpX k´1 `γk b k´1 qq `pϕpX k´1 `γk b k´1 q ´ϕpX k´1 qq ": T k´1,r pϕq `Tk´1,d pϕq, (3.35) in order to split the random and deterministic contributions in the transitions of the scheme (1.2).

We then perform a Taylor expansion with integral remainder at order 2 for the function ϕ in the two terms of the r.h.s. of (3.35). Namely, with the above notations:

T k´1,d pϕq " γ k b k´1 ¨∇ϕpX k´1 q `γk ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt, T k´1,r pϕq " ? γ k σ k´1 U k ¨∇ϕpX k´1 `γk b k´1 q `γk ż 1 0 p1 ´tqTr ´D2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1 ¯dt.
Hence,

ϕpX k q´ϕpX k´1 q " γ k AϕpX k´1 q `γk ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt `?γ k σ k´1 U k ¨∇ϕpX k´1 `γk b k´1 q `γk ż 1 0 p1´tqTr ´D2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1 ´D2 ϕpX k´1 qΣ k´1 ¯dt " γ k AϕpX k´1 q `γk ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt `γk ż 1 0 p1 ´tqTr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σk´1 ¯dt `ψk pX k´1 , U k q ": γ k AϕpX k´1 q `Dk 2,b `Dk 2,Σ `ψk pX k´1 , U k q, (3.36)
where

ψ k pX k´1 , U k q " ? γ k σ k´1 U k ¨∇ϕpX k´1 `γk b k´1 q `γk ż 1 0 p1 ´tqTr ´D2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1
´D2 ϕpX k´1 `γk b k´1 qΣ k´1 ¯dt.

(3.37)

Observe now that, conditionally to F k´1 , the mapping u Þ Ñ ψ k pX k´1 , uq is Lipschitz continuous: indeed, the innovation U k does not appear in the other contributions of the right side of (3.36). Consequently, as ϕ is Lispchitz continuous we derive, for all pu, u 1 q P pR d q 2 :

|ψ k pX k´1 , uq ´ψk pX k´1 , u 1 q| ď ? γ k }σ k´1 } }∇ϕ} 8 |u ´u1 |.
The result is obtained by summing up the previous identities from k " 1 to n, observing, with the notations of (3.19), that L n " ř n k"1 ϕpX k q ´ϕpX k´1 q, D 2,b,n "

ř n k"1 D k 2,b , D 2,Σ,n " ř n k"1 D k 2,Σ , G n :" ř n k"1 ψ k pX k´1 , U k q. Proof of Lemma 2.
The idea is to use conditionally and iteratively the Gaussian concentration property (GC) of the innovation. Let us note that this strategy was already the key ingredient in [START_REF] Frikha | Concentration bounds for stochastic approximations[END_REF]. In the current framework, we exploit that the functions u Þ Ñ ∆ k pX k´1 , uq :" ψ k pX k´1 , uq ´E rψ k pX k´1 , U k q|F k´1 s are conditionally independent w.r.t. F k´1 and Lips-chitz continuous with constant ? γ k }σ} 8 }∇ϕ} 8 by Lemma 1. We thus write:

E exp ´´qλ Γ n M n ¯" E exp ˜´qλ Γ n n ÿ k"1 ∆ k pX k´1 , U k q " E " exp ´´qλ Γ n n´1 ÿ k"1 ∆ k pX k´1 , U k q ¯E" exp ´´qλ Γ n ∆ n pX n´1 , U n q ¯|F n´1 ıı ď E " exp ´´qλ Γ n n´1 ÿ k"1 ∆ k pX k´1 , U k q ¯exp ´q2 λ 2 2Γ 2 n γ n }σ} 2 8 }∇ϕ} 2 8 ¯ı, (3.38)
where we used (GC) in the third line recalling as well that E r∆ n pX n´1 , U n q|F n´1 s " 0.

Iterating the process over k, we obtain:

ˆE exp ´´qλ Γ n M n ¯˙1 q " ˜E exp ´´qλ Γ n n ÿ k"1 ∆ k pX k´1 , U k q ¯¸1 q ď exp ´qλ 2 }σ} 2 8 }∇ϕ} 2 8 2Γ n ¯. (3.39) Finally, exp ´´λa ? Γ n ¯ˆE exp ´´qλ Γ n M n ¯˙1 q ď exp ´gpλq ? Γ n ¯,
where g : R `Ñ R is defined by gpλq " ´a ? Γn λ `qλ 2 2Γn }σ} 2 8 }∇ϕ} 2 8 . As a ą 0, the function attains its minimum at λ n given in (3.24). This eventually yields the expected bound.

Proof

of Lemma 3.
From the definition in (3.37) and the Fubini theorem, we have that for all k P rr1, nss:

E rψ k pX k´1 , U k q|F k´1 s " γ k ż 1 0 p1 ´tqTr ´E" D 2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1 ´D2 ϕpX k´1 `γk b k´1 qΣ k´1 |F k´1 ‰ ¯dt. (3.40)
Recalling that U k has the same moments as the standard Gaussian random variable up to order three (see (GC)) and is independent of F k´1 , a Taylor expansion yields:

E " Tr ´D2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1 ´D2 ϕpX k´1 `γk b k´1 qΣ k´1 ¯ˇˇF k´1 ı " Tr ´D2 ϕpX k´1 `γk b k´1 qσ k´1 E rU k b U k sσ k´1 ż 1 0 E " Tr ´`D 3 ϕpX k´1 `γk b k´1 `ut ? γ k σ k´1 U k qt ? γ k σ k´1 U k ˘`σ k´1 U k b U k σ k´1 ˘¯ˇˇˇF k´1 ı du ´Tr ´D2 ϕpX k´1 `γk b k´1 qΣ k´1 " Tr ´D2 ϕpX k´1 `γk b k´1 qσ k´1 pE rU k b U k s ´Iq looooooooomooooooooon "0 σ k´1 t? γ k ż 1 0 E " Tr ´`rD 3 ϕpX k´1 `γk b k´1 `ut ? γ k σ k´1 U k q ´D3 ϕpX k´1 `γk b k´1 qsσ k´1 U k σk´1 U k b U k σ k´1 ˘¯ˇˇˇF k´1 ı du,
recalling from (GC) that for all pi, j, lq P rr1, rss,

E rU i k U j k U l k |F k´1 s " E rU i 1 U j 1 U l 1 s " 0 (cancellation argument). Hence, |E rψ k pX k´1 , U k q|F k´1 s| ď γ k ż 1 0 p1 ´tqt 1`β rϕ p3q s β E " γ 1`β 2 k }σ k´1 } 3`β |U k | 3`β ż 1 0 u β du ˇˇF k´1 ı dt " rϕ p3q s β γ 3`β 2 k }σ k´1 } 3`β E r|U k | 3`β s p1 `βqp2 `βqp3 `βq ,
recalling that the third derivatives of ϕ are β-Hölder continuous for the first inequality. We thus derive:

|E β n | " | Ḡn | ? Γ n ď 1 ? Γ n n´1 ÿ k"1 ˇˇE " ψ k pX k´1 , U k q|F k´1 ‰ˇˇď rϕ p3q s β }σ} 3`β 8 E r|U 1 | 3`β s p1 `βqp2 `βqp3 `βq Γ p 3`β 2 q n ? Γ n ": a n .
Proof of Proposition 1.

First of all, let us decompose the Lyapunov function V with a Taylor expansion like in Lemma 1. We again use a splitting between the deterministic contributions and those involving the innovation. We write for all n P N:

V pX n q ´V pX n´1 q " γ n AV pX n´1 q `γ2 n ż 1 0 p1 ´tqTr ´D2 V pX n´1 `tγ n b n´1 qb n´1 b b n´1 ¯dt ´γn 2 Tr `D2 V pX n´1 qqΣ n´1 ˘`? γ n σ n´1 U n ¨∇V pX n´1 `γn b n´1 q `γn ż 1 0 p1 ´tqTr ´D2 V pX n´1 `γn b n´1 `t? γ n σ n´1 U n qσ n´1 U n b U n σ n´1 ¯dt ď ´γn α V V pX n´1 q `γn β V `CV γ 2 n 2 }D 2 V } 8 V pX n´1 q `γn 2 }D 2 V } 8 }σ} 2 8 `?γ n σ n´1 U n ¨∇V pX n´1 `γn b n´1 q `γn 2 }D 2 V } 8 }σ} 2 8 |U n | 2 ď γ n ´´α V 2 V pX n´1 q `r c ¯`? γ n σ n´1 U n ¨∇V pX n´1 `γn b n´1 q `γn 2 }D 2 V } 8 }σ} 2 8 |U n | 2 (3.41)
for a constant r c :" r cpV, σ, β V q. We have in fact considered the time steps sufficiently small (in (S), we have chosen for all n P N,

γ n ă minp 1 2 ? C V c , α V 2C V }D 2 V }8 q).
The two terms involving the innovation U n in the above decomposition can be controlled thanks to the Gaussian concentration hypothesis (GC). Let us define for all x P R d and all γ, λ ą 0 the quantities:

I 1 pγ, λ, xq :" E " exp `λ? γσpxqU 1 ¨∇V px `γbpxqq ˘ı, I 2 pγ, λq :" E " exp `λγ 2 }D 2 V } 8 }σ} 2 8 |U 1 | 2 ˘ı.
The first one is directly controlled owing to hypothesis (GC):

I 1 pγ n , λ, xq ď exp ´λ2 γ n |σ ˚pxq∇V px `γn bpxqq| 2 2 ¯ď pL V q exp ´λ2 γ n C V }σ} 2 8 V px `γn bpxqq 2 ¯. (3.42) Furthermore, under (GC), for all c ă 1 2 , I c :" E rexppc|U n | 2 qs ă `8. Hence, for all λ ă 2c }D 2 V }8}σ} 2
8 γ 1 , Jensen's inequality yields:

I 2 pγ n , λq ď " E exp `c|U n | 2 ˘ıλγn}D 2 V }8}σ} 2 8 2c " exp ´γn lnpI c q λ}D 2 V } 8 }σ} 2 8 2c ¯. (3.43)
These controls allow to prove the integrability statement of the proposition by induction. For n " 0, recalling from assumption (C1) that for all λ ă λ 0 , E exppλ|X 0 | 2 q ă `8 and from (L V ), i) that V pxq ď c|x| 2 outside of a compact set, we derive that for all λ P p0, λ 0 c q, there exists C 0 V,λ P p1, `8q such that

E exp `λV pX 0 q ˘ď C 0 V,λ . Set now r β V :" r c `lnpI c q }D 2 V }8}σ} 2 8 2c and r α V :" min `1 γ 1 , α V 2 ´λC V }σ} 2 8 p1 `γ1 C V r1 γ1 }D 2 V }8 2 sq ˘P p0, 1 γ 1 s, for λ ă α V 2C V }σ} 2 8 p1`γ 1 C V r1`γ 1 }D 2 V }8 2 sq .
Let us assume that for all λ ă λ V :" min ´λ0 2c ,

α V 2C V }σ} 2 8 p1`γ 1 C V r1`γ 1 }D 2 V }8 2 sq , c }D 2 V }8}σ} 2 8 γ 1 ¯,
the property

@k P rr0, n ´1ss, E exp `λV pX k q ˘ď C V,λ :" C 0 V,λ _ exp ´λ r β V r α V ¯, (P n´1 )
holds for a fixed n ´1 P N 0 and let us prove pP n q. By inequalities (3.41), (3.42) and (3.43) and the Cauchy-Schwarz inequality, we derive that for all λ ă λ V ,

E exp `λV pX n q " E " exp `λV pX n´1 q ˘E" exp `λpV pX n q ´V pX n´1 qq ˘ˇF n´1 s ı ď E " exp `λrV pX n´1 qp1 ´αV 2 γ n q `r cγ n s ˘I1 pγ n , 2λ, X n´1 q 1{2 I 2 pγ n , 2λq 1{2 ı " exp `λγ n r β V ˘E" exp ´λ`1 ´αV 2 γ n ˘V pX n´1 q `λ2 γ n C V }σ} 2 8 V pX n´1 `γn b n´1 q ¯ı.
Recall now that

V pX n´1 `γn b n´1 q ď V pX n´1 q `γn |∇V pX n´1 q||b n´1 | `γ2 n 2 }D 2 V } 8 |b n´1 | 2 pL V q,iiq ď V pX n´1 qp1 `γn C V r1 `γn }D 2 V } 8 2 sq.
Thus,

E " exp `λV pX n q ˘‰ ď exp `λγ n r β V ˘E" exp `λ p1 ´γn r α V q looooomooooon Pr0,1q V pX n´1 q ˘ı pJensenq ď exp `λγ n r β V ˘E" exp `λV pX n´1 q ˘ıp1´γn r α V q ď exp `λγ n r β V ˘Cp1´γn r α V q V,λ
using (P n´1 ) for the last inequality. From the above equation and the previous definition of C V,λ we have:

exp `λγ n r β V ˘Cp1´γn r α V q V,λ ď C V,λ ðñ C V,λ ě exp ´λ r β V r α V ¯.
Hence, pP n q holds. Taking c V ă λ V completes the proof. Remark 9 Noting that v ˚:" inf xPR d V pxq ą 0, we get that for all pn, ξq P N ˆr0, 1s, and for all λ ă λ V pv ˚q1´ξ :

E exppλV ξ n q " E exp ´λpv ˚qξ ´Vn v ˚¯ξ loomoon ě1 ¯ď E exp `λpv ˚qξ´1 V n ˘ď C V,λpv ˚qξ´1 ă `8.
Thus, we readily get as a by-product of Proposition 1 that, for all ξ P r0, 1s, λ ă λ V pv ˚q1´ξ , sup nPN E exppλV ξ n q ă `8. We refer to Lemaire (see e.g. Theorem 17 in [START_REF] Lemaire | An adaptive scheme for the approximation of dissipative systems[END_REF]) for additional results in that direction.

Proof of Lemma 4.

Recalling from (G V ) that there exists C V,ϕ ą 0 such that for all x P R d , |ϕpxq| ď C V,ϕ `1 `aV pxq ˘, we get for j P t1, 2u:

" E exp ´jpλ n |ϕpX 0 q ´ϕpX n q| Γ n ¯ 1 jp ď « E exp ´jpλ n C V,ϕ p2 `aV pX 0 q `aV pX n qq Γ n ¯ff 1 jp ď exp ´2C V,ϕ λ n Γ n ¯«E exp ´2jp C V,ϕ λ n a V pX 0 q Γ n ¯ff 1 2jp « E exp ´2jp C V,ϕ λ n a V pX n q Γ n ¯ff 1 2jp
.

Write now for i P t0, nu by the Young inequality:

2jpC V,ϕ λ n a V pX i q Γ n ď c V V pX i q `pjpq 2 C 2 V,ϕ λ 2 n c V Γ 2 n ,
where c V is the positive real constant such that

I 1 V " sup ně0 E rexppc V V pX n qqs ă `8 (see Proposition 1). We then get " E exp ´jpλ n |ϕpX 0 q ´ϕpX n q| Γ n ¯ 1 jp ď exp ´2C V,ϕ λ n Γ n ¯exp ´jpC 2 V,ϕ λ 2 n c V Γ 2 n ¯´E exppc V V pX 0 qq ¯1 2jp ´E exppc V V pX n qq ¯1 2jp ď exp ˆpj `1qpC 2 V,ϕ λ 2 n c V Γ 2 n ˙exp ˆcV p ˙pI 1 V q 1 jp .
Proof of Lemma 5.

' Proof of inequalities (3.27) and (3.28).

-

If x Þ Ñ x∇ϕpxq, bpxqy is Lipschitz continuous. We first rewrite from the definition of D 2,b,n in (3.19): D 2,b,n " n ÿ k"1 γ k ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt " n ÿ k"1 γ k " ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q, b k´1 ´bpX k´1 `tγ k b k´1 qydt `ż 1 0 `x∇ϕ, bypX k´1 `tγ k b k´1 q ´x∇ϕ, bypX k´1 q ˘dt ı .
From the boundedness of ∇ϕ, and the Lipschitz property of the mappings x Þ Ñ bpxq (which has been assumed from the very beginning) and x Þ Ñ x∇ϕpxq, bpxqy (assumed for the current inequality), recalling that b k´1 " bpX k´1 q, one derives that :

|D 2,b,n | ď n ÿ k"1 γ 2 k ´}∇ϕ} 8 rbs 1 `rx∇ϕ, bys 1 ¯|b k´1 | 2 ď C n ÿ k"1 γ 2 k |b k´1 |, C :" Cpb, ϕq. (3.44)
From this inequality, assumption (L V ), ii) and the Jensen inequality (applied to the exponential function for the measure

1 Γ p2q n ř n k"1 γ 2 k δ k ), we derive: ˆE exp ´4pλ n Γ n |D 2,b,n | ¯˙1 4p ď ˜1 Γ p2q n n ÿ k"1 γ 2 k E exp ´4pλ n Γ p2q n Γ n C a C V a V k´1 ¯¸1 4p .
From the Young inequality we obtain:

E exp ´4pλ n Γ p2q n Γ n C a C V a V k´1 ¯ď exp ´´2 ? 2pλ n Γ p2q n Γ n C a C V ? c V ¯2¯E rexppc V V k´1 qs.
We finally derive with the notations of Proposition 1:

ˆE exp ´4pλ n Γ n |D 2,b,n | ¯˙1 4p ď exp ´2pλ 2 n pΓ p2q n q 2 Γ 2 n pC a C V q 2 c V ¯pI 1 V q 1 4p ď exp ´C3.27 pλ 2 n pΓ p2q n q 2 Γ 2 n ¯pI 1 V q 1 4p , setting C 3.27 :" 2 pC ? C V q 2 c V with C " 1 2 `}∇ϕ} 8 rbs 1 `rx∇ϕ, bys 1 ˘as in (3.44).
-

If a ď c V q 4C V p }σ} 2 8 }∇ϕ} 2 8 }D 2 ϕ} 2 8 ? Γn Γ p2q n " χ n ? Γn Γ p2q n
with the notation introduced in (3.34). Write first from (3.19) (definition of D 2,b,n ), using a Taylor expansion on ∇ϕ:

ˆE exp ´4pλ n Γ n |D 2,b,n | ¯˙1 4p ď ˜E exp ´4pλ n Γ n n ÿ k"1 γ 2 k ż 1 0 p1 ´tq ˇˇTr ´D2 ϕpX k´1 `tγ k b k´1 qb k´1 b b k´1 ¯ˇˇd t ¯¸1 4p . (3.45)
We first easily get from the assumptions on ϕ and point ii) of (L V ) that:

ˆE exp ´4pλ n Γ n |D 2,b,n | ¯˙1 4p ď ˜E exp ´2pλ n Γ n n ÿ k"1 γ 2 k C V V k´1 }D 2 ϕ} 8 ¯¸1 4p .
From the Jensen inequality,we derive:

ˆE exp ´4pλ n Γ n |D 2,b,n | ¯˙1 4p ď ˜1 Γ p2q n n ÿ k"1 γ 2 k E exp ´2pλ n Γ p2q n Γ n }D 2 ϕ} 8 C V V k´1 ¯¸1 4p .
We then have from the definition of λ n in (3.24) that:

vn :" 2pλ n Γ p2q n Γ n }D 2 ϕ} 8 C V c V " Γ p2q n ? Γ n 2C V p c V q }D 2 ϕ} 8 }σ} 2 8 }∇ϕ} 2 8 a ď 1.
The Jensen inequality for concave functions yields for all k P rr1, nss:

E exp ´2pλ n Γ p2q n Γ n }D 2 ϕ} 8 C V V k´1 ¯" E exp ´v n c V V k´1 ¯ď ´E exp ´cV V k´1 ¯¯vn .
Thus, setting

v n :" vn 4p " λ n Γ p2q n 2Γ n }D 2 ϕ} 8 C V c V , (3.46)
we finally derive,

" E exp ´4pλ n Γ n |D 2,b,n | ¯ 1 4p ď « 1 Γ p2q n n ÿ k"1 γ 2 k ´sup lě1 E " exppc V V l´1 q ‰ ¯vn ff 1 4p " pI 1 V q vn ": C n ,
using again the notations of Proposition 1. This gives (3.28).

' Proof of inequality (3.26). We proceed as for the proof of (3.28) and (3.27). Write:

ˆE exp ´4pλ n Γ n |D 2,Σ,n | ¯˙1 4p ď ˜E exp ´4pλ n Γ n n ÿ k"1 γ k 2 ˇˇTr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σk´1 ¯ˇˇ¯¸1 4p ď ˜E exp ´2pλ n Γ n }σ} 2 8 rϕ p2q s 1 n ÿ k"1 γ 2 k |b k´1 | ¯¸1 4p ď ˜E exp ´2pλ n Γ n }σ} 2 8 rϕ p2q s 1 C 1 2 V n ÿ k"1 γ 2 k |V k´1 | 1 2 ¯¸1 4p ď ˜1 Γ p2q n n ÿ k"1 γ 2 k E exp ´2pλ n Γ p2q n Γ n }σ} 2 8 rϕ p2q s 1 C 1 2 V |V k´1 | 1 2 ¯¸1 4p .
Using once again the Young inequality and setting C 3.26 :"

}σ} 4 8 rϕ p2q s 2 1 4 C V c V , we obtain: ˆE exp ´4pλ n Γ n |D 2,Σ,n | ¯˙1 4p ď exp ´pλ 2 n 4 ´Γp2q n Γ n ¯2}σ} 4 8 rϕ p2q s 2 1 C V c V ¯pI 1 V q 1 4p ď exp ´C3.26 pλ 2 n ´Γp2q n Γ n ¯2¯p I 1 V q 1 4p
.

A refinement when ~σ~2 ´νp~σ~2q is a Coboundary

We will assume in this section that there exists a solution ϑ of the Poisson problem Aϑ " ~σ~2 ´νp~σ~2q, where ~¨~is a matrix norm such that } ¨} ď ~¨~, satisfying the assumptions stated for ϕ in Theorem 2. This is in particular the case for the Fröbenius norm } ¨}F under the assumptions of the previous Theorem 4.

In this special case, we have a slightly different concentration result improving our previous ones for a certain deviation range.

Theorem 8 Under the assumptions of Theorem 2 and with the notations introduced therein, we have that:

(a) For (β P p0, 1s and θ P p 1 2`β , 1s), there exist two explicit monotonic sequences cn ď 1 ď Cn , n ě 1, with lim n Cn " lim n cn " 1 such that for all n ě 1 for all a ą 0:

P " | a Γ n ν n pAϕq| ě a ‰ ď 2 Cn exp ˆ´c n 2νp~σ~2q}∇ϕ} 2 8 Φ n paq ˙, Φ n paq :" » -˜a2 ´1 ´2 1 `b1 `4 c3 n Γn a 2 ¯¸_ ˜a 4 3 Γ 1 3 n cn ˜1 ´2 3 cn ˆΓn a 2 ˙1 3 ¸`¸fi fl ,
where x `" maxpx, 0q and cn :" ´rϕs 1 rϑs 1

¯2{3 νp~σ~2q~σ~´2

{3 8 čn with čn being an explicit positive sequence s.t. čn Ó n 1.

(b) For β P p0, 1s, θ " 1 2`β , there exist two explicit monotonic sequences cn ď 1 ď Cn , n ě 1, with lim n Cn " lim n cn " 1 such that for all n ě 1 for all a ą 0:

P " | a Γ n ν n pAϕq `Bn,β | ě a ‰ ď 2 r C n exp ˆ´r c n 2νp~σ~2q}∇ϕ} 2 8 Φ n paq ˙.
Remark 10 (About deviation rates) Observe that in order to derive global deviation bounds (valid for every a ą 0) two concentration regimes appear in the previous bounds.

For an arbitrary fixed a ą 0, we have that for n large enough (depending on a), the Gaussian concentration regime will give the fastest decay, since

2 1`b1`4c 3 n Γn a 2 Ñ n 0. Also, when a - ?
Γ n the two above contributions give a Gaussian bound, with suboptimal constants. Eventually, when a " ? Γ n , for a fixed n, we have that the first term is "stuck" at the threshold Γ n whatever level a is considered, i.e. a 2 `1

´2 1`b1`4c 3 n Γn a 2 ˘ÝÑ aÑ8 c3
n Γ n whereas the second clearly becomes bigger.

To summarize, when the Gaussian regime prevails (i.e. when a ? Γn is small), the results of Theorem 2 have been improved in the sense that the variance in the deviations is a sharper upper bound of the "carré du champ" ş R d |σ ˚∇ϕpxq| 2 νpdxq appearing in the asymptotic Theorem 1. Indeed, we managed to replace the supremum norm ~σ~2 8 deriving from Theorem 2 and the domination condition on the matrix norms by νp~σ~2q. However, our martingale approach naturally leads to a bound in }∇ϕ} 2 8 .

On the other hand, the global double regime seems to be the price to pay to benefit from the better approximation of the "carré du champ" in the Gaussian regime.

Eventually, Theorem 3 is a direct consequence of the previous theorem in the Gaussian regime.

Proof: We focus on case (a) for β P p0, 1q, θ P p1{p2 `βq, 1s. Case (b) could be derived similarly following the proof of Theorem 2. We restart from the computations of Section 3.1 that give for all λ ą 0 the control in equation (3.21). Let us now deal with the term giving the concentration and write for all ρ ą 1:

E exp ´´qλ Γ n M n ¯ď ˜E exp ´´ρ qλ Γ n M n ´ρ2 pqλq 2 rϕs 2 1 2Γ 2 n n ÿ k"1 γ k AϑpX k´1 q ¯¸1 ρ ˆ˜E exp ´ρ2 pqλq 2 rϕs 2 1 2pρ ´1qΓ 2 n n ÿ k"1 γ k AϑpX k´1 q ¯¸1´1 ρ ": T 1 ρ 1 T 1´1 ρ 2 . (4.1)
Since for all x P R d , Aϑpxq " ~σpxq~2 ´νp~σ~2q, we obtain:

T 1 " exp ´ρ2 pqλq 2 rϕs 2 1 νp~σ~2q 2Γ n ¯E exp ´´ρ qλ Γ n M n ´ρ2 pqλq 2 rϕs 2 1 2Γ 2 n n ÿ k"1 γ k ~σpX k´1 q~2 ¯.
The key idea is that we have exploited the Poisson equation solved by ϑ to replace the previous rough control exp ´pqλq 2 rϕs 2 1 ~σ~2 8 2Γn ¯, coming from the martingale increment obtained in equation (3.18) and the domination condition on the matrix norms, by the above term exp ´ρ2 pqλq 2 rϕs 2 1 νp~σ~2q 2Γn ¯. This last contribution will be part of the optimization procedure over λ. This improvement will be all the more significant that neighborhoods of the points where the norm of the diffusion coefficient σ attains its supremum are not very much charged by the invariant distribution. The point for T 1 is then to prove that the remaining expectation is less than 1. It will be shown by exhibiting an appropriate underlying supermartingale.

Set to this end Ă T 1 :" exp ´´ρ 2 pqλq 2 rϕs 2 1 νp~σ~2q 2Γn ¯T1 . Define now, for a given n P N and

m P N 0 , S m :" exp ´´ρ qλ Γn M m ´ρ2 pqλq 2 rϕs 2 1 2Γ 2 n ř m k"1 γ k ~σpX k´1 q~2 ¯.
From the definition of the martingale pM k q kě1 in (3.14) and the controls of the Lipschitz constants of the functions `ψk pX k´1 , ¨q˘k Prr1,nss in Lemma 1, we get by iterated conditioning:

Ă T 1 ď E " S n´1 exp ´´ρ 2 pqλq 2 rϕs 2 1 2Γ 2 n γ n ~σpX n´1 q~2 ¯E" exp ´´ρ qλ Γ n pM n ´Mn´1 q ¯ˇˇF n´1 ıı ď pGCq E " S n´1 exp ´´ρ 2 pqλq 2 rϕs 2 1 2Γ 2 n γ n ~σpX n´1 q~2 ¯expp ρ 2 pqλq 2 2Γ 2 n γ n rϕs 2 1 ~σpX n´1 q~2q ı ď E rS n´1 s ď 1.
In other words, pS m q mě0 is a positive supermartingale. We finally get that, for all ρ ą 1:

T 1 ρ 1 ď exp ´ρpqλq 2 rϕs 2 1 νp~σ~2q 2Γ n ¯. (4.2)
For the term T 2 , we have that setting µ :" µpq, n, ρ, λq "

pqλq 2 ρ 2 rϕs 2 1 2pρ´1qΓn , T 2 " E exp ´µ Γ n n ÿ k"1 γ k AϑpX k´1 q ¯,
so that this contribution can be controlled from the previous expansion of Lemma 1 exploiting the technical lemmas of Section 3.1 replacing λ by µ and ϕ by ϑ.

In case (a), for θ P p1{p2 `βq, 1s, β P p0, 1s, the Hölder inequalities yield that for all µ P R `and all p, q P p1, `8q, 1 p `1 q " 1, similarly to (3.21),

T 2 " E exp ´µ Γ n n ÿ k"1 γ k AϑpX k´1 q ¯ď ˆE exp ´´qµ Γ n M ϑ n ¯˙1 q ˆˆE exp ´2pµ Γ n |L ϑ n | ¯˙1 2p ˆE exp ´4pµ Γ n |D ϑ 2,b,n | ¯˙1 4 p ˆE exp ´4pµ Γ n |D ϑ 2,Σ,n | ¯˙1 4 p , (4.3)
where the superscripts in ϑ emphasize that the contributions to be analyzed are those associated with the solution ϑ of the Poisson problem with source ~σ~2 ´νp~σ~2q.

Still for simplicity, we assume as well (case (i)) that the mapping x Þ Ñ xbpxq, ∇ϑpxqy is Lipschitz continuous. Plugging in (4.3) the controls established in Lemma 4 (with j " 2), Lemma 5 (equations (3.26) and (3.27)) and (3.39), then replacing λ n by µ, we get similarly to the first inequality of (3.30) and with the notations of Lemma 3:

T 2 ď exp ´qµ 2 ~σ~2 8 rϑs 2 1 2Γ n ¯exp ˆµ2 2Γ n p `pa 2 n 2 ˙exp ˆ3pC 2 V,ϑ µ 2 c V Γ 2 n `cV p ˙pI 1 V q 1 2 p ˆexp ˜C3.26 pµ 2 pΓ p2q n q 2 Γ 2 n ¸pI 1 V q 1 4 p ˆexp ˜C3.27 ´3pµ 2 pΓ p2q n q 2 2Γ 2 n `1 2p ¯¸pI 1 V q 1 4 p . ď exp ´µ2 Γ n ´q~σ~2 8 rϑs 2 1 2 `p ´pΓ p2q n q 2 Γ n rC 3.26 `3 2 C 3.27 s `3C 2 V,ϑ c V Γ n ¯`1 2p ¯ēxp ´1 p `cV `C3.27 2 ˘`pa 2 n 2 ¯pI 1 V q 1 p .

Set now

Cn :" exp ´1 p `cV `C3.27 2 ˘`pa 2 n 2 ¯pI 1 V q 1 p , ēn :" p´p Γ p2q n q 2 Γ n rC 3.26 `3 2 C 3.27 s `3C 2 V,ϑ c V Γ n ¯`1 2p . (4.4)
In the considered case, the exponent p :" pn can again be taken such that pn Ñ We derive from the above control and (4.2) that for all q, ρ ą 1:

ˆE exp ´´λq Γ n M n ¯˙1 q ď ´T 1 ρ 1 T 1´1 ρ 2 ¯1 q ď exp ´ρqλ 2 rϕs 2 1 νp~σ~2q 2Γ n ¯C ρ´1 ρq n exp ´ρ ´1 ρq µ 2 Γ n ´q~σ~2 8 rϑs 2 1 2 `ē n ¯¯.
Plugging this bound in (3.21), using again the controls of Lemmas 4 and 5, eventually yields:

P " a Γ n ν n pAϕq ě a ı ď exp ˆ´aλ ? Γ n ˙exp ´λ2 2Γ n `ρqrϕs 2 1 νp~σ~2q `1 p ˘¯C ρ´1 ρq n exp ´ρ ´1 ρq µ 2 Γ n ´q~σ~2 8 rϑs 2 1 2 `ē n ¯ēxp ´λ2 Γ n p ´pΓ p2q n q 2 Γ n `C3.26 `3 2 C 3.27 ˘`3C 2 V,ϕ c V Γ n ¯¯exp ´1 p `cV `C3.27 2 ˘`pa 2 n 2 ¯pI 1 V q 1 p .
Choosing p :" p n Ñ n `8 and such that p n pΓ p2q n q 2 Γn Ñ n 0, we get by a standard symmetry and with the notations introduced in the proof of Theorem 2:

P "ˇˇˇa Γ n ν n pAϕq ˇˇě a ı ď 2 C n C ρ´1 ρq n exp ˆ´aλ ? Γ n ˙exp ´λ2 Γ n `ρqrϕs 2 1 νp~σ~2q 2 `en ˘ēxp ´ρ ´1 ρq µ 2 Γ n ´q~σ~2 8 rϑs 2 1 2 `ē n ¯¯,
where e n is defined similarly to ēn in (4.4) replacing p by p. In particular e n Ñ n 0. Note that for the previous choices of p, p, we have that r

C n :" C n C ρ´1 ρq n Ñ n 1 uniformly in ρ ą 1.
Recalling that µ " pqλq 2 ρ 2 rϕs 2 1 2pρ´1qΓn , we are thus led to minimize the polynomial function

P : λ Þ ÝÑ ´aλ ? Γ n `λ2 Γ n A n `λ4 Γ 3 n B n , where A n " A n pρq " ρ r A n and B n " B n pρq " ρ 3 ρ´1 r B n with r A n :" qrϕs 2 1 νp~σ~2q 2 `en and r B n :" q 3 rϕs 4 1 4 ´q~σ~2 8 rϑs 2 1 2 `ē n ¯. (4.5)
Note that both sequences p r A n q ně1 and p r B n q ně1 are bounded and bounded away from zero sequences (and do not depend on ρ). The function P is clearly convex and coercive so it attains its minimum at λ min , unique zero of the equation P 1 pλ min q " 0. This equation reads

λ 3 `An Γ 2 n 2B n λ ´aΓ 5 2 n 4B n " 0 (4.6)
which is the canonical form of this third degree equation to apply the Cardan-Tartaglia formula ( * ) so that

λ min pρq " Γ n 2 » -˜a ? Γ n B n `d´2 A n 3B n ¯3 `a2 Γ n B 2 n ¸1 3 `˜a ? Γ n B n ´d´2 A n 3B n ¯3 `a2 Γ n B 2 n ¸1 3 fi fl .
(4.7) In order to derive our non-asymptotic bound, we select two "regimes" based on a first order expansion of λ min in two cases a Bn ?

Γn Ñ 0 and Bn ? Γn a Ñ 0, assuming that the free parameter ρ " ρ n to be specified later on remains bounded, e.g. ρ P p1, 3s (which implies that both quantities An Bn and 1 Bn remain bounded as well). Also, note that if ρ Ñ 1, then 1 Bn and An Bn Ñ 0. First, one easily checks that if px n q ě1 and pa n q ně1 are two sequences of positive real numbers where pa n q ně1 is bounded, then

´xn `aa 3 n `x2 n ¯1 3 `´x n ´aa 3 n `x2 n ¯1 3 " # 2 3 xn an if x n " o `a 3 2 n ˘pthen x n Ñ 0q, p2x n q 1 3 if a n " o `x 2 3 n ˘pthen x n Ñ `8q. (4.8) ' If a Bn ? Γn " o ´´An Bn ¯3 2 ¯(hence goes to 0), setting then x n " a Bn ?
Γn and a n " 2An 3Bn yields

λ min pρq " λ ˚pρq :" a ? Γ n 2A n as n Ñ `8.
Note that λ ˚:" λ ˚pρq corresponds to the optimization of the quadratic part of P . Then

P pλ ˚q " ´a2 4A n ´1 ´a2 4A 3 n B n Γ n ¯" ´a2 4 r A n ρ ´1 ´a2 4 r A 3 n pρ ´1q r B n Γ n ¯.
Set now ξ n :" αnpaq ρ´1 with α n paq "

r Bn 4 r A 3 n a 2
Γn . Then

P pλ ˚q " ´a2 4 r A n 1 ´ξn 1 `αnpaq ξn . It remains to maximize the mapping ξ Þ Ñ 1´ξ 1`αnpaqξ ´1 over p0, 1q. Its optimum is attained for ξ n " 1 1`b1`1 αnpaq
, which in turn yields

P pλ ˚q " ´a2 4 r A n ¨1 ´2 1 `b1 `4 r A 3 n Γn r Bna 2 '. (4.9)
Note that, with the resulting specification of ρ " ρ n :" 1 `αnpaq

ξ n P p1, 3s (at least for large enough n), the above condition x n " o `a 3 2 n ˘in (4.8) is satisfied a posteriori. * . If the equation z 3 `pz `q " 0 has a unique real zero z ˚then its discriminant ∆ " 4p 3 `27q 2 ą 0 and z ˚" ´1 2 `´q `b ∆ 27 ˘¯1 3 `´1 2 `´q ´b ∆ 27 ˘¯1 3 .

' If a

Bn ?

Γn Ñ `8, then, still owing to (4.8),

λ min pρq " λ˚p ρq " Γ n 2 ´2a B n ? Γ n ¯1 3 " ˆaΓ n 4B n ˙1 3 a Γ n as n Ñ `8.
The value λ˚p ρq corresponds to the quartic pseudo-optimum of P (i.e. obtained by neglecting the quadratic term). This yields, when reintroducing the parameter ρ,

P `λ ˚pρq ˘" ´a 4 3 Γ 1 3 n pρ ´1q 1 3 ρp4 r B n q 1 3 ˜3 4 ´r A n p4 r B n q 1 3 Γ 1 3 n a 2 3
pρ ´1q

1 3 ¸.
The right hand side of this equality is a function of ρ P p1, `8q. Its analysis yields that the optimum is attained in p1, 3{2s and that it tends asymptotically in n to 3{2 in our considered regime. Taking as suboptimal ρ " 3{2 gives:

P `λ ˚pρq ˘ď ´a 4 3 4 ˆΓn r B n ˙1 3 ˜1 ´2 3 r A n r B n ´Γn a 2 ¯1 3 ¸. (4.10)
From (4.10), (4.9) and (4.5), we conclude the proof of case (a) by setting cn :" r

A n r B ´1 3 n
which matches with the definition in the statement of the theorem.

In the biased case, the result follows similarly from the corresponding analysis performed in Section 2 taking A n pρq "

ρqrϕs 2 1 νp~σ~2q 2 . Remark
11 When a -? Γ n , one checks that λ min pρq -Γ n and P pλ min pρqq -´Γn . This behavior is consistent with our non-asymptotic bound. However, for practical and numerical purposes observe that the optimum can be estimated. Namely, plugging the identity (4.6) satisfied by λ min pρq in (4.7) into the definition of P , yields

P `λmin pρq ˘" ´λmin pρq 2 ? Γ n ˜3a 2 ´λmin pρqρ r A n ? Γ n " ´?Γ n 4 pρ ´1q 1 3 ρ Φ n pa, ρq ˆ3a 2 ´?Γ n 2 pρ ´1q 1 3 r A n Φ n pa, ρq ˙,
where

Φ n pa,ρq " ˜a ? Γ n r B n `˜pρ´1q ´2 r A n 3 r B n ¯3`a 2 r B 2 n Γ n ¸1 2 ¸1 3 `˜a ? Γ n r B n ´˜pρ ´1q ´2 r A n 3 r B n ¯3`a 2 r B 2 n Γ n ¸1 2 ¸1 3 .
Then, an optimization in ρ P p1, `8q for given a, Γ n can be performed (noting that ρ Þ Ñ pρ ´1q i{3 ρ ´1, i P t1, 2u are bounded functions over p1, `8q).

Smoothness Results for the Poisson Problems (Proof of Theorem 4)

We first prove here Theorem 4 which allows to derive from the deviation results of Theorems 2 and 3 the practical deviation bounds of Section 2.3 (i.e. Theorems 5, 6 and 7). We recall that we work in the confluent setting of (D p α ) and that we additionally consider two main types of assumptions:

-Strong confluence conditions and smoothness (C R ). Namely, assumptions (L V ), (D p α ) and (R 3,β ) introduced in Sections 1.2 and 2.2 with the condition }Dσ} 2 8 ď 2α 2p3`βq´p .

-Mild confluence conditions and non-degeneracy (C UE ). Namely, assumptions (L V ), (D p α ), (R 1,β ) and (UE) introduced in Sections 1.2 and 2.2 together, when d ą 1, with the condition }Dσ} 2 8 ď 2α 2p1`βq´p and the technical structure assumption on the diffusion coefficient that for all pi, jq P rr1, dss 2 , Σ i,j pxq " Σ i,j px i^j , ¨¨¨, x d q.

It is well known that when (C R ) or (C UE ) are in force, there exists a unique invariant distribution for (1.1), i.e. assumption (U) holds. We refer to [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF], [START_REF] Pagès | A mixed-step algorithm for the approximation of the stationary regime of a diffusion[END_REF], [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF], [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF] for proofs of this assertion. The next step consists precisely in investigating the smoothness of the corresponding Poisson problem as well as some associated quantitative pointwise bounds on the gradient of its solution, which is one of the key terms appearing in the deviation bounds of Theorems 2 and 3.

Let us indicate that the conditions appearing in (C R ) depend on pure pathwise properties, whereas the case (C UE ) takes advantage of the regularity of the underlying semigroup which allows to alleviate some smoothness assumptions on the coefficients and some restrictions on the variations of σ. When the dimension increases, it becomes useful to benefit from the smoothing effects of a non-degenerate semi-group, especially if we keep in mind that one of our goals is to handle Lipschitz continuous sources.

Proof of Theorem 4

Under (C UE ) or (C R ), it is well known that the Poisson equation (2.5) that we now recall:

@x P R d , Aϕpxq " f pxq ´νpf q,
admits a unique solution centered w.r.t. ν and with linear growth, in W 2 p,loc pR d , Rq for any p ą 1 under (C UE ) (see [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF]), or in C 3,β pR d , Rq under (C R ) (see Proposition A.8 in [START_REF] Pagès | A mixed-step algorithm for the approximation of the stationary regime of a diffusion[END_REF]). In both cases, we have the following representation: ϕpxq " ´żR ``P t f pxq ´νpf q ˘dt where P t f pxq :" E rf pY 0,x t qs (5.1) and Y 0,x t solves (1.1) with Y 0,x 0 " x. To comply with the framework of the above Theorems 5 and 6, the first step is to establish a pointwise gradient control.

Gradient Control

Under (C UE ) or (C R ) we manage to obtain pointwise gradient bounds for ϕ. In our current confluent setting, these estimates are obtained through controls on the tangent flow, again without any a priori uniform ellipticity condition of type (UE). Lemma 6 (Pointwise Gradient Bounds) Assume that (C UE ) or (C R ) holds. Then

}∇ϕ} 8 ď rf s 1 α ,
with α as in (D p α ).

Proof: Gradient Control in the Confluent framework Assume now that (D p α ) holds. Observe that, as soon as (R 1,β ) holds, it is well known that that ∇ x Y 0,x t is well defined and belongs to L 2 pPq, see [START_REF] Ikeda | Stochastic differential equations[END_REF]. Hence, for t ą 0, i P rr1, dss:

B x i E rf pY 0,x t qs " E rx∇f pY 0,x t q, B x i Y 0,x t ys, B x i Y 0,x t " e i `ż t 0 DbpY 0,x s qB x i Y 0,x s ds `d ÿ j"1 ż t 0 Dσ ¨j pY 0,x s qB x i Y 0,x s dW j s ,
where e i stands for the i th canonical vector and Db, Dσ ¨j P R d b R d . Let p P p1, 2s be given such that (D p α ) holds. Considering the mapping y P R d Þ Ñ |y| p , where | ¨| stands for the Euclidean norm of R d , it is easily seen from Itô's formula that:

|B x i Y 0,x t | p " 1 `p ż t 0 A B x i Y 0,x s |B x i Y 0,x s | , DbpY 0,x s q B x i Y 0,x s |B x i Y 0,x s | E |B x i Y 0,x s | p ds `p d ÿ j"1 ż t 0 A B x i Y 0,x s |B x i Y 0,x s | , Dσ ¨j pY 0,x s q B x i Y 0,x s |B x i Y 0,x s | E |B x i Y 0,x s | p dW j s `p 2 d ÿ j"1 ż t 0 ´|Dσ ¨j pY 0,x s qB x i Y 0,x s | 2 |B x i Y 0,x s | 2 `pp ´2q |xB x i Y 0,x s , Dσ ¨j pY 0,x s qB x i Y 0,x s y| 2 |B x i Y 0,x s | 4 ¯|B x i Y 0,x s | p ds " exp ˜p ż t 0 A B x i Y 0,x s |B x i Y 0,x s | , DbpY 0,x s q B x i Y 0,x s |B x i Y 0,x s | E ds ¸ˆE `M ˘t ˆexp ˜p 2 d ÿ j"1 ż t 0 ´|Dσ ¨j pY 0,x s qB x i Y 0,x s | 2 |B x i Y 0,x s | 2 `pp ´2q |xB x i Y 0,x s , Dσ ¨j pY 0,x s qB x i Y 0,x s y| 2 |B x i Y 0,x s | 4 ¯ds ¸, (5.2)
where pM t q tě0 :"

`p ř d j"1 ş t 0 @ Bx i Y 0,x s |Bx i Y 0,x s | , Dσ ¨j pY 0,x s q Bx i Y 0,x s |Bx i Y 0,x s | D dW j
s ˘tě0 is a square integrable martingale with bounded integrand and EpM q t :" exppM t ´1 2 xM y t q denotes the associated Doléans exponential martingale. From condition (D p α ), we thus get:

|B x i Y 0,x t | p ď expp´αptq ˆEpM t q. (5.3)
We eventually derive:

ż `8 0 |E rx∇f pY 0,x t q, B x i Y 0,x t ys|dt ď rf s 1 ż `8 0 E r|B x i Y 0,x t | p s 1{p dt ď rf s 1 ż `8 0 exp p´αtq dt " rf s 1 α .
From the above control and equation ( 5.1), we thus derive:

@i P rr1, dss, @x P R d , |B x i ϕpxq| ď rf s 1 α . (5.4)
Similarly, for all x P R d , ∇ϕpxq " ş `8 0 Erp∇Y 0,x t q ˚∇f pY t,x 0 qsdt where we have

∇Y 0,x t " `Bx 1 Y 0,x t ¨¨¨B x d Y 0,x t ˘so that p∇Y 0,x t q ˚" ¨pB x 1 Y 0,x t q . . . pB x d Y 0,x t q ˚‹ '.
Hence, recalling that | ¨| stands for the Euclidean norm, |∇ϕpxq| ď ş `8 0 Er}p∇Y 0,x t q ˚}|∇f pY 0,x t q|sdt where we recall that for

A P R d b R d , }A} :" sup |z|ď1,zPR d |Az| denotes the operator (or spectral) matrix norm. Thus, |∇ϕpxq| ď }∇f } 8 ş `8 0 Er}p∇Y 0,x t q ˚}p s 1{p dt " }∇f } 8 ş `8 0 Er}∇Y 0,x t } p s 1{p dt. Now, }∇Y 0,x t } " sup |z|ď1 |∇Y 0,x t z|.
For any z P R d , |z| ď 1, setting Z 0,x,z t :" ∇Y 0,x t z, one has the following dynamics for the R d -valued process pZ 0,x,z s q sPr0,ts :

Z 0,x,z t :" z `ż t 0 DbpY 0,x s qZ 0,x,z s ds `d ÿ j"1 ż t 0
Dσ ¨j pY 0,x s qZ 0,x,z s dW j s .

Hence, we derive similarly to (5.3) that |Z 0,x,z t | p ď |z| p expp´pαtqEpM t q, where EpM t q does not depend on z. Write now,

Er}∇Y 0,x t } p s 1{p " Ersup |z|ď1 |Z 0,x,z t | p s 1{p ď Ersup |z|ď1 |z| p expp´pαtqEpM t qs 1{p ď expp´αtq. (5.5)
This eventually proves the claim }∇ϕ} 8 :" sup xPR d |∇ϕpxq| ď }∇f }8 α . Additional smoothness -Theorem 4 can be derived under (C R ), by iterating computations similar to the ones performed in Lemma 6. On the other hand, to have the required smoothness, since we cannot expect some smoothing effect from a non-degenerate diffusion coefficient, we have to impose that b, σ, f themselves lie in C 3,β pR d , Rq and the restriction on the variations of σ which ensures exponential integrability in time for the expectations of the iterated tangent flows, see Lemma A.8 in [START_REF] Pagès | A mixed-step algorithm for the approximation of the stationary regime of a diffusion[END_REF] for details (see the parallel between the above condition on Dσ and assumption (AC p ) appearing p. 559 in [START_REF] Pagès | A mixed-step algorithm for the approximation of the stationary regime of a diffusion[END_REF]).

-Proving Theorem 4 under (C UE ) requires more sophisticated tools (Schauder estimates for operators with unbounded coefficients).

Proof: Theorem 4 under (C UE )

Let us begin with the scalar case. For d " 1, set for all x P R,

vpxq :" ´ż `8 0 dt ErΨpY 0,x t qB x Y 0,x t s " ´ż `8 0 dt E " ΨpY 0,x t q exp ´ż t 0 b 1 pY 0,x s qds ¯E´ż t 0 σ 1 pY 0,x s qdW s ¯ , (5.6)
where for all y P R, Ψpyq :" B y f pyq. We observe that B x ϕpxq " vpxq. Also, from our assumptions on f , b, σ, we have that Ψ, b 1 , σ 1 P C 0,β b pR d , Rq. Theorems 2.4-2.6 in Krylov and Priola, [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF] then yield the existence of a unique solution to the PDE: r Awpxq `b1 pxqwpxq " Ψpxq, where r Awpxq " Awpxq `σσ 1 pxqw 1 pxq, (

belonging to C 2,β b pR d , Rq and such that the following Schauder estimate holds:

D C ě 1, }w} 2,β ď Cp1 `}Ψ} β q.
(5.8) Indeed, from (D p α ), we get that b 1 pxq ď ´α ă 0 and the potential in (5.7) has the good sign. From (5.6) and the Girsanov theorem, we also get:

vpxq " ´ż `8 0 dt E " Ψp r Y 0,x t q exp ´ż t 0 b 1 p r Y 0,x s qds ¯ ,
where d r Y 0,x s " `bp r Y 0,x s q `σσ 1 p r Y 0,x s q ˘ds `σp r Y 0,x s qdW s . Note that r Y has generator r A. A simple identification procedure, similar to the proof of Theorem II.1.1 in Bass [START_REF] Bass | Diffusions and Elliptic Operators[END_REF] then gives v " w. The result follows from (5.8). Let us emphasize that this is a quite deep and involved result for unbounded coefficients.

In the multi-dimensional setting, recalling the technical condition that for all i P rr1, dss, j ě i, Σ i,j pxq " Σ i,j px i , ¨¨¨, x d q, we have that differentiating formally the PDE (2.5) in the space variable x i , i P rr1, dss yields that B x i ϕ " v i should satisfy:

r A w i pxq `Bx i b i pxqw i pxq " Ψ i pxq ´ÿ jPrr1,dssztiu B x i b j pxqv j pxq ´1 2 ÿ jPrr1,i´1ss B x i Σ j,j pxqB x j v j pxq ´ÿ jPrr1,i´1ss ÿ kPrrj`1,dssztiu B x i Σ j,k pxqB x j v k pxq, (5.9)
with Ψ i pxq :" B x i f pxq and

r Aw i pxq :" A w i pxq`1 2 B x i Σ i,i pxqB x i w i pxq `ÿ jPrr1,dssztiu B x i Σ i,j pxqB x j w i pxq.
We would now like to enter the previous framework of Schauder estimates. To do so, we first observe from (D p α ) and the Cauchy-Schwarz inequality that B x i b i pxq ď ´α ă 0. Consider now i " 1 in (5.9). From our current assumptions on f , b and the previous computations on the gradient for the multi-dimensional case, it remains to prove r Ψ 1 pxq :" Ψ 1 pxq ´řj‰1 B x 1 b j pxqv j pxq P C 0,β b pR d , Rq. This will be the case, once we will

Partie II: Non-asymptotic concentration inequalities have proved that ∇ϕ is β-Hölder continuous, which is a priori not direct. This property is assumed for the remaining of the proof and shown below. In particular, it leads to the restriction concerning the variations of σ when d ą 1. Hence, Theorems 2.4-2.6 in Krylov and Priola, [KP10] still apply and give that there exists a unique solution w 1 P C 2,β b pR d , Rq to (5.9) which also satisfies:

D C ě 1, }w 1 } 2,β :" ÿ α,|α|Prr0,2ss }D α w 1 } 8 `rD p2q w 1 s β ď Cp1`} r Ψ 1 } β q ": CppL V q, pR 1,β q, pUEqq.
(5.10) The identification w 1 " B x 1 ϕ " v 1 is standard. The control (5.10) allows to iterate, since it gives that ∇w 1 " pB

x 1 v 1 , ¨¨¨, B x d v 1 q " pB x 1 ,x 1 ϕ 1 , ¨¨¨, B x d ,x 1 ϕq is β-Hölder.
We thus get by induction, from the specific chosen structure on σ and by Theorems 2.4-2.6 in Krylov and Priola, [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF], that for all i P rr1, dss there exists a unique solution w i P C 2,β b pR d , Rq to (5.9) such that:

D C ě 1, }w i } 2,β ď Cp1 `} r Ψ i } β q ": CppL V q, pR 1,β q, pUEqq, r Ψ i pxq :" Ψ i pxq ´ÿ jPrr1,dssztiu B x i b j pxqv j pxq ´1 2 ÿ 1 ď j ă i, k P rr1, dssztiu B x i Σ j,k pxqB x j v k pxq.
(5.11)

The Lipschitz property of the mapping x Þ Ñ x∇ϕpxq, bpxqy is eventually derived following the procedure described in Remark 7.

Remark 12 (Structure of σ) We emphasize that the structure condition on σ assumed in Theorem 4 under (C UE ) is mainly technical. It is of course always verified in dimension d " 1. For d ą 1 it is motivated by the fact that, differentiating (2.5) without this assumption yields to consider a system of coupled linear PDEs with growing coefficients for which the Schauder estimates have not been established yet. Following the existing literature for Schauder estimates for systems (see e.g. Boccia [START_REF] Boccia | Schauder estimates for solutions of higher-order parabolic systems[END_REF]), we think that the results of Krylov and Priola should extend to this case. This would allow to get rid of the indicated condition. Here, the condition simply allows to decouple the system.

Let us mention too that the results by Priola [Pri09] could also be a starting point to investigate the smoothness of the Poisson problem for degenerate kinetic models.

These aspects will concern further research.

Additional Smoothness continued: β-Hölder continuity of the gradient through pathwise analysis. We control here, under (D p α ), p P p1, 2s and (R 1,β ), β P p0, 1s, the β-Hölder modulus of continuity of the gradient. We will progressively see how the restrictions on Dσ come out. For px, x 1 q P R 2d , write for all i P rr1, dss:

ˇˇB x i ϕpxq ´Bx i ϕpx 1 q ˇ" ˇˇˇż `8 0 ´E rx∇f pY 0,x t q, B x i Y 0,x t ys ´E rx∇f pY 0,x 1 t q, B x i Y 0,x 1 t ys ¯dt ˇˇď ˇˇˇż `8 0 ´r∇f s β E r|Y 0,x t ´Y 0,x 1 t | β |B x i Y 0,x t |s `}∇f } 8 E r|B x i Y 0,x t ´Bx i Y 0,x 1 t |s ¯dt ˇˇ" : pG β 1 `Gβ 2 qpx, x 1 q.
(5.12)

Let us first deal with the expectation in G β 1 . Namely, write

E r|Y 0,x t ´Y 0,x 1 t | β |B x i Y 0,x t |s ď E r|Y 0,x t ´Y 0,x 1 t | pβ s 1 p Er|B x i Y 0,x t | qs 1 q , p, q ą 1, p´1 `q ´1 " 1.
Take now pβ " q ðñ p " 1`β β , q " 1 `β which leads to the same integrability constraints on the flows.

If β `1 ď p in (D p α ), then we readily get similarly to (5.3) that Er|B x i Y 0,x t | qs 1 q ď expp´αtq.

If now β `1 ą p, as soon as (D 1`β ᾱ ) holds for some ᾱ ą 0, which is actually the case provided that

}Dσ} 2 8 ď 2α 1 `β ´p , (5.13)
for q " 1 `β, we again get similarly to (5.3) that Er|B x i Y 0,x t | qs 1 q ď expp´ᾱtq. On the other hand, the mean value theorem yields:

E r|Y 0,x t ´Y 0,x 1 t | pβ s 1 p ď |x ´x1 | β E r ż 1 0 dλ}∇Y 0,x 1 `λpx´x 1 q t } pβ s 1 p ď |x ´x1 | β ´ż 1 0 dλE r}∇Y 0,x 1 `λpx´x 1 q t } pβ s ¯1 p ď |x ´x1 | β rexpp´αβtqI 1`βďp `expp´ᾱβtqI 1`βąp s ,
exploiting (5.5) for the last inequality provided that (5.13), which in turn implies that (D 1`β ᾱ ) for some ᾱ ą 0, holds if 1 `β ą p. Plugging these bounds in (5.12) gives that:

@px, x 1 q P pR d q 2 , |G β 1 px, x 1 q| ď r∇f s β p1 `βq " I 1`βďp α `I1`βąp ᾱ  |x ´x1 | β . (5.14)
We already see that, when 1`β ą p, for the parameter p of the initial confluence condition (D p α ), a first constraint on the variations of σ, namely (5.13) appears.

Let us now turn to G β 2 . Following the expansion of (5.2) write:

|B x i Y 0,x t ´Bx i Y 0,x 1 t | 2 " 2 ż t 0 A B x i Y 0,x s ´Bx i Y 0,x 1 s , DbpY 0,x s qB x i Y 0,x s ´DbpY 0,x 1 s qB x i Y 0,x 1 s E ds `2 d ÿ j"1 ż t 0 A B x i Y 0,x s ´Bx i Y 0,x 1 s , Dσ ¨j pY 0,x s qB x i Y 0,x s ´Dσ ¨j pY 0,x 1 s qB x i Y 0,x 1 s E dW j s `d ÿ j"1 ż t 0 |Dσ ¨j pY 0,x s qB x i Y 0,x s ´Dσ ¨j pY 0,x 1 s qB x i Y 0,x 1 s | 2 ds. Let uptq :" E |B x i Y 0,x t ´Bx i Y 0,x 1 t | 2 , t ě 0.
First note that up0q " 0. Taking now the expectation and interchanging expectation and time integration yields uptq "

ż t 0 E Ξ s ds
where pΞ t q tě0 is a pathwise continuous process clearly determined by the terms inside the above time integrals. One readily checks that, t Þ Ñ E Ξ s is continuous so that u is continuously differentiable and satisfies

u 1 ptq " 2 E A B x i Y 0,x t ´Bx i Y 0,x 1 t , DbpY 0,x t qB x i Y 0,x t ´DbpY 0,x 1 t qB x i Y 0,x 1 t E `d ÿ j"1 E |Dσ ¨j pY 0,x t qB x i Y 0,x t ´Dσ ¨j pY 0,x 1 t qB x i Y 0,x 1 t | 2 .
Using the Young inequality for a parameter ε P p0, 1s, small enough and to be chosen further, we derive:

u 1 ptq ď 2 E « A B x i Y 0,x t ´Bx i Y 0,x 1 t |B x i Y 0,x t ´Bx i Y 0,x 1 t | , DbpY 0,x t q B x i Y 0,x t ´Bx i Y 0,x 1 t |B x i Y 0,x t ´Bx i Y 0,x 1 t | E |B x i Y 0,x t ´Bx i Y 0,x 1 t | 2 `ż t 0 }DbpY 0,x t q ´DbpY 0,x 1 t q}|B x i Y 0,x 1 t ||B x i Y 0,x t ´Bx i Y 0,x 1 t | ff `E« p1 `εq d ÿ j"1 ˇˇˇD σ ¨j pY 0,x t q B x i Y 0,x t ´Bx i Y 0,x 1 t |B x i Y 0,x t ´Bx i Y 0,x 1 t | ˇˇˇ2 |B x i Y 0,x t ´Bx i Y 0,x 1 t | 2 `p1 `ε´1 q d ÿ j"1 }Dσ ¨j pY 0,x t q ´Dσ ¨j pY 0,x 1 t q} 2 |B x i Y 0,x 1 t | 2 ff .
From this computation, the point is now to make the confluence condition (D p α ) appear and to separate the components for which we will exploit the β-Hölder continuity, namely Db, pDσ ¨j q jPrr1,nss . To do so we first observe that:

A B x i Y 0,x t ´Bx i Y 0,x 1 t |B x i Y 0,x t ´Bx i Y 0,x 1 t | , DbpY 0,x t q B x i Y 0,x t ´Bx i Y 0,x 1 t |B x i Y 0,x t ´Bx i Y 0,x 1 t | E `1 2 d ÿ j"1 ˇˇˇD σ ¨j pY 0,x t q B x i Y 0,x t ´Bx i Y 0,x 1 t |B x i Y 0,x t ´Bx i Y 0,x 1 t | ˇˇˇ2 ď ´α `p2 ´pq 1 2 }Dσ} 2 8 " ´α,
where we suppose from now on that

´α :" ´α `p2 ´pq 1 2 }Dσ} 2 8 ă 0 ðñ }Dσ} 2 8 ă 2α 2 ´p .
(5.15)

Hence,

u 1 ptq ď 2E " p´α `ε 2 }Dσ} 2 8 q|B x i Y 0,x t ´Bx i Y 0,x 1 t | 2  `2E " }DbpY 0,x t q ´DbpY 0,x 1 t q}|B x i Y 0,x 1 t ||B x i Y 0,x t ´Bx i Y 0,x 1 t |  `E" p1 `ε´1 q d ÿ j"1 }Dσ ¨j pY 0,x t q ´Dσ ¨j pY 0,x 1 t q} 2 |B x i Y 0,x 1 t | 2  .
Using now again the Young inequality, with η P p0, 1s small enough, for the middle term of the above r.h.s., we obtain:

u 1 ptq ď 2 ´´α `ε 2 }Dσ} 2 8 `η 2 ¯E" |B x i Y 0,x t ´Bx i Y 0,x 1 t | 2 ı `η´1 E " }DbpY 0,x t q ´DbpY 0,x 1 t q} 2 |B x i Y 0,x 1 t | 2 ‰ `p1 `ε´1 q d ÿ j"1 E " }Dσ ¨j pY 0,x t q ´Dσ ¨j pY 0,x 1 t q} 2 |B x i Y 0,x 1 t | 2 ı ď 2 ´´α `ε 2 }Dσ} 2 8 `η 2 ¯E" |B x i Y 0,x t ´Bx i Y 0,x 1 t | 2 ı `η´1 rDbs 2 β E " |Y 0,x t ´Y 0,x 1 t | 2β |B x i Y 0,x 1 t | 2 ‰ `p1 `ε´1 qrDσs 2 β E " |Y 0,x t ´Y 0,x 1 t | 2β |B x i Y 0,x 1 t | 2 ı .
(5.16) Denote:

´α ε,η,σ :" ´α `ε 2 }Dσ} 2 8 `η 2 ă 0,
for ε, η small enough. Setting for every t ě 0,

rptq :" η ´1rDbs 2 β E " |Y 0,x t ´Y 0,x 1 t | 2β |B x i Y 0,x 1 t | 2 ‰ `p1 `ε´1 qDσs 2 β E " |Y 0,x t ´Y 0,x 1 t | 2β |B x i Y 0,x 1 t | 2 ı , equation (5.16
) reads an ordinary differential inequation:

u 1 ptq ď ´2α ε,η,σ uptq `rptq, up0q " 0.

We derive from the Gronwall lemma that

uptq " Er|B x i Y 0,x t ´Bx i Y 0,x 1 t | 2 s ď expp´2α ε,η,σ tq ż t 0 expp2α ε,η,σ sqrpsqds.
Reproducing as well the computations that led to (5.14), we derive:

uptq ď C η,ε,β |x ´x1 | 2β ż t 0 exp `´2α ε,η,σ pt ´sq ˘ˆE " |B x i Y 0,x 1 s | 2p1`βq ı 1 1`β `ż 1 0 dλE " }∇Y 0,x 1 `λpx´x 1 q s } 2p1`βq ı β 1`β ˙ds.
From the analysis leading to (5.3), (5.5) we now derive:

Er|B x i Y 0,x t ´Bx i Y 0,x 1 t | 2 s ď C η,ε,β 2 |x ´x1 | 2β expp´2α ε,η,σ tq " exp `2pα ε,η,σ ´α 2p1`βq qt αε,η,σ ´α 2p1`βq `exp `2pα ε,η,σ ´β α2p1`βq qt αε,η,σ ´β α2p1`βq  ,
and

´α 2p1`βq ď ´α ``2p1 `βq ´p˘1 2 }Dσ} 2 8 .
Thus, α2p1`βq ă 0 as soon as

}Dσ} 2 8 ă 2α 2p1 `βq ´p, (5.17)
which is precisely the restriction on the variations of σ appearing in (C UE ) when d ą 1, then α2p1`βq ą 0 and:

Er|B x i Y 0,x t ´Bx i Y 0,x 1 t | 2 s ď Cη,ε,β |x ´x1 | 2β expp´α 2p1`βq tq.
This last control then gives the expected bound for the β-Hölder modulus of the gradient. Namely, from (5.12), (5.14),

rB x i ϕs β ă r∇f s β p1 `βq " I 1`βďp α `I1`βąp ᾱ ı `}∇f } 8 Cη,ε,β α2p1`βq .

Proof of the Practical Results of Section 2.3

We first begin with the proof of the

Slutsky like Theorem 6

We keep here for simplicity the generic notation } ¨} for any admissible matrix norm according to the assumptions of the theorem. We first write:

P « a Γ n ν n pf q ´νpf q a ν n p}σ} 2 q ě a ff " P " ν n pAϕq ě a ? Γ n a ν n p}σ} 2 q  .
(5.18)

We then proceed similarly to Theorem 8, with an exponential Bienaymé-Tchebychev inequality, for all λ ą 0 we have:

P « a Γ n ν n pf q ´νpf q a ν n p}σ} 2 q ě a ff ď E " exp ˆ´aλ ? Γ n a ν n p}σ} 2 q ˙exp pλν n pAϕqq ı " exp ˆ´aλ ? Γ n a ν p}σ} 2 q ˙E" exp ˆ´aλ ? Γ n " a ν n p}σ} 2 q ´aν p}σ} 2 q ı ˙exp pλν n pAϕqq ı " exp ˆ´aλ ? Γ n a ν p}σ} 2 q ˙E" exp ˜´aλ ? Γ n ν n p}σ} 2 q ´ν p}σ} 2 q a ν n p}σ} 2 q `aν p}σ} 2 q ¸exp pλν n pAϕqq ı " exp ˆ´aλ ? Γ n a ν p}σ} 2 q ˙E" exp ˜´aλ ? Γ n ν n pAϑq a ν n p}σ} 2 q `aν p}σ} 2 q ¸exp pλν n pAϕqq ı .
By the Hölder inequality, for r p, r q ą 1, such that 1 r p `1 r q " 1:

P « a Γ n ν n pf q ´νpf q a ν n p}σ} 2 q ě a ff ď exp ˆ´aλ ? Γ n a ν p}σ} 2 q «E exp ˜´ar pλ ? Γ n ν n pAϑq a ν n p}σ} 2 q `aν p}σ} 2 q ¸ff1{r p " E exp pλr qν n pAϕqq ı 1{r q .
The proof of Theorem 8 yields:

P « a Γ n ν n pf q ´νpf q a ν n p}σ} 2 q ě a ff ď R n exp ˆ´aλ ? Γ n a ν p}σ} 2 q ˙exp ˆρr qλ 2 Γ n r A n `ρ3 r q 3 λ 4 pρ ´1qΓ n r B n Ė exp ˜´ar pλ ? Γ n ν n pAϑq a ν n p}σ} 2 q `aν p}σ} 2 q ¸¸1{r p , (5.19)
where we recall from identity (4.5):

r A n " qrϕs 2 1 νp}σ} 2 q 2 `en and r B n " q 3 rϕs 4 1 4 ´q}σ} 2 8 rϑs 2 1 2 `ē n ¯.
Also, R n Ñ n 1 denotes a "generic" remainder. Observe that thanks to the bounds of Theorem 4 (stated in the above Lemma 6), we get:

r A n ď qrf s 2 1 νp}σ} 2 q 2α `en .
(5.20)

Let us now handle the remainder

" E exp ˆ´ar pλ ? Γn νnpAϑq ?
νnp}σ} 2 q`?νp}σ} 2 q ˙1{r p :

« E exp ˜´ar pλ ? Γ n ν n pAϑq a ν n p}σ} 2 q `aν p}σ} 2 q ¸ff1{r p " « E ˜exp « ´ar pλ ? Γ n ν n pAϑq a ν n p}σ} 2 q `aν p}σ} 2 q ¸`1 νnpAϑqě0 `1νnpAϑqă0 ˘¸ff 1{r p ď ˜«E exp ˜ar p 2 λ ? Γ n ν n pAϑq a ν p}σ} 2 q ¸ff1{r p P " ν n pAϑq ě 0 ‰ 1{r q `«E exp ˜´ar p 2 λ ? Γ n ν n pAϑq a ν p}σ} 2 q ¸ff1{r p P " ν n pAϑq ă 0 ‰ 1{r q ¸1{r p .
Let us mention that we introduced the above partition in order to get a sharper constant in the final inequality, 1 below instead of 2, which would follow getting rid of the indicator functions. Now, by Theorem 2 we easily get:

E « exp ˜´ar pλ ? Γ n ν n pAϑq a ν n p}σ} 2 q `aν p}σ} 2 q ¸ff1{r p ď R n exp ˆa2 r p 2 Γ n νp}σ} 2 q λ 2 Γ n ´}σ} 2 8 }∇ϑ} 2 8 2 `en ¯"P " ν n pAϑq ě 0 ‰ 1{r q `P" ν n pAϑq ă 0 ‰ 1{r q ı 1{r p . (5.21) We choose r p :" r ppnq Ñ `8, such that r p 2 a 2
Γn Ñ 0, and so pP " ν n pAϑq ě 0 ‰ 1{r q `Prν n pAϑq ă 0s 1{r q q 1{r p ď 2 1{r p Ñ 1. Moreover, exploiting again that for the Gaussian regime, r p 2 a 2 Γn Ñ 0, we obtain by (5.21) and (5.19):

P « a Γ n ν n pf q ´νpf q a ν n p}σ} 2 q ě a ff ď R n exp ˆ´aλ ? Γ n a ν p}σ} 2 q ˙exp ˆρr qλ 2 Γ n p r A n `en q `ρ3 r q 3 λ 4 pρ ´1qΓ n r B n ˙. (5.22)
From identity (5.22), the optimization over λ is similar to the one performed in the proof of Theorem 8. This yields the deviation bound (2.10). The non-asymptotic confidence interval in (2.11) is derived as for Theorem 5 from the gradient bounds of Theorem 4 and (2.10).

Regularization of Lipschitz Sources

We assume here that assumptions (C2), (L V ), (UE) are in force. We suppose as well that the following smoothness holds for b, σ:

(R b,σ ) Regularity and Structure. We assume that there exists β P p0, 1q such that b, σ

in (1.1) belong to C 1,β pR d , R d q and C 1,β b pR d , R d b R d q respectively.
Also, for all pi, jq P rr1, dss 2 , Σ i,j pxq " Σ i,j px i^j , ¨¨¨, x d q.

Importantly, we are interested, under assumptions (C2), (L V ), (UE), (R b,σ ), in giving controls for the estimation of νpf q when the source f is simply Lipschitz continuous. This is indeed the natural framework for the source which can be handled through functional inequality techniques, see [START_REF] Malrieu | Concentration inequalities for Euler Schemes[END_REF], [START_REF] Boissard | Simple bounds for the convergence of empirical and occupation measures in 1-Wasserstein distance[END_REF].

To comply with our previous framework, namely to exploit the smoothness result of Theorem 4 under (C UE ), we need to regularize the source. Let η be a mollifier (i.e. a non-negative compactly supported function such that ş R d ηpxqdx " 1). Define for δ ą 0, η δ pxq " 1 δ d ηp x δ q. We regularize f introducing f δ :" f ‹η δ where ‹ stands for the convolution on R d . From usual estimates, we obtain:

D C η ą 0, @x P R d , |f δ pxq ´f pxq| ď C η δrf s 1 , @β P p0, 1q, r∇f δ s β ď C η rf s 1 δ ´β . (5.23)
We emphasize here that we will choose β later in order to be compatible with a certain range of step sequences. We assume for simplicity that θ P p1{3, 1s (no bias). Recall that we want to investigate:

Pr a Γ n pν n pf q ´νpf qq ě as " P " pν n pf δ q ´νpf δ qq `Rn,δ pf q ě a ? Γ n ı , R n,δ pf q :" rpν n pf q ´νpf qq ´pν n pf δ q ´νpf δ qqs.

(5.24)

From (5.23), one readily gets:

|R n,δ pf q| ď 2C η δrf s 1 . (5.25)
On the other hand, the coefficients b, σ and the source f δ satisfy assumption (R 1,β ) (observe indeed that the mollified function f δ P C 1,β pR d , Rq). Hence, Theorem 4 yields that there exists a unique solution ϕ δ P C 3,β pR d , Rq to the equation:

Aϕ δ " f δ ´νpf δ q.
(5.26)

Observe from the proof of Theorem 4 under (C UE ) (see equations (5.4) and (5.11)) and (5.23) that:

}∇ϕ δ } 8 ď α ´1rf s 1 , @β P p0, 1q, D C β ą 0, @i P t1, 2u, rϕ piq δ s 1 ď C β p1 `}∇f δ } C β q ď C β δ ´β , rϕ p3q δ s β ď C β δ ´β , rx∇ϕ δ , bys 1 ď C β δ ´β .
(5.27) Now, from (5.26) the deviation in (5.24) rewrites:

P " a Γ n pν n pf q ´νpf qq ě a ‰ " P " ν n pAϕ δ q `Rn,δ pf q ě a ? Γ n ı .
(5.28)

From (5.25), the term R n,δ pf q can be seen as a remainder as soon as a ?

Γn " 2C η δrf s 1 ě |R n,δ pf q|. On the other hand, the deviations of ν n pAϕ δ q can be analyzed as above, reproducing the proofs of Theorems 2 and 8, replacing the bounds on prϕ piq s 1 q iPt1,2u , rϕ p3q s β appearing therein by those of equation (5.27). Precisely, we get from (5.25), similarly to (3.30) (replacing the controls on ϕ by those on ϕ δ in the proofs of Lemmas 3 and 5):

P " ˇˇν n pAϕ δ q `Rn,δ pf q ˇˇě a ? Γ n  ď 2 « E exp ´´qλ n Γ n M n ¯ff1 q exp ´´aλ n ? Γ n p1 ´?Γ n 2C η rf s 1 δ a q ēxp ´λ2 n 2Γ n p `ppa δ n q 2 2 ¯exp ˆ3pC 2 V,ϕ λ 2 n c V Γ 2 n `cV p ˙pI 1 V q 1 2p ˆexp ˜Cδ 3.26 pλ 2 n pΓ p2q n q 2 Γ 2 n ¸pI 1 V q 1 4p ˆexp ˜Cδ 3.27 pλ 2 n pΓ p2q n q 2 Γ 2 n ¸pI 1 V q 1 4p , (5.29)
where

C δ 3.27 :" C β δ ´β pC ? C V q 2 c V and C δ 3.26 :" }σ} 4 8 C 2 β δ ´2β 4 C V c
V precisely correspond to the modifications of the constants C 3.27 and C 3.26 :"

}σ} 4 8 rϕ p2q s 2 1 4 C V c V introduced in the proof of Lemma 5 when replacing }D 2 ϕ} 8 by }D 2 ϕ δ } 8 ď C β δ ´β and rx∇ϕ, bys 1 ď C by rx∇ϕ δ , bys 1 ď C β δ ´β C. Similarly, a δ n :" rϕ p3q δ s β › › σ › › p3`βq 8 E " |U 1 | 3`β ‰ p1 `βqp2 `βqp3 `βq Γ p 3`β 2 q n ? Γ n ď C β δ ´β › › σ › › p3`βq 8 E " |U 1 | 3`β ‰ p1 `βqp2 `βqp3 `βq Γ p 3`β 2 q n ? Γ n ,
is obtained from the definition of a n in Lemma 3 replacing rϕ p3q s β by rϕ p3q δ s β . From the above equation and Lemma 2 we get:

P " |ν n pf q ´νpf q| ě a ? Γ n  " P " ˇˇν n pAϕ δ q `Rn,δ pf q ˇˇě a ? Γ n  ď 2pI 1 V q 1 p exp ´cV p `ppa δ n q 2 2 ēxp ´´a 2 2q}σ} 2 8 }∇ϕ} 2 8 ´1 ´?Γ n 4C η rf s 1 δ a ´1 q}σ} 2 8 }∇ϕ} 2 8 ! p Γ n ´6C 2 V,ϕ c V `2" C δ 3.26 `Cδ 3.27 spΓ p2q n q 2 ¯`1 p )¯¯.
The Young inequality yields that for all ε n ą 0:

P " |ν n pf q ´νpf q| ě a ? Γ n  ď 2pI 1 V q 1 p exp ´cV p `ppa δ n q 2 2 `ε´1 n Γ n δ 2 ēxp ´´a 2 2q}σ} 2 8 }∇ϕ} 2 8 ´1 ´1 q}σ} 2 8 }∇ϕ} 2 8 ! 2ε n C 2 η rf s 2 1 `p Γ n ´6C 2 V,ϕ c V `2" C δ 3.26 `Cδ 3.27 spΓ p2q n q 2 ¯`1 p ) loooooooooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooooooooon ":d δ n ¯¯.
(5.30)

We now want to let p :" ppnq Ñ n `8, ε n Ñ n 0 so that the associated contributions in the above equation can be viewed as remainders. From the previous definitions of C δ 3.27 , C δ 3.26 , we see that, to achieve this goal, two constraints need to be fulfilled: namely, we must choose δ, p such that

ε ´1 n Γ n δ 2 Ñ n 0 and ppa δ n q 2 Ñ n 0.
Now, if θ P p1{2, 1s there exists β P p0, 1q such that Γ p 3`β 2 q ď C. In that case:

a δ n ď C ? Γn δ ´β " Γ ´p 1 2 p1´βq´βεq n Ñ n 0 for δ " Γ ´p 1 2 `εq n and ε ă 1´β 2β . Taking p :" ppnq " Γ p 1 2 p1´βq´βεq n yields ppa δ n q 2 Ñ n 0. On the other hand, ε n " Γ ´ε n also yields ε ´1 n Γ n δ 2 " Γ ´ε n Ñ n 0.
For θ P p1{3, 1{2q, Γ p 3`β 2 q n diverges for all β P p0, 1q, we then have

Γ p 3`β 2 q n ? Γn ď Cn 1 2 ´θp1`β 2 q .
Hence, there exists β P p0, 1q such that

Γ p 3`β 2 q n ? Γn ď Cn 1 2 ´θp1`β 2 q Ñ n 0. However, taking δ " Γ ´p 1 2 `εq n
, which seems to be an almost "necessary" choice to satisfy the first constraint

ε ´1 n Γ n δ 2 Ñ n 0, yields: a δ n -δ ´β Γ p 3`β 2 q n ? Γ n -n p1`βqp 1 2 ´θq`εβp1´θq Ñ n `8,
so that the second constraint cannot be fulfilled. This means that the regularization induces a constraint on the time steps which must not be too large. In other words, under the sole Lipschitz assumption on the source f , the fastest convergence regime is out of reach.

Summing up the previous computations, we complete the proof of Theorem 7.

Applications

6.1 Non-Asymptotic Deviation Bounds in the Almost Sure CLT

Let pU n q ně1 be an i.i.d sequence of centered d-dimensional random variables with unit covariance matrix. We define the sequence of normalized partial sums by Z 0 " 0 and

Z n :" ř n k"1 U k ? n , n ě 1.
The almost sure Central Limit Theorem (denoted from now on a.s. CLT) describes how the weighted sum of the renormalized sums Z n which appear in the usual asymptotic CLT, behaves viewed as a random measure. Precisely, it states that setting for k ě 1, γ k " 1{k:

ν Z n :" 1 Γ n n ÿ k"1 γ k δ Z k w, a.s. ÝÑ n G, Gpdxq :" exp ´´|x| 2 2 ¯dx p2πq d{2 . (6.1)
The above convergence had been established in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF], as a by-product of their results concerning the approximation of invariant distributions, under the minimal moment condition U i P L 2 pPq, thus weakening the initial assumptions by Brosamler and Schatte (see [START_REF] Brosamler | An almost everywhere central limit theorem[END_REF] and [START_REF] Schatte | On strong versions of the Central Limit Theorem[END_REF]). The underlying idea is to use a reformulation of the dynamics of pZ n q ně0 in terms of a discretization scheme appearing as a perturbation of (1.2). One indeed easily checks that, for n ě 0:

Z n`1 " Z n ´γn`1 2 Z n `?γ n`1 U n`1 `rn Z n , r n :" c 1 ´1 n `1 ´1 `1 2pn `1q " O ´1 n 2 ¯.
(6.2) Thus, the sequence pZ n q ně0 appears as a perturbed Euler scheme with decreasing step γ n " 1 n of the Ornstein-Uhlenbeck process dX t " ´1 2 X t dt `dW t whose invariant distribution is G. Then the regular Euler scheme

X n`1 " X n ´γn`1 2 X n `?γ n`1 U n`1 , (6.3)
satisfies (1.4) with ν " G. The a.s. weak convergence (6.1) established in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] follows as a consequence of the (fast enough) convergence of Z n towards X n as n goes to infinity. Moreover, this rate is fast enough to guarantee that the sequence ν Z n satisfies the conclusion of Theorem 1 point (a) (when γ n " 1 n , Γ p2q n ?

Γn Ñ n 0), i.e. its convergence rate is ruled by a CLT at rate a logpnq. In fact this holds under a lower moment assumption U 1 P L 3 pPq.

Let us mention that the convergence rates related to the a.s. CLT had already been investigated by several authors. Let us quote among relevant works, Csorgo and Horváth [START_REF] Csorgo | Invariance principles for logarithmic averages[END_REF], for real valued i.i.d. random variables, Chaâbane and Maâouia [START_REF] Chaabane | Théorèmes limites avec poids pour les martingales vectorielles[END_REF], who investigate the convergence rate of the strong quadratic law of large numbers for some extensions to vector-valued martingales, and Heck [START_REF] Heck | The principle of large deviations for the almost everywhere central limit theorem[END_REF], for large deviation results. As an application of our previous results, we will derive some new non-asymptotic Gaussian deviation bounds for the a.s. CLT, when the involved random variables pU n q ně1 satisfy (GC). We insist here that the sub-Gaussianity of the innovations is crucial to get a nonasymptotic Gaussian deviation bound. The result readily extends to the wider class of innovations satisfying the general sub-Gaussian exponential deviation inequality (1.5). Also, we slightly weaken the regularity assumptions needed on the function f in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] for the associated a.s. CLT to hold.

Non-Asymptotic Deviation Bounds.

Theorem 9 Assume the innovation sequence pU n q ně1 satisfies (GC) and let f be a Lipschitz continuous function such that Gpf q " ş R d f pxqGpdxq " 0. Then, there exist two explicit monotonic sequences c n ď 1 ď C n , n ě 1, with lim n C n " lim n c n " 1 such that for all a ą 0 and n ě 1:

P " a logpnq `1|ν Z n pf q| ě a ı ď 2C n exp ˆ´c n a 2 2}∇ϕ} 2 8 ˙, (6.4)
where ϕ denotes the solution of the Poisson equation:

@x P R d , 1 2 ∆ϕpxq ´1 2 x ¨∇ϕpxq " f pxq, (6.5)
which, under the current assumptions, is unique and belongs to W 2 p,loc pR d , Rq, for any p ą 1, with }∇ϕ} 8 ď 2rf s 1 .

Proof. For pZ n q ně0 as in (6.2), and pX n q ně0 as in (6.3) we introduce:

∆ n :" Z n ´Xn .
With the definition of ν Z n in (6.1), write ν Z n pf q " 1 Γn ř n k"1 γ k f pZ k´1 q. We also have similarly ν X n pf q :" 1 Γn ř n k"1 γ k f pX k´1 q. For all λ ą 0, we derive similarly to (3.21) (see as well (5.29)) and with the notations of (5.24):

P " a Γ n |ν Z n pf q| ě a ‰ " P " a Γ n ˇˇ1 Γ n n ÿ k"1 γ k `f pZ k´1 q ´f pX k´1 q ˘`ν X n pf q ˇˇě a ı ď P " a Γ n ˇˇ1 Γ n n ÿ k"1 γ k `f pZ k´1 q ´f pX k´1 q ˘`ν X n pAϕ δ q `Rn,δ pf q ˇˇě a ı ď 2 exp ´´λa ? Γ n ´1 ´2? Γ n C η rf s 1 δ a ¯¯´E exp ´prf s 1 λν ∆ n p| ¨|q ¯¯1 p ˆˆE exp ´´q qλ Γ n M n ¯˙1 q q ˆE exp ´2pqλ Γ n p|L n | `| Ḡn |q ¯˙1 2pq ˆE exp ´4pqλ Γ n |D 2,b,n | ¯˙1 4pq ˆˆE exp ´4pqλ Γ n |D 2,Σ,n | ¯˙1 4pq (6.6)
for q, q P p1, `8), p " q q´1 , p " q q´1 . Also, ϕ δ corresponds to the solution of the Poisson equation (6.5) obtained replacing f by its mollified version f δ . Now, we need the following lemma to control ν ∆ n p| ¨|q :"

1 Γn ř n k"1 γ k |∆ k´1 |.
Lemma 7 There is a non-negative constant C 6.7 such that for all λ ą 0:

E exp ´λν ∆ n p| ¨|q ¯" E exp ´λ Γ n n ÿ k"1 γ k |∆ k´1 | ¯ď exp ˜C6.7 λE r|U 1 |sΓ p 3 2 q n Γ n `C2 6.7 λ 2 Γ p3q n 2Γ 2 n ¸. (6.7)
For clarity, we postpone the proof to the end of the current section. On the other hand, from Section 5.3 we have that ϕ δ P C 3,β pR d , Rq for all β P p0, 1q. We derive from (6.6), (6.7) similarly to the proof of Theorem 7 by setting λn :" a

? Γn q q}∇ϕ} 2 8 : P " a Γ n |ν Z n pf q| ě a ‰ ď 2 exp ´´a 2 2q q}∇ϕ} 2 8 ´1 ´4? Γ n C η rf s 1 δ a ¯¯exp ´C6.7 λn rf s 1 E r|U 1 |sΓ p 3 2 q n Γ n ēxp ´C2 6.7 prf s 2 1 λ2 n Γ p3q n 2Γ 2 n ¯pI 1 V q 1 pq exp ´1 pq `cV `Cδ 3.27 2 ˘`ppa δ n q 2 2q ēxp ´λ 2 n ´pq ´3C 2 V,ϕ c V Γ 2 n `"C δ 3.26 `3 2 C δ 3.27 ‰ pΓ p2q n q 2 Γ 2 n ¯`1 2pq ¯ď 2pI 1 V q 1 pq exp ´1 pq `cV `Cδ 3.27 2 ˘`ppa δ n q 2 2q `ε´1 n Γ n δ 2 ēxp ¨´a 2 2q q}∇ϕ} 2 8 ´1 ´dδ n ´p q q}∇ϕ} 2 8 rf s 2 1 C 2 6.7 ´Γp3q n `E r|U 1 |s 2 pΓ p 3 2 q n q 2
Γn ¯' , for ε n ą 0 and d δ n as in (5.30). Choose again pp n q ně1 and δ as in Section 5.3 so that q n Ñ n 1, d δ n Ñ n 0 with the indicated monotonicity for n large enough. We can now take p :" pn Ñ n `8 such that p Γn Ñ n 0. The above inequality then gives the result up to a direct modification of the sequences pC n q ně1 , pc n q ně1 . Proof of Lemma 7

The definition of ∆ n implies:

∆ n`1 " ∆ n ´1 ´γn`1 2 ¯`r n Z n ,
where we recall from (6.2) that r n :"

b 1 ´1 n`1 ´1 `1 2pn`1q " Op 1 n 2 q.
In particular, there exists C1 ą 0 such that for all n ě 1,

|r n | ď C1 n 2 . (6.8)
Setting now ρ 0 " 1 and for n ě 1:

ρ n :" " n ź k"1 p1 ´γk 2 q ı ´1 " n ź k"1 2k 2k
´1 , a direct induction on ∆ n yields:

∆ n " 1 ρ n n ÿ k"1 r k ρ k Z k " 1 ρ n n ÿ k"1 r k ρ k ´k ÿ l"1 U l ? k ¯" 1 ρ n n ÿ l"1 ´n ÿ k"l r k ρ k ? k ¯Ul . (6.9)
Also, from the Wallis formula ρ n " n ? πn, which implies that there exists C2 ě 1 such that for all n ě 1: C´1

2 ? n ď ρ n ď C2 ? n. (6.10)
We now get from (6.9) and the Fubini theorem:

Γ n ν ∆ n p| ¨|q " n ÿ k"1 γ k |∆ k´1 | ď n ÿ k"1 γ k ρ k´1 k´1 ÿ l"1 ´k´1 ÿ m"l |r m |ρ m ? m ¯|U l | " n´1 ÿ l"1 " n ÿ k"l`1 γ k ρ k´1 `k´1 ÿ m"l |r m |ρ m ? m ˘ı|U l |. (6.11)
Combining (6.8) and (6.10), we get that there exist constants C3 , C4 ą 0 such that for all k P rrl `1, nss.

γ k ρ k´1 k´1 ÿ m"l |r m |ρ m ? m ď C3 k 3{2 l , n ÿ k"l`1 γ k ρ k´1 k´1 ÿ m"l |r m |ρ m ? m ď C4 l 3{2 . (6.12)
Plugging this inequality in (6.11), we derive:

ν ∆ n p| ¨|q ď 1 Γ n n´1 ÿ l"1 " n ÿ k"l`1 γ k ρ k´1 k´1 ÿ m"l |r m |ρ m ? m ı |U l | ď C4 Γ n n´1 ÿ l"1 |U l | l 3{2 . (6.13)
For any λ ą 0, Equation (6.13) and the Gaussian concentration property (GC) of the innovation entail:

E exp ´λν ∆ n p| ¨|q ¯ď n´1 ź k"1 E exp ´C4 λ Γ n k 3 2 |U k | ¯ď n´1 ź k"1 exp ´C 4 λ Γ n k 3 2 E r|U 1 |s `1 2 `C 4 λ Γ n k 3 2 ˘2" exp ´C 4 λE r|U 1 |sΓ p 3 2 q n Γ n `C 2 4 λ 2 Γ p3q n 2Γ 2 n ¯.
This completes the proof.

6.2

Numerical Results

We present in this section numerical results associated with the computation of the empirical measure ν n illustrating our previous theorems.

Sub-Gaussian tails

We first consider d " r " 1. Also, for simplicity, the innovations pU i q iě1 and X 0 are Bernoulli variables with PpU 1 " ´1q " PpU 1 " ´1q " 1 2 . We illustrate here Theorem 2 taking bpxq " ´x 2 , and σpxq " cospxq in (1.1). This is a (weakly) hypoelliptic example. Indeed, setting for x P R, X 1 pxq " cospxqB x and X 0 pxq " ´x 2 B x , we have spantX 1 , rX 1 , X 0 su " R. We choose as well to compute ν n pAϕq for ϕpxq " x `ε cospxq for ε " 0.01, and ϕpxq " cospxq. The function ϕ is here given. The assumptions of Theorem 2 follow from Theorem 18 in Rotschild and Stein [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF] (up to the introduction of a suitable partition of unity). From Theorem 2, for steps of the form pγ k q kě1 " pk ´θq kě1 , θ P r1{3, 1s (corresponding to β " 1 in Theorem 2), the function

a P R `Þ Ñ g n,θ paq :" # log `Pr| ? Γ n ν n pAϕq| ě as ˘, θ P p1{3, 1s, log `Pr| ? Γ n ν n pAϕq `pB n,1 ´E1 n q| ě as ˘, θ " 1{3,
is such that for a ą a n :" a n pθq where for θ P p1{3, 1s, a n pθq " 0 and for θ " 1{3, a n pθq "

rϕ p3q s β }σ} p3`βq 8 E " |U 1 | 3`β ‰ p1`βqp2`βqp3`βq Γ p 3`β 2 q n ? Γn : g n,θ paq ď ´cn pa ´an q 2 2}σ} 2 8 }∇ϕ} 2 8 `logp2C n q.
We plot in Figure 3.1 the curves of g n,θ for θ varying as θ j " 1 3 `p1 ´1 3 q j 5 , for j P rr1, 5ss, ϕpxq " x `ε cospxq and in Figure 3.2 the curve of g n,θ for θ " θ 0 " 1 3 and ϕpxq " cospxq. The simulations have been performed for n " 5 ˆ10 4 in Figure 3.1, n " 5 ˆ10 6 in Figure 3.2, and the probability estimated by Monte Carlo simulation for M C " 10 4 realizations of the random variable | ? Γ n ν n pAϕq| in the unbiased case and in the biased case of the random variable | ?

Γ n ν n pAϕq `pB n,1 ´E1 n q M |, where pB n,1 ´E1 n q M is obtained from B n,1
´E1 n replacing the integral over [0,1], that needs to be evaluated at every time step, by a quantization of the uniform law on r0, 1s with M " 10 points. We refer to [START_REF] Graf | Foundations of quantization for random vectors[END_REF] or [START_REF] Pagès | A space vector quantization method for numerical integration[END_REF] for details on quantization. We point out that this is one drawback that appears to obtain the fastest convergence rate, the bias needs to be estimated and therefore the function ϕ in some sense known (since the approximation of the bias requires to compute its derivatives). The corresponding 95% confidence intervals have size at most of order 0.016. To compare with, we also introduce the functions S n,θ paq :" ´pa´anpθqq 2

2}σ} 2 8 }∇ϕ} 2 8 , S n,θ,c paq :" ´pa´anpθqq 2 2νn c pσ 2 q}∇ϕ} 2 8 , S n,θ,A paq :" ´pa´anpθqq 2 2νn c p|σ∇ϕ| 2 q
and the optimal concentration P pλ min qpn, θ, a, ρq, obtained in Remark 11, optimizing numerically in ρ. The quantities ν nc pσ 2 q, ν nc p|σ∇ϕ| 2 q in the previous expressions actually correspond to the numerical estimation, for n c " 10 4 and pγ c k q kě1 " pk ´θc q kě1 with θ c " 1 3 `10 ´3, of νpσ 2 q, νp|σ∇ϕ| 2 q appearing respectively in the sharper concentration bound of Theorem 8 when σ 2 ´νpσ 2 q is a coboundary and in the asymptotic Theorem 1. In the unbiased case of Figure 3.1, we plot the maximum in j of the pS n,θ j q jPrr1,5ss , pS n,θ j ,c q jPrr1,5ss , pS n,θ j ,A q jPrr1,5ss , `P pλ min qpn, θ j , a, ρq ˘jPrr1,5ss corresponding to j " 1. The associated curves are denoted by S n , S n,c , S n,A and P pλ min qpnq.

The Figures 3.1 and 3.2 correspond to the unbiased and biased cases respectively. In the unbiased case, we observe that the curves almost overlay, the optimal deviation rate P pλ min q is very close to the empirical data. It is also below the numerical estimation of the asymptotic threshold given by S n,θ,A which is, for our considered example, almost indistinguishable from the coboundary S n,θ,c (indeed, since ε " 0.01, }∇ϕ} 2 8 ď 1 `ε2 and νpσ 2 q}∇ϕ} 2 8 » νp|σ∇ϕ| 2 q) and far below from the bounds of S n,θ . In the biased case, P pλ min q stays very close to the theoretical asymptotic bound given by S n,θ,A up to a certain deviation level a, namely for a P r0, 0.5s. It then remains the best bound provided by our results. In this example, the improvement associated with S n,θ,c is also notable. It is precisely because the source term has a more oscillating gradient that we have also considered a larger running time, corresponding to n " 10 6 , for the empirical curves. For this choice, we see relatively good agreement w.r.t. to the asymptotic deviation bounds of S n,θ 0 ,A .

The figures below thus illustrate that the explicit optimal rate of Remark 11 seems rather appropriate to capture the deviations of the empirical random measures.

-10

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 0 0.5 1 1.5 2 a P(λ min (n)) g n,θ 1 g n,θ 2 g n,θ 3 g n,θ 4 g n,θ 5 S n S n,A S n,c Figure 3.1 -Unbiased Case. Plot of a Þ Ñ g n,
θ paq, for pθ k q kPrr1,5ss , with ϕpxq " σpxq " x `ε cospxq, ε " 0.01.

We eventually plot below the deviation curves with source ϕpxq " cospxq adding a last curve obtained replacing in the formula for P pλ min q of Remark 11 the }∇ϕ} 2 8 νpσ 2 q by νp|σ∇ϕ| 2 q. For practical purposes, this last quantity is again estimated numerically with the same previous parameters. Even if the analysis of Theorem 8 cannot be extended to justify such a choice, the empirical evidence is rather striking.

-12

-10 -8 -6 -4 -2 0 0 0.5 1 1.5 2 a P(λ min (n, θ 0 )) g n,θ 0 S n,θ 0 S n,θ 0 ,C S n,θ 0 ,A Figure 3.2 -Biased Case. Plot of a Þ Ñ g n,θ paq, for θ 0 " 1 3 , with ϕpxq " σpxq " cospxq. -14 -12 -10 -8 -6 -4 -2 0 0 0.5 1 1.5 2 a P(λ min (n)) P(λ min (n)) with carre du champ coboundary g n,θ 1 g n,θ 2 g n,θ 3 g n,θ 4 g n,θ 5 S n S n,A S n,c Figure 3.3 -Plot of a Þ Ñ g n,θ paq, for pθ k q kPrr1,
5ss , with ϕpxq " σpxq " cospxq.

Slutsky like result

In this paragraph, we illustrate our results from Theorem 6, which can be viewed as an extension of the usual Slustky's Lemma to our current framework, for a multidimensional process, precisely for r " d " 2 in the case β P p0, 1q. In order to converge as fast as possible without bias, we take θ " 1 2`β `1 1000 . We also choose a model which satisfies the assumptions of Theorem 4 under (C UE ) and Lemma 6. We consider:

f pxq " |x| 1`β 1 `|x| β , bpxq " ˆ´4x 1 `6x 2 ´5x 1 ´5x 2 ˙, σσ ˚pxq " ˜cospx 1 `x2 q 2 `1 sinpx 1 q sinpx 2 q 4 sinpx 1 q sinpx 2 q 4 1 ´sinpx 2 q 2 ¸.
Remark that the non-degeneracy condition (UE) is fulfilled by Σ " σσ ˚, as well as the condition set in Theorem 4 under (C UE ), Σ i,j pxq " Σ i,j px i , . . . , x d q, for all 1 ď i ď j ď d. Furthermore, from the Cholesky decomposition, we write:

σpxq " ¨bcospx 1 `x2 q 2 `1 0 sinpx 1 q sinpx 2 q 4 b cospx 1 `x2 q 2 `1 c ´sinpx 1 q 2 sinpx 2 q 2 16p cospx 1 `x2 q 2 `1q `1 ´sinpx 2 q 2 ‹ '.
Let us check that (D p α ) is satisfied. Firstly, remark that Db`Db 2 is a constant matrix whose eigenvalues are t´? 2`9 2 , ? 2´9 2 u. Direct computations yield that, for all x P R d , ξ P R d :

B Dbpxq `Dbpxq 2 ξ, ξ F `1 2 r ÿ j"1 |Dσ ¨j pxqξ| 2 ď ´3.085|ξ| 2 .
It can be checked similarly that the condition }Dσ} 2 8 ď 2α 2p1`βq´p is satisfied for α " 3.085 and β " .5 which we consider below. Also, the condition (R 1,β ) clearly holds. In other words, all assumptions of Theorem 6 are in force. We set for the following plot:

g σ n paq " log Pr a Γ n |ν n f q| ě as, S σ paq " ´a2 α 2 2rf s 2 1 ,
with α " 3.085, and rf s 1 " 1. Unlike in the previous simulations, we do not know here the value of νpf q. In fact, in paragraph 6.2 we had chosen to compute the deviation of Aϕ from 0 " νpAϕq. Here, we estimate from the ergodic theorem νpf q, taking β " .5, by ν n c pf q « 0.71308 for n c " 5 ¨10 5 . Running M C " 10 2 samples, we find that the size of the associated 95% confidence interval is 3.208 ¨10 ´4. Finally, the simulations are performed for n " 5 ˆ10 4 , and the probability is calculated by Monte Carlo algorithm for M C " 10 3 realizations. The maximum size of the associated 95% confidence interval is 4.75054 ¨10 ´5. The innovations are Gaussian random variables.

-7 -6 -5 -4 -3 -2 -1 0 0 0.5 1 1.5 2 a g σ n S σ Figure 3.4 -Plot of a Þ Ñ g n paq with f pxq " |x| β 1`|x| β , β " .5.
In Figure 3.4, we observe that the curve S σ stays above g σ n as proved in Theorem 6. However, remark that the graphs are quite spaced. This can be explained, among other things, by the difference between νp}σ} 2 q rf s 1 α and the asymptotic variance νp|σ ˚∇ϕ| 2 q. Furthermore we have represented S σ which is a kind of asymptotic version of P pλ min pnqq in the previous plots.

Chapter 4 A sharp non-asymptotic concentration

Abstract : For an ergodic Brownian diffusion with invariant distribution ν, we consider a sequence of empirical distributions pν n q ně1 associated with an approximation scheme with decreasing time step pγ n q ně1 along an adapted regular enough class of test functions f such that f ´νpf q is a coboundary of the infinitesimal generator A. Denote by σ the diffusion coefficient and ϕ the solution of the Poisson equation Aϕ " f ´νpf q. When the square norm |σ ˚∇ϕ| 2 lies in the same coboundary class as f , we establish sharp nonasymptotic concentration bounds for suitable normalizations of ν n pf q ´νpf q. Our bounds are optimal in the sense that they match the asymptotic limit obtained by Lamberton and Pagès in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF], for a certain large deviation regime. In particular, this allows us to derive sharp non-asymptotic confidence intervals. Eventually, we are able to handle, up to an additional constraint on the time steps, Lipschitz sources f in an appropriate non-degenerate setting.

Introduction

Statement of the problem

Consider the stochastic differential equation

dY t " bpY t qdt `σpY t qdW t , (1.1)
where pW t q tě0 stands for a Wiener process of dimension r P N on a given filtered probability space pΩ, G, pG t q tě0 , Pq, b : R d Ñ R d , and σ : R d Ñ R d b R r are Lipschitz continuous functions and satisfy a Lyapunov condition (see further Assumption L V ) which provides the existence of an invariant distribution ν. Throughout the chapter, uniqueness of the invariant distribution ν is assumed. The purpose of this work is to estimate the invariant distribution of the diffusion equation (1.1).

In order to make a clear parallel with the objects we will introduce for the approximation of ν, let us first recall some basic facts on pY t q tě0 and ν.

Introduce for a bounded continuous function f and t P R `the average occupation measure:

ν t pf q :" 1 t ż t 0 f pY s qds. (1.2)
Foremost, bear in mind the usual ergodic theorem which holds under appropriate Lyapunov conditions (see e.g. [START_REF] Khasminskii | Stochastic Stability of Differential Equations[END_REF]):

ν t pf q a.s. ÝÑ tÑ`8 νpf q :" ż f dν. (1.3)
Under suitable stability and regularity conditions, Bhattacharya [START_REF] Bhattacharya | On the functional central limit theorem and the law of the iterated logarithm for Markov processes[END_REF] then established a corresponding Central Limit Theorem (CLT). Namely, for all smooth enough function f , ? t `νt pf q ´νpf q ˘L ÝÑ tÑ`8

N ˆ0, ż R d |σ ˚∇ϕpxq| 2 νpdxq ˙, (1.4)
where ϕ is the solution of the Poisson equation Aϕ " f ´νpf q and A stands for the infinitesimal operator of the diffusion (1.1) (see (2.3) below for more details). In the following, we say that f is coboundary when there is a smooth solution ϕ to the Poisson equation Aϕ " f ´νpf q.

Identity (1.4) is a Central Limit Theorem (CLT) whose asymptotic variance is the integral of the well known carré du champ (called energy but we will say, from now on, by abuse of terminology, carré du champ), for more precision, see [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] and [START_REF] Ledoux | Concentration of measure and logarithmic Sobolev inequalities[END_REF]. The carré du champ is actually a bilinear operator defined for any smooth functions ϕ, ψ by Γpϕ, ψq :" Apϕ ¨ψq ´Aϕ ¨ψ ´ϕ ¨Aψ, and so:

ν pΓ pϕ, ϕqq " ż R d Γpϕ, ϕqνpdxq " ´2 ż R d Aϕ ¨ϕ νpdxq " ż R d |σ ˚∇ϕ| 2 νpdxq.
Indeed, observe that ν pA pϕ ¨ψqq " 0. This is a consequence of the fact that ν solves in the distributional sense the Fokker-Planck equation A ˚ν " 0. Also, this observation yields that, in order to bypass solving a Poisson equation, a common trick consists in dealing with smooth functions of the form Aϕ.

From a practical point of view, several questions appear: how to approach the process pY t q tě0 , the integral ν t , and the deviation from the asymptotic measure appearing in (1.4)? Here, the first question is addressed by considering a suitable discretization scheme with decreasing time steps, pγ k q kě1 . The integral ν t can then be approximated by the associated empirical measure, whose deviations will be controlled in our main results. In particular, we take advantage of the discrete analogue to (1.4) established by Lamberton and Pagès in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] for the current approximation scheme, to derive sharp non-asymptotic bounds for the empirical measure.

We propose an approximation algorithm based on an Euler like discretization with decreasing time step, first introduced by Lamberton and Pagès in [LP02] who derived related asymptotic limit theorem in the spirit of (1.4), and exploited as well in Chapter 3 where some corresponding non-asymptotic bounds are obtained.

For the decreasing step sequence pγ k q kě1 and n ě 0, the scheme deriving from (1.1) is defined by: "

X n`1 " X n `γn`1 bpX n q `?γ n`1 σpX n qU n`1 , X 0 P L 2 pΩ, F 0 , Pq, (1.5)
where pU n q ně1 is an i.i.d. sequence of random variables on R r , independent of X 0 , and whose moments match with the Gaussian ones up to order 3. In particular, more general innovations than the Brownian increments can be used.

Intuitively, the decreasing steps in (1.5) allow to be more and more precise when time grows.

The empirical (random) occupation measure of the scheme is defined for all A P BpR d q (where BpR d q denotes the Borel σ-field on R d ) by: ν n pAq :" ν n pω, Aq :"

ř n k"1 γ k δ X k´1 pωq pAq ř n k"1 γ k . (1.6)
We are interested in the long time approximation, so we need to consider steps pγ k q kě1 such that Γ n :"

ř n k"1 γ k Ñ n `8.
Under suitable Lyapunov like assumptions, Lamberton and Pagès in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] first proved the following ergodic result: for any ν ´a.s. continuous function f with polynomial growth, ν n pf q a.s.

ÝÑ n νpf q " ş R d f pxqνpdxq, which is the discrete analogue of (1.3). The main benefit of decreasing steps instead of constant ones is thus that the empirical measure directly converges towards the invariant one. Otherwise, taking γ k " h ą 0 in (1.5), the previous ergodic theorem must be changed into:

ν n pf q a.s. ÝÑ n ν h pf q " ş R d f pxqν h pdxq,
where ν h is the invariant distribution of the scheme. So, an extra study must be carried out, namely the difference ν ´νh should be estimated. For more details about this approach we refer to [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF], [START_REF] Talay | Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme[END_REF] and the work of Malrieu and Talay [START_REF] Malrieu | Concentration inequalities for Euler Schemes[END_REF]. This work first addressed the issue of deriving non-asymptotic controls for the deviations of empirical measure of type (1.6) when γ k " γ ą 0 (constant step). The backbone of their approach consisted in establishing a Log Sobolev inequality, which implies Gaussian concentration, for the Euler scheme. In whole generality, functional inequalities (such as the Log Sobolev one) are a powerful tools to get simple controls on the invariant distribution associated with the diffusion process (1.1), see e.g. Ledoux [Led99] or Bakry et al. [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]. Withal Log Sobolev, and Poincaré inequalities turn out to be quite rigid in the framework of discretization schemes like (1.5) with or without decreasing steps.

For the CLT associated with stationary Markov chains, we refer to Gordin's Theorem (see [START_REF] Gordin | On the central limit theorem for stationnary markov processes[END_REF]). Note as well that the variance of the limit Gaussian law is also the carré du champ for discrete Poisson equation associated with the generator of the chain.

Let us mention as well some related works. In [START_REF] Blower | Concentration inequalities on product spaces with applications to Markov processes[END_REF], Blower and Bolley establish Gaussian concentration properties for deviations of functional of the path in the case of metric space valued homogeneous Markov chains. Non-asymptotic deviation bounds for the Wasserstein distance between the marginal distributions and the stationary law, in the homogeneous case can be found in [START_REF] Boissard | Simple bounds for the convergence of empirical and occupation measures in 1-Wasserstein distance[END_REF] (see also Boissard and Le Gouic in [START_REF] Boissard | On the mean speed of convergence of empirical and occupation measures in Wasserstein distance[END_REF] for controls on the expectations of this Wasserstein distance). The key point of these works is to demonstrate contraction properties of the transition kernel of the homogeneous Markov chain for a Wasserstein metric, which requires some continuity in this metric for the transition law involved, see e.g. [START_REF] Blower | Concentration inequalities on product spaces with applications to Markov processes[END_REF].

In the current work, we aim to establish an optimal non-asymptotic concentration inequality for ν n pf q ´νpf q. When |σ ˚∇ϕ| 2 ´νp|σ ˚∇ϕ| 2 q is a coboundary, we manage to improve the estimates in Chapter 3. Insofar, we better the variance in the upper-bound obtained in Theorem 3 therein: when the time step is such that γ n -n ´θ, for θ ą 1 3 , for all n P N, for a smooth enough function ϕ s.t. Aϕ " f ´νpf q, under suitable assumptions (further called Assumptions (A)), and if }σ} 2 is coboundary then there exist explicit non-negative sequences pr c n q ně1 and p r C n q ně1 , respectively increasing and decreasing for n large enough, with lim n r C n " lim n r c n " 1 s.t. for all n ě 1 and 0 ă a " op ? Γ n q:

P " a Γ n |ν n pf q ´νpf q| ě a ‰ " P " | a Γ n ν n pAϕq| ě a ‰ ď 2 r C n exp `´r c n a 2 2νp}σ} 2 q}∇ϕ} 2 8 ˘.
In fact, we get below the optimal variance bound, namely the carré du champ νp|σ ˚∇ϕ| 2 q, instead of the expression νp}σ} 2 q}∇ϕ} 2 8 as in the previous inequality. Up to the same previously indicated deviation threshold, a " op ? Γ n q, we derive the optimal Gaussian concentration. Consequently, we are able to derive directly some sharp non-asymptotic confidence intervals.

To establish our non-asymptotic results, we use martingale increment techniques which turn out to be very robust in a rather large range of application fields. Let us for instance mention the work of Frikha and Menozzi [START_REF] Frikha | Concentration bounds for stochastic approximations[END_REF] which establishes non-asymptotic bounds for the regular Monte Carlo error associated with the Euler discretization of a diffusion until a finite time interval r0, T s and for a class of stochastic algorithms of Robbins-Monro type. Still with martingale approach, Dedecker and Gouëzel [START_REF] Dedecker | Subgaussian concentration inequalities for geometrically ergodic markov chains[END_REF] have obtained nonasymptotic deviation bounds for separately bounded functionals of geometrically ergodic Markov chains on a general state space. Eventually, we can refer again to the work in Chapter 3 in the current setting.

The chapter is organized as follows. In Section 2, we state our notations and assumptions as well as some known and useful results related to our approximation scheme. Section 3 is devoted to our main concentration results (for a certain deviation regime that we will call Gaussian deviations), we also state therein several technical lemmas whose proofs are postponed to Section 4. Importantly, we also provide a user's guide to the proof which emphasizes the key steps in our approach. Section 4 is the technical core of the chapter. We then discuss in Section 5 some regularity issues for the considered test functions. Namely, we recall some assumptions introduced in Chapter 3 which yield appropriate regularity concerning the solution of Poisson equation. We also extend there our main results to test functions f that are Lipschitz continuous, up to some constraints on the step sequence.

We proceed in Section 6 to the explicit optimization of the constants appearing in the concentration bound deriving from our approach. Intrinsically, this procedure conducts to two deviation regimes, the Gaussian one up to a " op ? Γ n q and the super Gaussian one for a " ? Γ n which deteriorates the concentration rate. Even though awkward at first sight (see e.g. Remark 4 below), this refinement turns out to be useful for some numerical purposes, as it emphasized in Section 7. We conclude there with some numerical results associated with a degenerate diffusion.Some additional technical details needed in Section 6 are gathered in Appendix 8.

Assumptions and Existing Results

General notations

For all step sequence pγ n q ně1 , we denote:

@ P R, Γ p q n :" n ÿ k"1 γ k , Γ n :" n ÿ k"1 γ k " Γ p1q n .
Practically, the time step sequence is assumed to have the form: γ n -1 n θ with θ P p0, 1s, where for two sequences pu n q nPN , pv n q nPN the notation u n -v n means that Dn 0 P N, DC ě 1 s.t. @n ě n 0 , C ´1v n ď u n ď Cv n .

We will denote by C a non negative constant, and by pe n q ně1 , pR n q ně1 deterministic generic sequences s.t. e n Ñ n 0 and R n Ñ n 1, that may change from line to line. The constant C depends, uniformly in time, as well as the sequences pe n q ně1 , pR n q ně1 , on known parameters appearing in the assumptions introduced in Section 2.2 (called (A) throughout the document). Other possible dependencies will be explicitly specified.

In the following, for any smooth enough function f , for k P N we will denote D k f the tensor of the k th derivatives of f . Namely

D k f " pB i 1 . . . B i k f q 1ďi 1 ,...,i k ďd . However, for a multi-index α P N d 0 :" pN Y t0uq d , we set D α f " B α 1 x 1 . . . B α d x d f : R d Ñ R.
For a β-Hölder continuous function f : R d Ñ R, we introduce the notation rf s β :" sup

x‰x 1 |f pxq ´f px 1 q| |x ´x1 | β ă `8,
for its Hölder modulus of continuity. Here, |x ´x1 | stands for the Euclidean norm of x ´x1 P R d . We denote, for pp, mq P N 2 , by C p pR d , R m q the space of p-times continuously differentiable functions from R d to R m . Besides, for f P C p pR d , R m q, p P N, we define for β P p0, 1s the Hölder modulus:

rf ppq s β :" sup x‰x 1 ,|α|"p |D α f pxq ´Dα f px 1 q| |x ´x1 | β ď `8,
where α (viewed as an element of N d ) is a multi-index of length p, i.e. |α| :" ř d i"1 α i " p. Hence, in the above definition, the | ¨| in the numerator is the usual absolute value. We will as well use the notation rrn, pss, pn, pq P pN 0 q 2 , n ď p, for the set of integers being between n and p.

From now on, we introduce for k P N 0 , β P p0, 1s and m P t1, d, d ˆru the Hölder spaces , where for M P R m b R m , TrpM q stands for the trace of M . Hence } ¨} is the Fröbenius norm * . With these notations, C k,β pR d , R m q stands for the subset of C k pR d , R m q whose elements have bounded derivatives up to order k and β-Hölder continuous k th derivatives. For instance, the space of Lipschitz continuous functions from R d to R m is denoted by

C k,β pR d , R m q :" tf P C k pR d , R m q: @α P N d , |α| P rr1, kss, sup xPR d |D α f pxq| ă `8, rf pkq s β ă `8u, C k,β b pR d , R m q :" tf P C k,β pR d , R m q: }f } 8 ă `8u. ( 2
C 0,1 pR d , R m q.
Eventually, for a given Borel function f :

R d Ñ E, where E can be R, R d , R d b R r , R d b R d , we set for k P N 0 : f k :" f pX k q.
For k P N 0 , we denote by F k :" σ `pX j q jPrr0,kss ˘the σ-algebra generated by the pX j q jPrr0,kss .

Hypotheses

(C1) The first term of the random sequence X 0 is supposed to be sub-Gaussian, i.e. there is a threshold λ 0 ą 0 such that:

@λ ă λ 0 , Erexppλ|X 0 | 2 qs ă `8.
(GC) The innovations pU n q ně1 form an i.i.d. sequence with law µ, we also assume that ErU 1 s " 0 and for all pi, j, kq P t1, ¨¨¨, ru 3 , ErU i 1 U j 1 s " δ ij , ErU i 1 U j 1 U k 1 s " 0. Moreover, pU n q ně1 and X 0 are independent. Eventually, U 1 satisfies the following standard Gaussian concentration property, i.e. for every 1´Lipschitz continuous function g : R r Ñ R and every λ ą 0:

E " exppλgpU 1 qq ‰ ď exp `λErgpU 1 qs `λ2 2 ˘.
In particular, Gaussian and symmetrized Bernoulli random variables (in short r.v.) satisfy this inequality. Pay attention that a wider class of sub-Gaussian distributions could be considered. Namely, random variables for which there exists ą 0 s.t. for all λ ą 0:

E " exppλgpU 1 qq ‰ ď exp `λErgpU 1 qs ` λ 2 4 ˘. (2.2)
It is well know that this assumption yields that for all r ě 0, Pr|U 1 | ě rs ď 2 expp´r 2 q.

(C2) There is a positive constant κ s.t., defining for all x P R d , Σpxq :" σσ ˚pxq:

sup xPR d
TrpΣpxqq " sup

xPR d
}σpxq} 2 ď κ. * . This notation allows to define similarly vector and matrix norms. In fact, R d vectors can be regarded as line vectors. Then we define similarly for both cases the uniform norm } ¨}8 (L V ) We consider the following Lyapunov like stability condition: There exists V : R d ÝÑ rv ˚, `8r with v ˚ą 0 s.t. i) V P C 2 pR d , Rq, }D 2 V } 8 ă 8, and lim |x|Ñ8 V pxq " `8. ii) There exists C V P p0, `8q s.t. for all x P R d :

|∇V pxq| 2 `|bpxq| 2 ď C V V pxq.
iii) Let A be the infinitesimal generator associated with the diffusion equation (1.1), defined for all ϕ P C 2 0 pR d , Rq and for all x P R d by:

Aϕpxq " bpxq ¨∇ϕpxq `1 2 Tr `ΣpxqD 2 ϕpxq ˘, (2.3)
where, for two vectors v 1 , v 2 P R d , the symbol v 1 ¨v2 stands for the canonical inner product of v 1 and v 2 . There exist α V ą 0,

β V P R `s.t. for all x P R d ,
AV pxq ď ´αV V pxq `βV .

(U) There is a unique invariant distribution ν to equation (1.1).

For β P p0, 1s, we introduce:

(T β )
We choose a test function ϕ for which i) ϕ smooth enough, i.e. ϕ P C 3,β pR d , Rq, We further assume that:

ii) the mapping x Þ Ñ xbpxq, ∇ϕpxqy is Lipschitz continuous. iii) there exists C V,ϕ ą 0 s.t. for all x P R d |ϕpxq| ď C V,ϕ p1 `aV pxqq.

(S) We assume that the sequence pγ k q kě1 is small enough, Namely, we suppose that for all k ě 1:

γ k ď min ´1 2 ? C V c, α V 2C V }D 2 V } 8 ¯.
The constraint in (S) means that the time steps have to be sufficiently small w.r.t. the diffusion coefficients and the Lyapunov function.

Remark 1 The above condition (L V ) actually implies that the drift coefficient b lies, out of a compact set, between two hyperplanes separated from 0. Also, the Lyapunov function is lower than the square norm. In other words, there exist constants K, c ą 0 such that for all |x| ě K,

|V pxq| ď c|x| 2 , |bpxq| ď a C V c|x|. (2.4)
Actually, for σ bounded, b Lipschitz continuous, and if there is c ě 1 s.t. for all x P R d , ´c´1 |x| 2 ă xbpxq, xy ă ´c|x| 2 then the assumption (L V ) is satisfied.

Observe that we have supposed (U) without imposing any non-degeneracy conditions. Existence of invariant distribution follows from (L V ) (see [START_REF] Ethier | Markov Processes. Characterization and Convergence[END_REF]). For uniqueness, additional conditions need to be considered ((hypo)ellipticity [START_REF] Khasminskii | Stochastic Stability of Differential Equations[END_REF], [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF], [Vil09] or confluence [START_REF] Pagès | Ergodic approximation of the distribution of a stationary diffusion: rate of convergence[END_REF]).

Remark 2 In (T β ), condition ii) is direct if we consider Lyapunov function V pxq -1 `|x| 2 . Indeed, ϕ is supposed, in condition (T β ) i), to be Lipschitz continuous and so under a linear map. Hypothesis ii) is natural when there is a function f P C 1,β pR d , Rq with νpf q " 0 s.t.

Aϕ " f.

(2.5)

From the definition of A in (2.3), we rewrite:

x∇ϕpxq, bpxqy " f pxq ´νpf q ´1 2 Tr ´ΣpxqD 2 x ϕpxq ¯. (2.6)
Since the source f is Lipschitz continuous, and σ, D 2 x ϕ are bounded and Lipschitz continuous, the left hande side of the equation (2.6) is also Lispchitz continuous.

We say that assumption (A) holds whenever (C1), (GC), (C2), (L V ), (U), (T β ) for some β P p0, 1s and (S) are fulfilled. Except when explicitly indicated, we assume throughout the chapter that assumption (A) is in force.

Assume the step sequence pγ k q kě1 is chosen s.t. γ k -k ´θ, θ P p0, 1s. In particular, this implies that, for any ε ě 0, Γ

pεq n -n 1´εθ if εθ ă 1, Γ pεq n -lnpnq if εθ " 1 and Γ pεq n -1 if εθ ą 1.

On some Related Existing Result

In [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF], Lamberton and Pagès, proved an asymptotic result with the decreasing step scheme (1.5). Precisely, they obtain the discrete counterpart of (1.4) established in [START_REF] Bhattacharya | On the functional central limit theorem and the law of the iterated logarithm for Markov processes[END_REF], emphasizing as well some discretization effects, leading to a bias in the limit law, when the time step becomes too coarse. This last case is however the one leading to the highest convergence rates in the CLT. We recall here their main results, Theorem 10 of the above reference, for the sake of completeness.

Theorem 1 (Asymptotic Limit Results in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF]) Assume (C2), (L V ), (U) hold. If ErU 1 s " 0, ErU b3 1 s " 0, we get the following limit results where ν n stands for the empirical measure defined in (1.6).

(a) Fast decreasing step. If θ P p 1 3 , 1s and Er|U 1 | 6 s ă `8, then, for all function ϕ P C 2,1 pR d , Rq X C 3 pR d , Rq, one has: V pxq ă `8 (sublinear diffusion) in case (b). We refer to Theorems 9 and 10 in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] for additional details.

a Γ n ν n pAϕq L ÝÑ nÑ8 N `0, ż R d |σ ˚∇ϕ| 2 dν ˘. (
Remark 3 First of all, observe that the normalization is the same as for (1.4). It is the square root of the considered running time, namely t for the diffusion and Γ n for the scheme. In other words, a CLT is still available for the discretization procedure. However, by choosing a critical time step, i.e. for the fast convergence θ " 1 3 , a bias is begot. It can be regarded as a discretization effect. Note that, for all θ ě 1 3 , any step leads to the same asymptotic variance, namely the carré du champ, ş R d |σ ˚∇ϕ| 2 dν, like in [START_REF] Bhattacharya | On the functional central limit theorem and the law of the iterated logarithm for Markov processes[END_REF]. However, for the slow decreasing step, θ ă 1 3 , the discretization effect is prominent and "hides" the CLT.

Let us also mention the work of Panloup [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic differential equation driven by a lévy process[END_REF], where under similar assumptions for stochastic equation driven by a Lévy process, the convergence of the decreasing time step algorithm towards the invariant distribution of the stochastic process is established (see also [START_REF] Panloup | Computation of the invariant measure of a levy driven SDE: Rate of convergence[END_REF] for the CLT associated with square integrable Lévy innovations).

In the current diffusive context, i.e. under (A), some non-asymptotic results were successfully established in Chapter 3. It was as well observed there that if we slacken the regularity of the test function ϕ, a new bias looms. For ϕ P C 3,β pR d , Rq with β P p0, 1q, if θ " 1 2`β then ? Γ n ν n pAϕq exhibits deviations similar to the ones of a biased normal law with a different bias than in Theorem 1, c.f. Theorems 2, 3, 5, 7 and 9 in Chapter 3. When β " 1, the two biases correspond. We willingly shirk any discussion about bias appearance, which is discussed in the formerly mentioned chapter. Our target is to refine Theorem 4 in Chapter 3 that we recall: Theorem 2 (Non-asymptotic concentration inequalities in Chapter 3) Bhouououou Assume (A) holds, if there is ϑ P C 3,β pR d , Rq satisfying (T β ) and s.t.

Aϑ " }σ} 2 ´νp}σ} 2 q. †. With our tensor notations, D 2 ϕpxqbpxq b2 P pR d q b2 , D 3 ϕpxqbpxqpσpxquq b2 P pR d q b3 , and D 4 ϕpxqpσpxquq b4 P pR d q b4 . For β P p0, 1s and θ P p 1 2`β , 1s, there exist two explicit monotonic sequences cn ď 1 ď Cn , n ě 1, with lim n Cn " lim n cn " 1 such that for all n ě 1 and a ą 0: ˘. In other words, the empirical measure is sub-Gaussian with asymptotic variance equals to νp}σ} 2 q}∇ϕ} 2 8 which is an upper-bound of the carré du champ, νp|σ ˚∇ϕ| 2 q (asymptotic variance in the limit theorem). Thus, this is not fully satisfactory. Throughout the chapter, we refer to super Gaussian deviations when a ? Γn Ñ n `8. In this case, a subtle phenomenon appears: the right hand side gives a super Gaussian regime. In particular, the term in the exponential of the r.h.s. of (2.7) is bounded from above and below by a 4{3 Γ 1{3 n . We anyhow emphasize that Theorem 2 in Chapter 3 provides a non-asymptotic Gaussian concentration for all deviation regimes:

P " | a Γ n ν n pAϕq| ě a ‰ ď 2 r C n exp `´r c n 2νp}σ} 2 q}∇ϕ} 2 8 Φ n paq ˘, Φ n paq :" "´a 2 `1 ´2 1 `b1 `4 c3 n Γn a 2 ˘¯_ ´a4 3 Γ 1 3 n cn `1 ´2 3 cn `Γn a 2 ˘1 3 ˘`¯ı , (2.7
P " | a Γ n ν n pAϕq| ě a ‰ ď 2 r C n exp `´r c n a 2 2}σ} 2 8 }∇ϕ} 2 8 ˘.
(2.8)

In particular, for super Gaussian deviations, the deviation (2.8) is asymptotically better. However, it had already been observed in Chapter 3 that the bound (2.7) turned out to be useful for numerical purposes as it led to bounds closer to the empirical realizations. We will derive in Theorem 6 of Section 6 a deviation bound similar to (2.7) with an improved variance bound. Namely, we succeed to replace νp}σ} 2 q}∇ϕ} 2 8 by the carré du champ. We then observe in the numerical results of Section 7 that the associated deviation bounds match rather precisely those of the empirical realizations.

We will also employ the terminology of intermediate Gaussian deviations when a ?

Γn Ñ n C ą 0. For this regime, we keep a Gaussian regime with deteriorated constants. Again, we first deal with Gaussian deviations, and we postpone the study of super Gaussian deviations to Section 6.

Remark 5 Actually, in the proof of Theorem 2, we can only use a map ϑ satisfying assumption (T β ) s.t. Aϑ ě |σ ˚∇ϕ| 2 ´νp|σ ˚∇ϕ| 2 q. However, this inequality is equivalent to the coboundary condition ν ´a.s.. In fact, we set the function f :" Aϑ ´|σ ˚∇ϕ| 2 νp|σ ˚∇ϕ| 2 q ě 0, and νpf q " νpAϑq " 0.

(2.9)

As f is continuous and non negative, f " 0 ν ´a.s. Hence Aϑ " |σ ˚∇ϕ| 2 ´νp|σ ˚∇ϕ| 2 q ν á.s.

Main results

Our main contribution consists in establishing a concentration inequality whose variance matches asymptotically the carré du champ, see (1.4) and Theorem 1 in what we called the regime of Gaussian deviations. In the numerical part of Chapter 3, we see that changing the bound νp}σ} 2 q}∇ϕ} 2 8 by the carré du champ, leads to bounds much closer to the realizations. Here, we state a simple and "sharp" inequality.

Theorem 3 (Sharp non-asymptotic deviation results) Assume (A) is in force. Suppose that there exists ϑ P C 3,β pR d , Rq satisfying (T β ) for some β P p0, 1s s.t.

Aϑ " |σ ˚∇ϕ| 2 ´νp|σ ˚∇ϕ| 2 q.

(3.1)

Then, for θ P p 1 2`β , 1s, there exist explicit non-negative sequences pc n q ně1 and pC n q ně1 , respectively increasing and decreasing for n large enough, with lim n C n " lim n c n " 1 s.t. for all n ě 1, a ą 0 satisfying a ? Γn Ñ 0 (Gaussian deviations), the following bound holds:

P " | a Γ n ν n pAϕq| ě a ‰ ď 2 C n exp `´c n a 2 2νp|σ ˚∇ϕ| 2 q ˘, with C n " exp `rD 3 ϕs β }σ} 3`β 8 Er|U 1 | 3`β s p1`βqp2`βqp3`βq Γ p 3`β 2 q n ? Γn `pn Γ p2q n ? Γn ˘for p n ě 1 such that p n Ñ n `8 and p n Γ p2q n ? Γn Ñ n 0.
Remark 6 We obtain the optimal Gaussian bound with the carré du champ as variance which corresponds to CLT. This is asymptotically the sharpest result that we can expect. This inequality is very important for confidence intervals, as in this context a is supposed to be "small", i.e. bounded. Under suitable regularity assumptions on f , which guarantee that the function ϕ solving Aϕ " f ´νpf q satisfies (T β ) for some β P p0, 1s, it readily follows from Theorem 3 that:

P " νpf q P " ν n pf q ´a ? Γ n , ν n pf q `a ? Γ n ‰ ı ě 1 ´2C n exp `´c n a 2 2νp|σ ˚∇ϕ| 2 q ˘.
The conditions on f that lead to the required smoothness on ϕ and ϑ are discussed in Section 5 (see in particular Theorem 4 and Corollary 1). Briefly, it suffices to consider that, additionally to (C2) and

(L V ), Σ is also uniformly elliptic, b P C 1,β pR d , R d q, σ P C 1,β pR d , R d b R d q and that the source f P C 1,β pR d , R d q.
This last assumption on f can be weakened to Lipschitz continuous (see Theorem 5) with some restriction on the steps.

User's guide to the proof

Recall that, for a fixed given n P N and ϕ P C 3,β pR d , Rq, we want to estimate the quantity Pr a Γ n |ν n pAϕq| ě as, @a ą 0, where ν n pAϕq " 1 Γn ř n k"1 γ k AϕpX k´1 q. We focus below on the term Pr ? Γ n ν n pAϕq ě as. Indeed, the contribution Pr ? Γ n ν n pAϕq ď ´as can be handled by symmetry.

The first step of the proof consists in writing `AϕpX k´1 q ˘kPrr1,nss with a splitting method to isolate the terms depending on the current innovation U k for AϕpX k´1 q. This is done in Lemma 1 below. Precisely, for all k P rr1, nss and ϕ P C 3,β pR d , Rq we prove that:

ϕpX k q ´ϕpX k´1 q " γ k AϕpX k´1 q `γk ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt `1 2 γ k Tr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σk´1 ¯`ψ k pX k´1 , U k q, (3.2)
where

ψ k pX k´1 , U k q " ? γ k σ k´1 U k ¨∇ϕpX k´1 `γk b k´1 q `γk ż 1 0 p1 ´tqTr ´D2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1 ´D2 ϕpX k´1 `γk b k´1 qΣ k´1 ¯dt. (3.3)
Observe that ψ k pX k´1 , U k q in the r.h.s. of (3.2) is the only term containing the current innovation U k . Thus, the mapping u Þ Ñ ψ k pX k´1 , uq is Lipschitz continuous, because ϕ is.

This property is crucial to proceed with a martingale increment technique. Indeed, introducing the compensated increment ∆ k pX k´1 , U k q :" ψ k pX k´1 , U k q´Erψ k pX k´1 , U k q|F k´1 s, assumption (GC) allows to derive:

@λ ą 0, Erexpp´λ∆ k pX k´1 , U k qq|F k´1 s ď exp `λ2 rψpX k´1 , ¨qs 2 1 2 ˘. (3.4)
The corner stone of the proof is then to apply recursively this control to the martingale

M m :" ř m k"1 ∆ k pX k´1 , U k q, m P rr1, nss.
To control the deviation, the first step is an exponential inequality which combined to (3.2) yields:

Pr a Γ n ν n pAϕq ě as ď exp `´aλ ? Γ n ˘E" exppλν n pAϕqq ı ď exp `´aλ ? Γ n ˘E" exp `´λqM n Γ n ˘ı1{q R n , (3.5)
for λ ą 0, q ą 1 and R n is a remainder (whose behaviour is investigated in Lemma 6). The main contribution in the above equation is the one involving M n which can be analyzed thanks to (3.4). Namely: q, with C n , c n ą 0 respectively increasing and decreasing to 1 with n (see Theorem 2 of Chapter 3 for details).

E " exp `´qλ Γ n M n ˘ı " E " exp `´qλ Γ n M n´1 ˘E" exp `´qλ Γ n ∆ n pX n´1 , U n q ˘ˇˇF n´1 ‰ ı ď E " exp `´qλ Γ n M n´1 ˘exp `q2 λ 2 rψpX n´1 ,
To obtain the expected variance corresponding to the carré du champ νp|σ ˚∇ϕ| 2 q, the key point is to control finely the Lipschitz modulus of ψ k pX k´1 , ¨q in (3.4), (5.4).

From (3.2), we get the following simple expression of the derivative ∇ u ψ k pX k´1 , uq| u"U k " ? γ k σ k´1 ∇ϕpX k q. Hence, there is a remainder term Rpγ k , X k´1 , U k q and a constant

C (3.8) " C (3.8) ppAqq ą 0 s.t. |∇ u ψ k pX k´1 , uq| 2 | u"U k " γ k |σ k´1 ∇ϕ k´1 | 2 `C(3.8) γ 2 k a V k´1 `Rpγ k , X k´1 , U k q, (3.8)
for more details see (3.29) below.

In order to exhibit for each evaluation of the conditional expectations in (3.4), the contribution νp|σ ˚∇ϕ| 2 q, we use the auxiliary Poisson problem:

Aϑ " |σ ˚∇ϕ| 2 ´νp|σ ˚∇ϕ| 2 q.

(3.9)

We then write for the main term to control in (3.5),

Erexpp´λ qM n Γ n qs ď T 1 ρ 1 T ρ´1 qρ 2 T ρ´1 pρ 3 , (3.10)
for ρ ą 1, p, q ą 1 s.t. 1 p `1 q " 1 where:

T 1 :" E exp `´ρ qλ Γ n M n ´ρ2 q 2 λ 2 2Γ 2 n n ÿ k"1 γ k AϑpX k´1 q ´C(3.8) γ 2 k a V k´1 ˘, T 2 :" E exp `λ2 q 2 ρ 2 q 2pρ ´1qΓ 2 n n ÿ k"1 γ k AϑpX k´1 q ˘, T 3 :" E exp `λ2 q 2 ρ 2 p 2pρ ´1qΓ 2 n n ÿ k"1 C (3.8) γ 2 k a V k´1 ˘. (3.11)
Exploiting (3.9), we can now rewrite

T 1 " E exp ˆ´ρ qλ Γ n M n ´ρ2 q 2 λ 2 2Γ 2 n n ÿ k"1 ´γk " |σ ˚∇ϕpX k´1 q| 2 ´νp|σ ˚∇ϕ| 2 q ‰ `C(3.8) γ 2 k a V k´1 ¯" exp ´ρ2 q 2 λ 2 2Γ n νp|σ ˚∇ϕ| 2 q Ē exp ´´ρ qλ Γ n M n ´ρ2 q 2 λ 2 2Γ 2 n n ÿ k"1 `γk |σ ˚∇ϕpX k´1 q| 2 `C(3.8) γ 2 k a V k´1 ˘¯. (3.12)
The first term in the above r.h.s. yields the expected variance when we optimize over λ for q and ρ going to 1, which is the case in the regime of so called Gaussian deviations in Theorem 3. It improves the previous bound (3.7). Introduce now for m P rr1, nss,

S m :" exp ´´ρqλ Γ n M m ´ρ2 q 2 λ 2 2Γ 2 n m ÿ k"1 `γk |σ ˚∇ϕpX k´1 q| 2 `Cγ 2 k a V k´1 ˘¯. (3.13) Bringing to mind that M m " ř m k"1 ∆ k pX k´1 , U k q,
where Er∆ k pX k´1 , U k q|F k´1 s " 0 and r∆ k pX k´1 , ¨qs 1 " rψ k pX k´1 , ¨qs 1 , we get from (3.8), that, up to the remainder term pRpγ k , X k´1 , U k qq kPrr1,nss , S m can be viewed as a super martingale (see Lemma 5 for details). We actually rigorously show that, in the Gaussian regime (i.e. for a ? Γn Ñ 0), for θ P p1{3, 1q

ErS n s 1 ρq ď R n ÝÑ nÑ`8 1.
For θ " 1, or for super Gaussian deviations (i.e. for a ? Γn Ñ `8, see Section 6) with θ P p1{3, 1q we get:

ErS n s 1 ρq ď R n exp ´`ρqλ 2 Γ n `ρ3 q 3 λ 4 pρ ´1qΓ 3 n ˘en ¯,
where e n ą 0 decreases to 0 with n and R n ą 0 is still going to 1 with n. The difficulty in the above control is that the optimized λ also depends on n and ρ (see (3.33) below).

The second term T 2 is estimated directly repeating the arguments of the proof of Theorem 2 in Chapter 3 which are recalled above (see equations (3.5) to (3.7)). We apply the previous martingale increment technique that previously led to (3.7). Denoting by M ϑ n the martingale associated with the `ψϑ k pX k´1 , U k q ˘kPrr1,nss deriving from the expansion of Aϑ similarly to (3.2), we obtain:

T 2 ď E " exp `´λ 2 q 2 ρ 2 qM ϑ n 2pρ ´1qΓ 2 n ˘ı1{q R ϑ n ď exp ´λ4 q 4 ρ 4 q 8pρ ´1q 2 Γ 3 n }σ} 2 8 }∇ϑ} 2 8 ¯Rϑ n , (3.14)
for q ą 1 and where the superscript ϑ means that we only need to replace ϕ by ϑ in the previous definitions. Like in (3.5), R ϑ n is here a remainder. The third component T 3 is first controlled by Jensen inequality (over the exponential function and the measure is

1 Γ p3q n ř n k"1 γ 2 k δ k ): T 3 ď 1 Γ p2q n n ÿ k"1 γ 2 k E exp `λ2 q 2 ρ 2 pΓ p2q n 2pρ ´1qΓ 2 n C a V k´1 ˘. (3.15)
For the control of this term (as well for remainders from Taylor expansion in (3.2) and in Lemma 1), we recall a useful result from Chapter 3 (see Proposition 1 therein). Under (A), there is a constant c V :" c V ppAqq ą 0 such that for all λ P r0, c V s, ξ P r0, 1s:

I ξ V :" sup ně0 ErexppλV ξ n qs ă `8. (3.16)
We also refer to Lemaire (see [START_REF] Lemaire | An adaptive scheme for the approximation of dissipative systems[END_REF]) for additional integrability results of the Lyapunov functions in a more general framework. The identity (3.15) is handled by Young inequality

T 3 ď exp `1 2c V p λ 2 q 2 ρ 2 pΓ p2q n 2pρ´1qΓ 2 n Cq 2 Γp2q n n ÿ k"1 γ 2 k E exp `cV V k´1 ˘" exp `λ4 Γ 3 n e n Γp2q n n ÿ k"1 γ 2 k E exp `cV V k´1 ˘, with p Ñ n `8 s.t. for fixed ρ, q ą 1, e n " 1 2c V p q 2 ρ 2 p 16c 2 V pρ´1q 2 Γ p2q n Γ 2 n
Cq 2 Ñ n 0, note that for all

θ P p 1 3 , 1s, Γ p2q n ?
Γn Ñ n 0. We obtain then by (3.16):

T ρ´1 pρ 3 ď exp `λ4 Γ 3 n e n ˘pI 1 V q ρ´1 pρ " R n exp `λ4 Γ 3 n e n ˘, (3.17
Pr a Γ n ν n pAϕq ě as ď exp ´´aλ ? Γ n `λ2 Γ n A n pρq `λ4 Γ 3 n B n pρq ¯Rn , (3.18)
with R n Ñ 1, A n pρq :" ρp qνp|σ ˚∇ϑ| 2 q 2 `en q, B n :"

ρ 3 ρ´1 q 3 q 4 p q}σ} 2 8 }∇ϑ} 2 8 2
`en q for e n ą 0 decreasing to 0 with n.

We perform an optimization over λ with the Cardan method. However, the optimal choice of λ depends on ρ. So an optimization can be done for ρ too. In Lemma 4 below, we choose ρ for the regime of Gaussian deviations (i.e. a ? Γn Ñ n 0) which yields:

P " | a Γ n ν n pAϕq| ě a ‰ ď 2 C n exp ´´c n a 2 2νp|σ ˚∇ϕ| 2 q ¯,
for c n , C n ą 0 respectively decreasing and increasing (for n big enough) to 1 with n.

The optimal choices of λ and ρ for the regime of super Gaussian deviations (i.e.

a ?

Γn Ñ n `8
) is eventually discussed in Section 6. This leads to

P " | a Γ n ν n pAϕq| ě a ‰ ď 2 C n exp ´´c n a 4{3 Γ 1{3 n 2}σ} 2{3 8 }∇ϑ} 2{3 8
¯.

Technical lemmas and Proof of the Main Results

We first give a decomposition lemma of ν n pAϕq which is the starting point of our analysis. Its proof can be found in Chapter 3 (see Lemma 1 therein).

Lemma 1 (Decomposition of the empirical measure) For all n ě 1, k P rr1, nss and ϕ P C 2 pR d , Rq, the identity (3.2) holds and we have:

Γ n ν n pAϕq " ϕpX n q ´ϕpX 0 q ´" n ÿ k"1 γ k ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt `1 2 n ÿ k"1 γ k Tr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σ2 k´1 ¯`n ÿ k"1 ψ k pX k´1 , U k q ı , (3.19)
where ψ k pX k´1 , U k q is defined in (3.3).

Remark 7

In spite of the square terms in U k appearing in the r.h.s. of (3.3), we have that, conditionally to For notational convenience we introduce, for a given n P N ˚the following quantities:

F k´1 , u Þ Ñ ψ k pX k´1 ,
R n :" ϕpX n q ´ϕpX 0 q ´n ÿ k"1

γ k ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt ´1 2 n ÿ k"1 γ k Tr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σ2 k´1 ¯, M n :" n ÿ k"1 ∆ k pX k´1 , U k q, r R n :" R n ´n ÿ k"1 E " ψ k pX k´1 , U k q| F k´1 ‰ , (3.21)
where for all k P rr1, nss:

∆ k pX k´1 , U k q :" ψ k pX k´1 , U j q ´E" ψ k pX k´1 , U k q| F k´1 ‰ . (3.22)
From these definitions, Lemma 1 can be rewritten:

ν n pAϕq " 1 Γ n p r R n ´Mn q, (3.23)
where M n is a martingale. The key idea of the proof is to control more precisely the Lipschitz modulus of ψ n pX n´1 , ¨q than it was done in Chapter 3. From the definition in (3.2), let us write for all k P rr1, nss:

ψ k pX k´1 , U k q " ϕ k ´ϕk´1 `Rk´1,k , where R k´1,k :" ´γk AϕpX k´1 q ´γk ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt ´1 2 γ k Tr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σ2 k´1 ¯.
Hence, by derivation

∇ u ψ k pX k´1 , uq| u"U k " ? γ k σ k´1 ∇ϕpX k q. (3.24)
We will establish that ∇ u ψ k pX k´1 , uq| u"U k is not "too far" from ? γ k σ k´1 ∇ϕpX k´1 q.

Proof of Theorem 3 for bounded innovations

We first give the complete proof in this particular case. We will specify the additional required controls for possibly unbounded innovations in the next subsection.

A key tool in the derivation of our main results is the following lemma whose proof is postponed to Section 4 for the sake of clarity.

Lemma 2 (Remainders from Taylor decomposition) Under (A), for all q ě 1 and λ ą 0, we have:

P " a Γ n ν n pAϕq ě a ‰ ď exp `´aλ ? Γ n ˘´E exp `´qλ Γ n M n ˘¯1 q expp λ 2 Γ n e n qR n . (3.25)
We will now sharply control the Lipschitz constant of ψ k pX k´1 , ¨q, or equivalently ∆ k pX k´1 , ¨q, which appears iteratively to handle the martingale term in (3.25).

In case of bounded innovations, we see by assumption (T β ) i) (smoothness of ϕ) that:

|∇ u ∆ k pX k´1 , uq| u"U k | " | ? γ k σ k´1 ∇ϕpX k q| ď | ? γ k σ k´1 ∇ϕpX k´1 q| `|? γ k σ k´1 r∇ϕpX k q ´∇ϕpX k´1 `γk b k´1 qs | `|? γ k σ k´1 r∇ϕpX k´1 `γk b k´1 q ´∇ϕpX k´1 qs | ď ? γ k |σ k´1 ∇ϕpX k´1 q| `γk }σ k´1 } 2 }D 2 ϕ} 8 }U k } 8 `|? γ k σ k´1 r∇ϕpX k´1 `γk b k´1 q ´∇ϕpX k´1 qs |. (3.26)
Remark that we have both controls

| r∇ϕpX k´1 `γk b k´1 q ´∇ϕpX k´1 qs | ď γ k }D 2 ϕ} 8 |b k´1 | (L V ), ii) ď γ k a C V }D 2 ϕ} 8 ? V k´1 , | r∇ϕpX k´1 `γk b k´1 q ´∇ϕpX k´1 qs | ď p2}∇ϕ} 8 q 1 2 | r∇ϕpX k´1 `γk b k´1 q ´∇ϕpX k´1 qs | 1 2 (L V ), ii) ď p2}∇ϕ} 8 q 1 2 γ 1 k {2C 1{4 V }D 2 ϕ} 1{2 8 V 1{4 k´1 , (3.27)
in order to keep integrable powers of the Lyapunov function. We therefore eventually get from (3.28) and inequalities in (3.27):

|∇ u ∆ k pX k´1 , uq| u"U k | 2 ď γ k |σ k´1 ∇ϕpX k´1 q| 2 `2? γ k |σ k´1 ∇ϕpX k´1 q| `γk }σ k´1 } 2 }D 2 ϕ} 8 }U k } 8 `γ3{2 k }σ} 8 a C V }D 2 ϕ} 8 ? V k´1 γk }σ k´1 } 2 }D 2 ϕ} 8 }U k } 8 `}σ} 8 p2}∇ϕ} 8 q 1 2 γ k C 1{4 V }D 2 ϕ} 1{2 8 V 1{4 k´1 ˘2 ď γ k |σ k´1 ∇ϕpX k´1 q| 2 `C1,(3.28) γ 3{2 k }U k } 8 `C2,(3.28) γ 2 k }U k } 2 8 `C(3.8) γ 2 k ? V k´1 , (3.28)
with the constants C 1,(3.28) :" 2}σ} Recalling that we consider first }U k } 8 ď C 8 , we then derive:

r∆ k pX k´1 , ¨qs 2 1 ď γ k |σ ˚∇ϕ| 2 pX k´1 q `Cγ 3{2 k `C(3.8) γ 2 k ? V k´1 , ( 3
T m :" exp ´´ρqλ Γ n ∆ m pX m´1 , U m q´ρ 2 pqλq 2 2Γ 2 n γ m |σ ˚∇ϕpX m´1 q| 2 ´m ÿ k"1 C (3.8) γ 2 k V k´1 ¯. (3.30)
From the definition of S m in (3.13), we write S m :" ś m k"1 T k . The coefficients pT m q mě1 can be viewed as multiplicative increments of p Sm q mě0 .Inequality (3.29) precisely allows to quantify the martingality default for p Sm q mě0 . These factors appear when we exploit the auxiliary Poisson problem (3.9) in the definition of T 1 in (3.11).

T 1 " exp `ρ2 q 2 λ 2 2Γ n νp|σ ˚∇ϕ| 2 qs ˘E" S n´1 ErT n |F n´1 s ‰ .
Thereby, from the upper-bound (3.29) of the Lipschitz modulus, we directly obtain from (3.30) and (GC)

ErT n |F n´1 s " exp `´ρ 2 pqλq 2 2Γ 2 n γ n |σ ˚∇ϕpX n´1 q| 2 ˘E" exp `ρqλ Γ n ∆ n pX n´1 , U n q ´n ÿ k"1 C (3.8) γ 2 k V k´1 ˘ˇˇF n´1 ı ď exp `ρ2 pqλq 2 2Γ 2 n Cγ 3{2 n ´n´1 ÿ k"1 C (3.8) γ 2 k V k´1 ˘.
Hence, iterating:

T 1 ď exp `ρ2 pqλq 2 2Γ n νp|σ ˚∇ϕ| 2 q ˘ErS n´1 s exp `ρ2 pqλq 2 2Γ 2 n γ 3{2 n C ď exp `ρ2 pqλq 2 2Γ n νp|σ ˚∇ϕ| 2 q ˘exp `ρ2 pqλq 2 2Γ n Γ p3{2q n Γ n lo omo on "en C ˘" exp `ρ2 pqλq 2 2Γ n pνp|σ ˚∇ϕ| 2 q `en q ˘,
where e n Ñ 0. The controls for T 2 are deduced from (3.14), and T 3 from (3.17). We now gather the previous estimates into (3.10) (we recall Erexpp´λ qMn Γn qs ď T

1 ρ 1 T ρ´1 qρ 2 T ρ´1 pρ 2
) in the following lemma. Note also that the term R ϑ n appearing in (3.14) is controlled similarly to remainders in Lemma 2.

Lemma 3 (Gaussian concentration term) With notations of (3.21), under (A), for a bounded ρ ą 1, we have:

E exp `´λq Γ n M n ˘1 q ď exp `λ2 Γ n A n `λ4 Γ 3 n B n ˘Rn ,
where

A n :" ρ `qνp|σ ˚∇ϕ| 2 q 2 `en ˘and B n :" ρ 3 ρ ´1 q 3 q 4 `q}σ} 2 8 rϑs 2 1 2 `en ˘, (3.31)
for some 1 ă q :" qpnq Ñ n 1, and with:

e n ÝÑ nÑ`8 0, R n ÝÑ nÑ`8
1 uniformly in λ.

As a consequence of the previous Lemmas 2 and 3, we obtain (3.18), namely:

P `aΓ n ν n pAϕq ě a ˘ď C n exp `P pλq ˘, (3.32)
with P pλq :" ´aλ ?

Γn

`λ2

Γn A n `λ4

Γ 3 n B n , where A n " A n pρq :" ρ r A n and B n " B n pρq :" ρ 3 ρ´1 r B n with r A n " qνp|σ ˚∇ϕ| 2 q 2 `en and r B n " q 3 q 4 `q}σ} 2 8 }∇ϑ} 2 8 2 `en ˘.
Next, like enunciated at the end of the User's guide to the proof, we optimize a fourth order polynomial by the Cardan method, see (3.33) below (and Section 4 in Chapter 3). If λ n " arg min λ P pλq, then

P 1 pλ n q " ´a ? Γ n `2λ n Γ n A n `4λ 3 n Γ 3 n B n " 0.
The Cardan-Tartaglia formula yields only one positive real root. Namely, setting

Φ n pa, ρq " ´a ? Γ n Bn ``a 2 B2 n Γ n `pρ´1q `2 Ãn 3 Bn ˘3˘1 2 ¯1 3 `´a ? Γ n Bn ´`a 2 B2 n Γ n `pρ´1q `2 Ãn 3 Bn ˘3˘1 2 ¯1 3 ,
this conducts to take λ " λ n with:

λ n :" Γ n 2 pρ ´1q 1 3 ρ Φ n pa, ρq. (3.33)
Moreover, remark from the binomial Newton expansion that:

Φ n pa, ρq 3 " 2a ? Γ n Bn ´2pρ ´1q 1{3 Ãn Bn Φ n pa, ρq.
From (3.33) and the above expression, P pλ n q " P min pa, Γ n , ρq " λ n `´a ? Γn Ñ n 0 ´a2 2νp|σ ˚∇ϕ| 2 q p1 `op1qq.

Γn `λn Γn A n `λ3 n Γ 3 n B n ˘, we thus obtain P min pa, Γ n , ρq :" ´?Γ n pρ ´1q 1{3 Φ n pa, ρq 2 3 ρ `3a ´aΓ n pρ ´1q 1{3 Ãn Φ n pa, ρq ˘. ( 3
For the sake of clarity, the proof of Lemma 4 is postponed to Section 4.2. From (3.35) and Lemma 4, we conclude the proof of Theorems 3 and 7 for bounded innovations.

Proof of Theorems 3 for unbounded innovations

Switching to unbounded innovations requires additional technicalities. Our strategy consists in considering a truncation argument writing

∆ k pX k´1 , U k q " ∆ k pX k´1 , U k qr1 |U k |ď r k,n 2 `1|U k |ą r k,n 2 s,
to control the Lipschitz modulus of ψ k´1 where pr k,n q ně1,kďn is a suitable sequence specified in (3.40) below. In particular, r k,n :" r k,n ppAq, λ, ρq where λ ą 0, ρ ą 1 are as in the User's Guide to the Proof.

For our choice below, we will have that, for all k P rr1, nss, r k,n Ò n `8. That choice for r k,n also yields that when |U k | ď r k,n , our controls behave like for the bounded case. But when |U k | ą r k,n , we will handle this large deviation regime by assumption (GC). We indeed know that for all K ą 0:

µpt|x| ą Kuq ď 2 expp´K 2 2
q.

(3.36)

Let us recall from the definition of ∆ k pX k´1 , U k q in (3.3) and (3.21) that:

∆ k pX k´1 , U k q " ? γ k σ k´1 U k ¨∇ϕpX k´1 `γk b k´1 q `Ξk pX k´1 , U k q, (3.37)
where for all pk, uq P rr1, nss ˆRr :

Ξ k pX k´1 , uq :" γ k ż 1 0 p1 ´tqTr ´D2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 uqσ k´1 u b uσ k´1 ´E" D 2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1 |F k´1 ‰ ¯dt. (3.38)
For the terms pT k q kďn defined in (3.30), the lemma below controls the "super martingality" default of S n " ś n k"1 T k .

Lemma 5 For all k P rr1, nss

ErT k |F k´1 s ď ℵ k,n pλ, γ k , r k,n q exp `´k´1 ÿ i"1 C (3.8) γ 2 i V i´1 ˘,
with, as in (3.28), C (3.8) :" 6}σ} 2 8 }∇ϕ} 8 }D 2 ϕ} 8 ,

ℵ k,n pλ, γ k , r k,n q :" `1 `2 expp´Cr 2 k,n q ˘exp `ρqλ Γ n Cγ 1{2 k expp´r 2 k,n 4 q ˘exp `ρ2 q 2 λ 2 Γ 2 n Cγ 3{2 k r 2 k,n ˘,
(3.39) and r k,n " r k,n ppAq, λ, ρq :"

# r n " Cp1 `ρqλ Γn q `Γn Γ p3{2q n ˘1{4 , f or θ P p 1 3 , 1q, Cp1 `ρqλ Γn q lnpn `1q 1{4 lnpk `1q 1{2 , f or θ " 1, (3.40)
for C ą 0 s.t. r 1,1 ą c? γ 1 }σ} 8 }∇ϕ} 8 for c large enough (every c ą 8 works, see the proof of Lemma 5, and equation (4.21)). This choice is briefly explained in Remark 8 below.

Remark 8 The specific form of the truncation and of the time steps chosen yields

r k,n " $ ' & ' % O `p1 `ρqλ Γn qn 1´θ 4 ˘, if θ P p2{3, 1q, O `p1 `ρqλ Γn q lnpnq ´1{4 n 1 12 ˘, if θ " 2{3, O `p1 `ρqλ Γn qn θ 8 ˘, if θ P p1{3, 2{3q.
Anyhow, we always have for each k ď n, r k,n ÝÑ n `8. Identity (3.41) in the following lemma can give an intuition of our choice in (3.40). This result ensures that the terms ℵ k,n pλ, γ k , r k,n q can be viewed as remainders (observe indeed that some e n Ñ n 0 appear in the exponential (3.41) below). From the optimization in λ performed in the proof Lemma 4 (see equation (4.10)), the contribution ρλ Γn appearing in (3.39) will be large in the regime of super Gaussian deviations (see as well Remark 15). The above choice of r k,n actually permits to control the remainders in all the considered regimes.

For θ P p 1 3 , 1q, the choice in (3.40) can seem natural in order to absorb the term

ś n k"1 exp `ρqλ Γn Cγ 1{2 k expp´r 2 k,n 16 q ˘" exp `ρqλ Γn CΓ p1{2q n expp´r 2 n
16 q ˘coming from the iteration of (3.39). For θ " 1, the choice is a bit different due to the associated logarithmic explosion rates (i.e. Γ n -lnpnq). Actually, for the Gaussian deviations ( a ? Γn Ñ 0), we have ρqλ Γn Ñ n 0, see again (4.10) and Remark 15 below, and the term ρqλ Γn could be removed in (3.40). On the other hand, the contribution exp

`ρ2 q 2 λ 2 Γ 2 n Cγ 3{2 k r 2 k,n
˘will eventually yield a negligible contribution in the polynomial appearing in Lemma 3. We refer to the proof of Lemma 5 for details. Lemma 6 (Control of "super martingality default" of S n ) There exist non negative sequences pR n q ně1 , pe n q ně1 s.t. R n ÝÑ n 1, e n ÝÑ n 0, and for all n ě 1:

ErS n s ď n ź k"1 ℵ k,n pλ, γ k , r k,n q " R n exp ´`ρ 2 q 2 λ 2 Γ n `ρ4 q 4 λ 4 Γ 3 n ˘en ¯. (3.41)
Observe that a term in λ 4 appears here for the control of ErS n s. This is specifically due to the unbounded contributions. Namely, the exponential term in (3.41) comes from Note now carefully that, reproducing the arguments of the bounded case and using as well Lemma 6 to control ErS n s yields that Lemma 3 remains valid, up to a modification of the remainders e n . The proof then follows similarly to the bounded case. To sum up, the specificity of the unbounded innovations was to precisely control the Lipschitz constants, considering a suitable truncation, as well as the "martingality default" of S n appearing in T 1 .

expp ρ 2 q 2 λ 2 Γ 2 n Cγ 3{2 k r 2 k,n q in (3.

Proofs of technical lemmas

Remainders from the Taylor decomposition

Proof of Lemma 2

From the notations in (3.21), we recall (3.23): ν n pAϕq " 1 Γn p r R n ´Mn q. The idea is now to write for a, λ ą 0:

P " a Γ n ν n pAϕq ě a ‰ ď exp `´aλ ? Γ n ˘E" exp `λ Γ n p r R n ´Mn q ˘ı ď exp `´aλ ? Γ n ˘E" exp `´qλ Γ n M n ¯ı1{q E " exp `pλ Γ n | r R n | ˘ı1{p , (4.1)
for 1 p `1 q " 1, p, q ą 1. We rewrite the Taylor expansion with the same notations as in Chapter 3: r R n " L n ´pD 2,b,n `D2,Σ,n `Ḡ n q where: D 2,b,n :"

n ÿ k"1 γ k ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt, D 2,Σ,n :" 1 2 n ÿ k"1 γ k Tr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σ2 k´1 ¯, Ḡn :" n ÿ k"1 E rψ k pX k´1 , U k q|F k´1 s, L n :" ϕpX n q ´ϕpX 0 q. (4.2)
From (4.2), (4.1) and the Cauchy-Schwarz inequality, we get:

P " a Γ n ν n pAϕq ě a ‰ ď exp `´aλ ? Γ n ˘´E exp `´qλ Γ n M n ˘¯1 q ˆ´E exp `2pλ Γ n ˇˇL n ˇˇ˘¯1 2p ˆ´E exp `4pλ Γ n ˇˇḠ n ˇˇ˘¯1 4p ´E exp `8pλ Γ n ˇˇD 2,Σ,n ˇˇ˘¯1 8p ´E exp `8pλ Γ n ˇˇD 2,b,n ˇˇ˘¯1 8p . (4.3)
The term L n in (4.3) is controlled in Lemma 4 in Chapter 3 (for j " 2 therein):

´E exp `4pλ |L n | Γ n ˘¯1 4p ď pI 1 V q 1 4p exp `3pC 2 V,ϕ λ 2 c V Γ 2 n `cV p ˘" R n exp `λ2 n Γ n e n ˘, (4.4)
for p " p n Ñ n `8 s.t. p Γn Ñ n 0. Thanks to Lemma 3 in Chapter 3, we obtain:

| Ḡn | ? Γ n ď a n :" rϕ p3q s β › › σ › › 3`β 8 E " |U 1 | 3`β ‰ p1 `βqp2 `βqp3 `βq Γ p 3`β 2 q n ?
Γ n , a.s. .

Moreover, a n Ñ n a 8 " 0 for θ P p 1 2`β , 1s. Hence, for all p ą 1:

´E exp `4pλ Γ n | Ḡn | ˘¯1 4p ď exp `λ ? Γ n a n ˘ď exp `λ2 2Γ n p `a2 n p 2 ˘" R n exp `λ2 Γ n e n ˘. (4.5)
for p " p n Ñ n `8 s.t. a 2 n p Ñ n 0. We handle the term D 2,Σ,n in (4.3) by Lemma 5 in Chapter 3: there exists C 1 :"

C 1 ppAq, ϕq ą 0 such that ´E exp `4pλ n Γ n ˇˇD 2,Σ,n ˇˇ˘¯1 4p ď exp `C1 pλ 2 n pΓ p2q n q 2 Γ 2 n ˘pI 1 V q 1 4p " R n exp `λ2 n Γ n e n ˘, (4.6)
for p " p n Ñ n `8 s.t. p pΓ p2q n q 2 Γn Ñ n 0 (we recall that for all θ P p 1 3 , 1s, Γ p2q n ?

Γn Ñ n 0).
We deal with the term D 2,b,n from (4.3). Because x Þ Ñ x∇ϕpxq, bpxqy is Lipschitz continuous, thanks to Lemma 5 in Chapter 3, we know that there exists C 2 :" C 2 ppAq, ϕq ą 0 such that:

´E exp `4pλ n Γ n ˇˇD 2,b,n ˇˇ˘¯1 4p ď exp `C2 p 3pλ 2 n pΓ p2q n q 2 2Γ 2 n `1 2p q ˘pI 1 V q 1 4p " R n exp `λ2 n Γ n e n ˘, (4.7)
also for p " p n Ñ n `8 s.t. p pΓ p2q n q 2 Γn Ñ n 0. We gather (4.4), (4.5), (4.6) and (4.7) into (4.3), which allows us to control the remainder involving r R n previously decomposed in (4.2). There are non-negative sequences pR n q ně1 , pe n q ně1 s.t. lim nÑ`8 R n " 1, lim nÑ`8 e n " 0 and:

P " a Γ n ν n pAϕq ě a ‰ ď exp ´´aλ ? Γ n ¯ˆE exp ´´qλ Γ n M n ¯˙1 q exp ˆλ2 n Γ n e n ˙Rn . (4.8)

Asymptotics in the parameter ρ

We first begin with the proof of Lemma 4 which is purely analytical and rather independent of our probabilistic setting. We recall that we use it for both bounded and unbounded innovations.

Proof of Lemma 4

From the expression of Φ n pa, ρq in Theorem 7, remember that

Φ n pa, ρq " ´a ? Γ n Bn ``a 2 B2 n Γ n `pρ´1q
`2 Ãn 3 Bn

˘3˘1 2 ¯1 3 `´a ? Γ n Bn ´`a 2 B2 n Γ n `pρ´1q `2 Ãn 3 Bn ˘3˘1 2 ¯1 3 .
Here, ρ ą 1 is a free parameter. Let us set:

ρ ´1 :" ξ 27 8 Bn a 2 Ã3 n Γ n , (4.9)
for a parameter ξ :" ξpa, nq ą 0 to optimize. This choice yields

Φ n pa, ρq " a 1{3 B1{3 n Γ 1{6 n `p1 `a1 `ξq 1{3 `p1 ´a1 `ξq 1{3 ˘.
Hence, from the definition of λ n in (3.33):

ρλ n Γ n " 1 2 pρ ´1q 1{3 Φ n pa, ρq " 3 2 2 a Ãn ? Γ n ξ 1{3 `p1 `a1 `ξq 1{3 `p1 ´a1 `ξq 1{3 ˘. (4.10)
We point out that ξ Þ ÝÑ ξ 1{3 `p1 `?1 `ξq 1{3 `p1 ´?1 `ξq 1{3 ˘is a bounded function from r0, `8q to r0, 2 3 q. In fact,

ξ 1{3 `p1 `a1 `ξq 1{3 `p1 ´a1 `ξq 1{3 " ξÑ`8 ξ 1{3 p1 `ξq 1{6 `1 `1 3 ? 1 `ξ ´p1 ´1 3 ? 1 `ξ q `op 1 ? 1 `ξ q ˘" ξÑ`8 2 3 . (4.11)
From the definition of P min pa, ρ, Γ n q " P pλ n q " min xą0 P pxq in (

Ã3

n Γn `1q

ξ 1{3 `p1 `a1 `ξq 1{3 `p1 ´a1 `ξq 1{3 1 ´ξ1{3 2 `p1 `a1 `ξq 1{3 `p1 ´a1 `ξq 1{3 ˘": ´32 2 4 Ãn a 2 f Ψ pξq, (4.12)
for

f Ψ : ξ P R `Þ ÝÑ gpξq Ψξ `1 , (4.13)
where

g : ξ Þ ÝÑ ξ 1{3 `p1`a1 `ξq 1{3 `p1´a1 `ξq 1{3 ˘´1´ξ 1{3 2 `p1`a1 `ξq 1{3 `p1´a1 `ξq 1{3 ˘¯, and
Ψ :" 27 8 Bn a 2 Ã3 n Γ n
´(4.9)

" ρ ´1 ξ ¯. (4.14)
We bring to mind that we consider "Gaussian deviations", namely a ? Γn Ñ n 0.

From (6.11) and the asymptotic of Ãn defined in (6.24), i.e. 4 Ãn Ñ n 2νp|σ ˚∇ϕ| 2 q, we want to choose ξ :" ξpa, nq s.t. Λpξq :" We give here some useful estimates to control the remainder terms in the truncation procedure associated with unbounded innovations, see proof of Lemma 6. They specify the behaviour of the quantity ρλn Γn . In the regime of Gaussian deviations, we get from (4.10) and (4.11):

ρλ n Γ n ÝÑ a ? Γn Ñ0 0, λ n - a ? Γn Ñ0 a a Γ n . (4.16)

Technical Lemmas for Unbounded Innovations

We proceed with the proof of Lemmas 5 and 6 which are specifically needed for unbounded innovations.

Proof of Lemma 5

We use a partition at the threshold r k,n on the variable U k , to control finely the term ∆ k pX k´1 , U k q:

E " exp `´ρqλ Γ n ∆ k pX k´1 , U k q ˘|F k´1 ı " E " exp `´ρqλ Γ n ∆ n pX k´1 , U k q ˘1|U k |ďr k,n |F k´1 ı `E" exp `´ρqλ Γ n ∆ k pX k´1 , U k q ˘1|U k |ąr k,n |F k´1 ı ": T M k,s `T M k,l , (4.17)
where T M k,s and T M k,l stand for the contributions in ErT k |F k´1 s expp

ř k´1 i"1 C (3.8) γ 2 i V i´1
q associated with the martingale increment ∆pX k´1 , U k q for which the innovation is respectively small and large.

Let us first write:

T M k,s " E " exp `´ρqλ Γ n ∆ n pX k´1 , U k q ˘1|U k |ďr k,n |F k´1 ı " ż |u|ďr k,n exp `´ρqλ Γ n ∆ n pX k´1 , U k q ˘µpduq.
Observe now that, similarly to the computations of Section 3.2 for bounded innovations, see equation

(3.29), if |u| ď r k,n , u P R r Þ Ñ ∆ n pX k´1 , uq is s.t. r∆ n pX k´1 , ¨qs 2 1,Bp0,r k,n q ď γ k |σ k´1 ∇ϕpX k´1 q| 2 `Cγ 3{2 k r 2 k,n `C(3.8) γ 2 k V k´1 , (4.18)
where r∆ n pX k´1 , ¨qs 2 1,Bp0,r k,n q denotes the Lipschitz modulus of ∆ n pX k´1 , ¨q restricted on the ball Bp0, r k,n q of R r with radius r k,n .

Let us now extend u P Bp0, r k,n q Þ Ñ ∆ n pX k´1 , uq into a Lipschitz function on the whole set R r which globally verifies the bound of equation (4.18). The easiest way to do so is to consider:

u P R r Þ Ñ ∆pX k´1 , uq " ∆ `Xk´1 , Π Bp0,r k,n q puq ˘,
where Π Bp0,r k,n q p¨q denotes the projection on Bp0, r k,n q, namely for u P Bp0, r k,n q, we have Π Bp0,r k,n q puq " u, for u R Bp0, r k,n q, Π Bp0,r k,n q puq " u |u| r k,n . It is readily seen that:

r ∆pX k´1 , ¨qs 2 1 ď γ k |σ k´1 ∇ϕpX k´1 q| 2 `Cγ 3{2 k r 2 k,n `C(3.8) γ 2 k V k´1 . Hence: T M k,s " ż |u|ďr k,n exp `´ρqλ Γ n ∆n pX k´1 , U k q ˘µpduq ď E " exp `´ρqλ Γ n ∆n pX k´1 , U k q ˘|F k´1 ı ď pGCq exp `ρ2 q 2 λ 2 2Γ 2 n r ∆pX k´1,¨q s 2 1 ´ρqλ Γ n Er ∆pX k´1 , U k qs ď exp `ρ2 q 2 λ 2 2Γ 2 n pγ k |σ k´1 ∇ϕpX k´1 q| 2 `Cγ 3{2 k r 2 k,n `C(3.8) γ 2 k V k´1 q ´ρqλ Γ n Er ∆pX k´1 , U k qs ˘. (4.19)
Bearing in mind that Er∆pX k´1 , U k qs " 0, and observe now:

Er ∆pX k´1 , U k qs " Er∆pX k´1 , U k q1 |U k |ăr k,n s `Er ∆pX k´1 , U k q1 |U k |ěr k,n s " ´Er∆pX k´1 , U k q1 |U k |ěr k,n s `Er∆ `Xk´1 , Π Bp0,r k,n q pU k q ˘1|U k |ěr k,n s ď ´Er|∆pX k´1 , U k q| 2 s 1{2 `Er|∆ `Xk´1 , Π Bp0,r k,n q pU k qq| 2 s 1{2 ¯Pr|U k | ě r k,n s 1{2 ď Cγ 1{2 k expp´r 2 k,n 4
q, exploiting (3.36), (3.37) and (3.38) for the last inequality. Plugging this bound into (4.19) yields:

T M k,s ď exp `ρ2 q 2 λ 2 2Γ 2 n pγ k |σ k´1 ∇ϕpX k´1 q| 2 `Cγ 3{2 k r 2 k,n `C(3.8) γ 2 k V k´1 q`Cγ 1{2 k ρqλ Γ n expp´r 2 k,n 4 q ˘.
(4.20) The remaining term in (4.17), involving also the large deviations of the innovation, can be controlled as follows:

T M k,l " E " exp `´ρqλ Γ n ∆ k pX k´1 , U k q ˘1|U k |ąr k,n |F k´1 ı ď E " exp `´2ρqλ Γ n ∆ k pX k´1 , U k q ˘|F k´1 ı 1{2 E " 1 |U k |ąr k,n ı 1{2 ď 2E " exp `´2ρqλ Γ n ∆ k pX k´1 , U k q ˘|F k´1 ı 1{2 expp´r 2 k,n 4
q pGCq ď 2 exp `ρ2 q 2 λ 2 Γ 2 n γ k }σ} 2 8 }∇ϕ} 2 8 ´r2 k,n 4
˘.

Let us proceed from our definition of r k,n " r k,n ppAq, λ, ρq in (3.40) and write r k,n " p1 `ρqλ Γn qu k,n . One has that for all k ď n, u k,n Ñ n `8. In other words, with the previous inequality, recalling

r 2 k,n ě u 2 k,n p1 `ρ2 q 2 λ 2 Γ 2 n q: T M k,l ď 2 exp `q2 ρ 2 λ 2 Γ 2 n pγ k }σ} 2 8 }∇ϕ} 2 8 ´u2 k,n 4 q ´u2 k,n 4 ď 2 exp `´C q 2 ρ 2 λ 2 Γ 2 n u 2 k,n 8 ´u2 k,n 4
E " exp `´ρqλ Γ n ∆ k pX k´1 , U k q ˘|F k´1 ı ď exp ´ρ2 q 2 λ 2 2Γ 2 n `γk |σ k´1 ∇ϕ k´1 | 2 `Cγ 3{2 k r 2 k,n `C(3.8) γ 2 k V k´1 ˘`ρqλ Γ n Cγ 1{2 k expp´r 2 k,n 4 q 2 exp `´Cr 2 k,n ď exp `ρ2 q 2 λ 2 2Γ 2 n `γk |σ k´1 ∇ϕ k´1 | 2 `Cγ 3{2 k r 2 k,n `C(3.8) γ 2 k V k´1 ˘`ρqλ Γ n Cγ 1{2 k expp´r 2 k,n 4 q 1 `2 expp´Cr 2 k,n q ": exp `ρ2 q 2 λ 2 2Γ 2 n γ k |σ k´1 ∇ϕ k´1 | 2 `C(3.8) γ 2 k V k´1 ˘ˆℵ k,n pλ, γ k , r k,n q, where ℵ k,n pλ, γ k , r k,n q :" ´1`2 exp `´Cr 2 k,n ˘¯exp `ρqλ Γ n Cγ 1{2 k expp´r 2 k,n 4 q ˘exp `ρ2 q 2 λ 2 Γ 2 n Cγ 3{2 k r 2 k,n ˘.
We have thus isolated the "significant" term exp

`ρ2 q 2 λ 2 2Γ 2 n γ k |σ k´1 ∇ϕ k´1 | 2 ˘.
The result follows from the definition of T k in (3.30).

Proof of Lemma 6

We recall for convenience the definition of r k,n in (3.40):

r k,n " # r n " Cp1 `ρqλ Γn qp Γn Γ p3{2q n q 1{4 , f or θ P p 1 3 , 1q, Cpp1 `ρqλ Γn q lnpn `1q 1{4 lnpk `1q 1{2 , f or θ " 1.
From the above definition of ℵ k,n pλ, γ k , r k,n q, we introduce two remainders:

R 1 n :" n ź k"1 ´1 `2 exp `´Cr 2 k,n ˘¯ˆn ź k"1 exp `ρqλ Γ n Cγ 1{2 k expp´r 2 k,n 4 q "
: R 11 n ˆR12 n , R 2 n :" exp `ρ2 q 2 λ 2 Γ 2 n C n ÿ k"1 γ 3{2 k r 2 k,n ˘, (4.22)
that naturally appear when we iterate Lemma 5. Precisely:

ErS n s ď ErS n´1 sℵ n,n pλ, γ n , r n,n q ď n ź k"1 ℵ k,n pλ, γ k , r k,n q " R 1 n ˆR2 n . (4.23)
' For θ P p1{3, 1q:

We have chosen r k,n " r n in (3.40), so,

R 1 n " ´1 `2 exp `´Cr 2 n ˘¯n exp `ρqλ Γ n CΓ p1{2q n expp´r 2 n 4 q ˘" R 11 n R 12 n ,
and:

R 11 n " ´1 `2 exp `´Cr 2 n ˘¯n ď exp ´n ln `1 `2 expp´Cp Γ n Γ p3{2q n q 1{2 q ˘ď exp `2n expp´Cp Γ n Γ p3{2q n q 1{2 q ˘ÝÑ nÑ`8 1,
as, for θ P p1{3, 1q, ´Γn

Γ p3{2q n ¯1{2 ě ηpnq where ηpnq :" C `np1´θq{2 1 θPp2{3,1q `pn 1{3 lnpnq ´1q 1 2 1 θ"2{3 `nθ{4 1 θPp1{3,2{3q ˘, (4.24)
see also Remark 8. For the remaining of the proof, we will thoroughly exploit this kind of arguments. Precisely, we recall that:

@ζ P R `, DC ζ ě 1, s.t. @0 ď β ď ζ, @x P R `, x β expp´x 2 q ď C ζ expp´C ´1 ζ x 2 q. (4.25)
Thus, for the remaining term R 12 n in R 1 n , exploiting (4.25), up to a modification of the constant C ą 0 from line to line:

R 12 n ď exp `ρqλ Γ n CΓ p1{2q n expp´r 2 n 4 q ď exp ´ρqλ Γ n CΓ p1{2q n exp `´ρ 2 q 2 λ 2 Γ 2 n C 4 p Γ n Γ p3{2q n q 1{2 ˘exp `´C 4 p Γ n Γ p3{2q n q 1{2 ˘ď exp ´CΓ p1{2q n p Γ p3{2q n Γ n q 1{4 exp `´ρ 2 q 2 λ 2 Γ 2 n C 8 p Γ n Γ p3{2q n q 1{2 ˘expp´C 4 p Γ n Γ p3{2q n q 1{2 q ď exp `CΓ p1{2q n expp´C 8 p Γ n Γ p3{2q n
q 1{2 q ˘ď exp `Cn 1´θ{2 e ´ηpnq ˘(4.24),(4.25)

ÝÑ nÑ`8 1. Hence R 1 n " R 11 n R 12 n Ñ n 1. Introducing e θă1 n :" C d Γ p3{2q n Γ n ÝÑ nÑ`8 0,
we get from the definition of r k,n in (3.40) and (4.22)

R 2 n ď exp ´`ρ 2 q 2 λ 2 Γ n `ρ4 q 4 λ 4 Γ 3 n ˘eθă1 n ¯.
Recalling, from (4.23), that ErS n s ď R 1 n R 2 n , we thus get from the above computations:

ErS n s ď R n exp `p ρ 2 q 2 λ 2 Γ n `ρ4 q 4 λ 4 Γ 3 n qe n ˘,
which gives the result for θ P p 1 3 , 1q. ' For θ " 1, using the previous notations:

0 ď lnpR 11 n q ď ln ´n ź k"1 `1 `2 expp´Cr 2 k,n q ˘¯" n ÿ k"1 ln ´1 `2 exp `´Cr 2 k,n ˘ď C n ÿ k"1 exp `´Cr 2 k,n ˘ď C n ÿ k"1 exp `´C lnpn `1q 1{2 lnpk `1q " C n ÿ k"1 pk `1q ´C lnpn`1q 1{2 ď C2 ´C lnpn`1q 1{2 `C ż n 2 x ´C lnpn`1q 1{2 dx " C2 ´C lnpn`1q 1{2 `C 2 1´C lnpn`1q 1{2 ´n1´C lnpn`1q 1{2 C lnpn `1q 1{2 ´1 ÝÑ nÑ`8 0.
We deduce that:

R 11 n " n ź k"1 ´1 `2 exp `´Cr 2 k,n ˘¯ÝÑ nÑ`8 1.
Like for the case θ P p1{3, 1q, we get the control:

R 12 n ď n ź k"1 exp `ρqλ Γ n Cγ 1{2 k expp´r 2 k,n 4 q ď n ź k"1 exp ´ρqλ Γ n Cγ 1{2 k expp´C 4 r1 `p ρqλ Γ n q 2 s lnpn `1q 1{2 lnpk `1q ˘ď (4.25) n ź k"1 exp ´Cγ 1{2 k exp `´C 4 r1 `Cp ρqλ Γ n q 2 s lnpn `1q 1{2 lnpk `1q ˘ď n ź k"1 exp `C 1 k 1{2 pk `1q ´C lnpn`1q 1{2 ˘Ñ nÑ`8 1,
in fact, we have already established for the control of R 11 n that

ř n k"1 pk`1q ´C lnpn`1q 1{2 ÝÑ n 0.
Thus, we proved that

R 1 n " R 11 n R 12 n Ñ n 1.
Let us now turn to the other contribution in (4.22):

R 2 n " exp `ρ2 q 2 λ 2 Γ 2 n C n ÿ k"1 γ 3{2 k r 2 k,n ď exp ´2Cp ρ 2 q 2 λ 2 Γ 2 n `ρ4 q 4 λ 4 Γ 4 n q lnpn `1q 1{2 n ÿ k"1 γ 3{2 k lnpk `1q ď exp `2Cp ρ 2 q 2 λ 2 Γ n `ρ4 q 4 λ 4 Γ 3 n q ř n k"1 γ 3{2 k lnpk `1q ? Γ n " exp `p ρ 2 q 2 λ 2 Γ n `ρ4 q 2 λ 4 Γ 3 n qe θ"1 n ˘, with e θ"1 1,n :" 2C 1 ? Γn ř n k"1 lnpk`1q k 3{2 ÝÑ nÑ`8
0, and recalling that ? Γ n -lnpn `1q 1{2 for the last inequality.

To sum up, for all θ P p 1 3 , 1s, thanks to inequality (4.23), we know that there exist non negative sequences pR n q ně1 , pe n q ně1 , s.t. R n ÝÑ n 1, e n ÝÑ n 0, and

ErS n s ď R n exp ´`ρ 2 q 2 λ 2 Γ n `ρ4 q 4 λ 4 n Γ 3 n ˘en ¯.

Regularity Results and Consequences

This section is devoted to some regularity results for the Poisson problem

Aϕ " f ´νpf q.

(5.1)

In particular, we state below some Schauder like controls, which are, because of our methodology that requires pointwise controls of the derivatives, more adapted than the standard Sobolev estimates (see e.g. Pardoux and Veretennikov [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF]).

There are two kinds of assumptions that guarantee the solution ϕ of (5.1) enjoys the required smoothness of Theorems 3. "`f pY 0,x t q ´νpf q ˘‰dt of the solution of (5.1) can be differentiated using iterated tangent flows to establish that ϕ P C 3,β pR d , Rq, for some β P p0, 1s. It suffices for that to have f P C 3,β pR d , Rq and b P C 3,β pR d , R d q, σ P C 3,β b pR d , R d b R r q. We refer to Section 2.2 and 5 of Chapter 3 for additional details. Importantly, under such assumptions, specific non-degeneracy conditions are not needed.

(b) If we do not assume such an a priori smoothness on f, b, σ, we need to make an extra non-degeneracy assumption which will allow some regularity gain (elliptic bootstrap) and a stronger confluence like condition (D p α ) with }Dσ} 2 8 ď 2α 2p3`βq´p for given pα, pq P p0, `8q ˆr1, 2q. Precisely, assuming that the bounded diffusion coefficient Σ is also uniformly elliptic, we exploit the results of Krylov and Priola [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF] (Theorems 2.4-2.6) to derive that, up to an additional technical condition on Σ when d ą 1, we actually have ϕ P C 3,β pR d , Rq for some β P p0, 1q, as soon as

f P C 1,β pR d , Rq and b P C 1,β pR d , R d q, σ P C 1,β b pR d , R d b R d q.
We point out that the second set of assumptions is very important in order to go towards the usual setting of functional/transport inequalities which typically involves Lipschitz continuous test functions. This is for instance the case for the controls of the Wasserstein distances between the law of Y t in (1.1) and the invariant distribution ν (see e.g. [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]). We are thus able, in the non-degenerate framework (b), to consider directly sources f P C 1,β pR d , Rq. Observe that, when β Ñ 0, we are almost Lipschitz. Actually, we can, in this setting, handle functions f P C 0,1 pR d , Rq up to a spatial regularization which leads to a constraint on the steps (see Theorem 5 below).

Assumptions and Regularity Results

The regularity results stated here can be found in Section 5 of Chapter 3. Let us now recall the useful assumptions needed.

(UE) Uniform Ellipticity. We assume that r ě d in (1.1) and that there is σ ą 0 such that @ξ P R d , xσσ ˚pxqξ, ξy ě σ|ξ| 2 .

For β P p0, 1q, we introduce the following condition.

(R 1,β ) Regularity Condition. From equation (1.1), we suppose b P C 1,β pR d , R d q, and Bhouou

σ P C 1,β b pR d , R d b R r q. (D p α ) Confluence Conditions.
We assume that there exists α ą 0 and p P r1, 2q such that for all

x P R d , ξ P R d A Dbpxq `Dbpxq 2 ξ, ξ E `1 2 r ÿ j"1 ´pp ´2q |xDσ ¨j pxqξ, ξy| 2 |ξ| 2 `|Dσ ¨j ξ| 2 ¯ď ´α|ξ| 2 , (5.2)
where Db stands here for the Jacobian of b, σ ¨j stands for the j th column of the diffusion matrix σ and Dσ ¨j for its Jacobian matrix.

There are others assumptions than (D p α ) which yield, in the non-degenerate setting, gradient control. This is the case for the so-called Bakry and Émery curvature criterion ([BE85, BGL14]) which is however pretty hard to check for general multidimensional diffusion coefficients. However for Hölder control of the gradient, this critetion seems to be not adapted, see Chapter 3 Section 2.2 for more details.

We eventually introduce, as in Chapter 3, a technical condition on the diffusion coefficient σ. It allows to prove that each partial derivative B x i ϕ of the solution of (5.1) satisfies an autonomous scalar Poisson problem. We suppose:

(Σ) for every pi, jq P rr1, dss 2 and x " px 1 , . . . , x d q P R d , Σ i,j pxq " Σ i,j px i^j , ¨¨¨, x d q.

We say that assumption (P β ) is satisfied if (UE), (D p α ) with }Dσ} 2 8 ď 2α 2p1`βq´p , (R 1,β ), and (Σ) are in force. From Section 5.3 of Chapter 3 we have the following result.

Theorem 4 (Elliptic Bootstrap in a non-degenerate setting) Assume (P β ) holds for some β P p0, 1q and that f P C 1,β pR d , Rq. Then, there is a unique ϕ P C 3,β pR d , Rq solving (5.1).

Note as well from Remark 2, that there exists C ą 0, |D 2 ϕpxq| ď Cp1 `|x|q ´1. In other words, the solution ϕ of (5.1) satisfies (T β ).

Remark 10 (On Schauder estimates for β " 1) We insist on the fact that β P p0, 1q in the above theorem. Indeed, it is well known that the Hölder exponent β cannot go to 1 in the Schauder estimates. Note that, for the particular case f P C 1,1 pR d , Rq, we also have f P C 1,β pR d , Rq for all β P p0, 1q. This means that, for such f , the elliptic bootstrap works up to an arbitrarily small correction.

From Theorem 4 we readily have:

Corollary 1 (Regularity for the Poisson problem with carré du champ source) Assume (P β ) holds for some β P p0, 1q and that f P C 1,β pR d , Rq. Then, there is a unique ϑ P C 3,β pR d , Rq solving Aϑ " |σ ˚∇ϕ| 2 ´νp|σ ˚∇ϕ| 2 q and satisfying (T β ).

Indeed, it suffices to observe from Theorem 4 and the assumption (R 1,β ) in (P β ) that f :" |σ ˚∇ϕ| 2 P C 1,β pR d , Rq and to apply again Theorem 4 for this source.

Concentration inequalities for a Lipschitz source in a nondegenerate setting

As indicated in the introduction of the Section, we aim at controlling deviations for Lipschitz sources. In the current Lipschitz framework we aim to address, we need a slightly different set of assumptions. Namely, we will assume that (C1), (GC), (C2), (L V ), (U), (S) and (P β ) are in force and we will say that (L β ) holds. Under this new assumption, we have the following result.

Theorem 5 (Non-asymptotic concentration bounds in a Lipschitz framework) Assume that (L β ) is in force. Let f be a Lipschitz continuous function. For a time step sequence pγ k q kě1 of the form γ k -k ´θ, θ P p1{2, 1s, we have that, there exist two explicit monotonic sequences c n ď 1 ď C n , n ě 1, with lim n C n " lim n c n " 1 such that for all n ě 1 and for every a ą 0:

P " | a Γ n `νn pf q ´νpf q ˘| ě a ‰ ď 2C n exp `´c n a 2 2νp|σ ˚∇ϕ| 2 q ˘, (5.3)
where ϕ P C 0,1 pR d , Rq X W 2 2,loc pR d , Rq is a weak solution of the Poisson equation Aϕ " f ´νpf q.

Sketch of the proof To prove the above result, the starting point consists in regularizing the source f by mollification. Namely, we consider f δ " f ‹ η δ , where ‹ denotes the usual convolution, for a suitable mollifier η δ p¨q :" 1 δ d ηp δ q, δ ą 0, where η is a compactly supported non-negative function s.t. ş R d ηpxqdx " 1. We then write ν n pf q´νpf q " ν n pf δ qν pf δ q `pν n ´νqpf ´fδ q ": ν n `fδ ´νpf δ q ˘`R n,δ . We aim at letting δ go to 0 so that R n,δ can be viewed as a remainder. On the other hand, we will apply the same strategy as in the proof of Theorem 3 to analyze the deviations of ν n `fδ ´νpf δ q ˘" ν n pAϕ δ q. Precisely, reproducing the arguments of Section 5.3 of Chapter 3 to equilibrate the explosions of the derivatives of ϕ δ in the proof of Theorem 3 yields that there exists two explicit monotonic sequences cn ď 1 ď C n , n ě 1, with lim n C n " lim n cn " 1 s.t.

P " | a Γ n `νn pf q ´νpf q ˘| ě a ‰ ď 2C n exp `´c n a 2 2νp|σ ˚∇ϕ δ | 2 q q.
(5.4)

From the previous Schauder estimates, we know that ϕ δ P C 3,β pR d , Rq for all δ ą 0 with explosive C 3,β norm in δ but with bounded gradient. Recall indeed that, for all β P p0, 1q,

}f δ } C 1,β ď Cδ ´β , |∇ϕ δ | ď rf δ s 1 α " rf s 1 α . (5.5)
We again refer to Lemma 6 and Section 5.3 in Chapter 3 for details. On the other hand, it is well known, see e.g. [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF], that ϕ δ pxq " ´ş`8 0 Erf δ pY 0,x t q ´νpf δ qsdt. From their Proposition 1, we have in our case that, denoting by ν Y 0,x t the law of Y 0,x t we have the following control for the total variation between ν Y 0,x t and ν. There exists constants pC, cq :" pC, cqp(L β )q s.t.

}ν Y 0,x t ´ν} T.V. ď C exppc|x|q expp´αtq.
(5.6)

Introducing now fδ " f δ ´f ´ν`f δ ´f ˘, we rewrite that, for all x P R d :

`ϕδ ´ϕ˘p xq " ´ż `8 0 Er fδ pY 0,x t qsdt, | `ϕδ ´ϕ˘p xq| ď ż `8 0 ´żR d | fδ pyq| 2 `νY 0,x t `ν˘p dyq ¯1{2 }ν Y 0,x t ´ν} 1{2 T.V. dt ď (5.6) C 1{2 expp c 2 |x|q} fδ } 8 ż `8 0 exp ´´α 2 t ¯dt.
Recalling that f is Lipschitz and that pf ´fδ qpxq " ş R d `f px ´yq ´f pxq ˘ηδ pyqdy , we actually have } fδ } 8 ď Crf s 1 δ, which establishes the pointwise convergence ϕ δ pxq ÝÑ δÑ0 ´ş`8 0 Erf pY 0,x t q´νpf qsdt ": ϕpxq which is the only weak solution of (5.1) in W 2 p,loc pR d , Rq, p ą 1 (see Theorem 1 in [START_REF] Pardoux | On the Poisson Equation and Diffusion Approximation[END_REF]).

Let us now prove that lim δÑ0 νp|σ ˚∇ϕ δ | 2 q " νp|σ ˚∇ϕ| 2 q.

(5.7)

For all ε ą 0, there is a compact K :" Kpεq such that, denoting by K c :" R d zK, we have:

ż R d ˇˇ∇ϕ δ ´∇ϕ ˇˇ2 pxq1 K c pxqνpdxq ď 4}∇ϕ} 2 8 ż R d 1 K c pxqνpdxq ď ε 2 .
Also, since ϕ P W 2 2,loc pR d , Rq, we write:

ż R d ˇˇ∇ϕ δ ´∇ϕ ˇˇ2 pxq1 K pxqνpdxq ď ż R d ˇˇż R d `∇ϕpx ´zq ´∇ϕpxq ˘ρδ pzqdz ˇˇ21 K pxqνpdxq " ż R d ˇˇż R d `ż 1 0 D 2 ϕpx ´λzqzdλ ˘ρδ pzqdz ˇˇ21 K pxqνpdxq ď ż 1 0 dλ ż R d ˇˇż R d D 2 ϕpx ´λzqzρ δ pzqdz ˇˇ21 K pxqνpdxq ď ż 1 0 dλ ´żR d dz|z| 2 ρ δ pzq ż R d |D 2 ϕpx ´λzq| 2 1 K pxqνpdxq ¯ď Cδ 2 }ϕ} 2 W 2 2 p K,Rq ă ε 2 ,
for δ small enough, using the Cauchy-Schwarz inequality for the penultimate control and denoting by K a compact set such that for all z P Bp0, Cδq Ą supppη δ q, x P K, x ´z P K. This in particular gives (5.7). Hence, setting

c n :" cn νp|σ ˚∇ϕ| 2 q νp|σ ˚∇ϕ δ | 2 q Ñ n 1,
and recalling from Section 5.3 in Chapter 3 that δ :" δpnq Ñ n 0 ‡ , we derive that (5.3) follows from (5.4) up to a modification of cn . Furthermore, let us point out that, the result can alternatively be stated replacing the carré du champ in (5.3) by the variance of the Lipschitz source under the invariant law. In fact, we can write by the dominated convergence theorem:

lim δÑ0 νp|σ ˚∇ϕ δ | 2 q " νp|σ ˚∇ϕ| 2 q.
(5.8)

Remark 11

The new threshold θ ą 1 2 comes from the specific Lipschitz regularity of the test function f . Intuitively, this threshold naturally appears when we consider β Ñ 0 in the previous condition θ P p 1 2`β , 1s induced by the regularity of ϕ P C 3,β pR d , Rq which holds, under (P β ), when f P C 1,β pR d , Rq. We underline anyhow that, for β " 0, the Schauder estimates do not directly apply.

6 Optimisation over ρ under Gaussian and super Gaussian deviations

In Lemma 4, we performed an asymptotic estimation of the upper-bound for Gaussian deviations. However, from a numerical point of view, it appears to be more significant to optimize over ρ in whole generality, i.e. not only when a ? Γn Ñ 0. This procedure conducts to deviations bounds that are much closer to the realizations.

In particular, in super Gaussian deviations framework (i.e. a ? Γn Ñ 0), we provide here a "weaker" concentration inequality than the Gaussian one, which precisely comes from the optimization over ρ for this regime. This loss of concentration, with the terminology of Remark 4, is intrinsic to our method, as it will be shown in the proofs of Theorem 6 and Lemma 7 below, see also Remark 17.

Theorem 6 (Deviations in the super Gaussian regime) Assume (A) is in force. If there exists ϑ P C 3,β pR d , Rq, β P p0, 1s satisfying (T β ) s.t.

Aϑ " |σ ˚∇ϕ| 2 ´νp|σ ˚∇ϕ| 2 q, (6.9)

then, for θ P p 1 2`β , 1s , there exist explicit non-negative sequences pc n q ně1 and pC n q ně1 , respectively increasing and decreasing for n large enough, with lim n C n " lim n c n " 1 s.t. ‡. which was anyhow constrained to go to 0 sufficiently slowly in order to balance the explosions in the derivatives coming from the Schauder estimates, see (5.5). It is specifically this feature that led to the condition γ n -n ´θ , θ P p1{2, 1s.

for all n ě 1, a ą 0, the following bounds hold. When a ? Γn Ñ `8 (Super Gaussian deviations):

P " | a Γ n ν n pAϕq| ě a ‰ ď 2 C n exp `´c n a 4{3 Γ 1{3 n 2}σ} 2{3 8 rϑs 2{3 1 ˘.
Remark 12 Observe from Corollary 1, that, the function ϑ enjoys the required smoothness as soon as assumption (P β ) introduced in Section 5 holds.

Remark

P " | a Γ n ν n pAϕq| ě a ‰ ď 2 C n exp `´c n a 2 2νp|σ ˚∇ϕ| 2 q ˘.
Observe that for such regimes, there is an equivalence, up to multiplicative constants, between the bounds in Theorem 3 and in Theorem 6.

The idea of the proof of Theorem 6 follows the same lines as for Theorem 3, except for the optimization over ρ which is more fussy, see Lemma 7 below.

We recall that the analysis in the proof of Theorem 3 leaving open a possible optimization over the parameter ρ which we now perform The next lemma indicates that this optimization implies a Gaussian regime for Γn which corresponds to our choice in Lemma 4. We then retrieve the Gaussian regime. In the super Gaussian deviations framework, the optimization over ρ leads to consider ρ ´1 " 1 2 `op1q which yields the loss in the concentration inequality.

Let us first continue with the proof of Lemma 7 which is purely analytical and rather independent of our probabilistic setting.

Proof of Lemma 7

We keep the notations of Lemma 4, that we bring to mind.

Φ n pa, ρq " ´a ?

Γ n Bn ``a 2 B2 n Γ n `pρ´1q
`2 Ãn 3 Bn

˘3˘1 2 ¯1 3 `´a ? Γ n Bn ´`a 2 B2 n Γ n `pρ´1q
`2 Ãn 3 Bn

˘3˘1 2 ¯1 3 " a 1{3 B1{3 n Γ 1{6 n `p1 `a1 `ξq 1{3 `p1 ´a1 `ξq 1{3 ˘, for ρ ´1 :" ξ 27 8 Bn a 2 Ã3 n Γ n , (6.10)
where ξ :" ξpa, nq ą 0 is a parameter that we are going to optimize. Furthermore:

P pλ n q " ´32 2 4 Ãn a 2 f Ψ pξq, (6.11)
for

f Ψ : ξ P R `Þ ÝÑ gpξq Ψξ `1 , (6.12)
where

g : ξ Þ ÝÑ ξ 1{3 `p1`a1 `ξq 1{3 `p1´a1 `ξq 1{3 ˘´1´ξ 1{3 2 `p1`a1 `ξq 1{3 `p1´a1 `ξq 1{3 ˘¯, and
Ψ :" 27 8 Bn a 2 Ã3 n Γ n `(4.9) " ρ ´1 ξ ˘. (6.13)
' Let us first focus on case (a), "Gaussian deviations" ( a ? Γn Ñ n 0). We, now anyhow, want to maximize Λ in ξ to obtain the best possible concentration bound. Let A :" tξ P r0, `8s : f Ψ pξq " }f Ψ } 8 u be the set of points where f Ψ reaches its maximum. Observe that for a fixed Ψ, f Ψ pξq Ñ ξÑ8 0. Thus, `8 R A . Let now ξ ˚be an arbitrary point in A .

From the smoothness of f Ψ , the optimality condition writes:

f 1 Ψ pξ ˚q " g 1 pξ ˚q pΨξ ˚`1q ´Ψgpξ ˚q pΨξ ˚`1q 2 " 0 ô g 1 pξ ˚q pΨξ ˚`1q " Ψf Ψ pξ ˚q pΨξ ˚`1q ô f Ψ pξ ˚q " g 1 pξ ˚q Ψ . (6.14)
Partie II: Non-asymptotic concentration inequalities

Recall now that we want to maximize over the ξ s.t. ξ Ñ `8, ξΨ Ñ a ? Γ n Ñ0 0. Indeed, from the proof of Lemma 4, we saw that for such a choice, we obtain the expected Gaussian concentration, namely P pλ n q " a ?

Γn Ñ n 0 ´a2 4 Ãn
.

From the computations of Lemma 8 in Appendix 8, we have: Observe as well that this choice yields:

g 1 pξq " ξÑ`8 8 3 5 ξ 2 `op 1 ξ 2 q. ( 6
ρ ´1 " 2 1{2 Ψ 3 3{2 ? Ψ " 1 2 B1{2 n a Ã3{2 n ? Γ n . ( 6

.18)

' Now we will study the the case (b) "super Gaussian deviations" for which a ? Γn Ñ n `8. Note that we cannot expect a Gaussian regime in this case. In fact, for Ψ going to infinity (see definition (6.13)), by (6.14), to get a Gaussian regime at a maximizer ξ ˚of Λ, we have from (6.11) that f Ψ pξ ˚q " g 1 pξ˚q Ψ has to remain separated from 0 when a ? Γn Ñ `8. Since, in this case Ψ Ñ n `8, this imposes to consider points ξ Ñ 0 in order to exploit the asymptotic behaviour (see again Lemma 8 for more details):

g 1 pξq " ξÑ0 2 1{3 3ξ 2 3 `1 `op1q ˘. (6.19)
In other words, from (6.19), we expect that there is a constant K ˚ą 0 s.t.

f Ψ pξ ˚q " ξ ˚Ñ0 2 1{3 3ξ 2{3 ˚Ψ ě K ˚. So ξ ˚ď C Ψ 3{2 Ñ ΨÑ`8 0. Now, |f Ψ pξ ˚q| " | gpξ˚q Ψξ`1 | ď |gpξ ˚q| Ñ ξ˚Ñ0 0.
This means that it is impossible to stay in a Gaussian regime.

We now still look at the optimal ξ ˚Ñ 0 which allows to stay at "the biggest possible regime". Thenceforth, we will estimate ξ ˚directly from the map f Ψ defined in (6.12):

f Ψ pξq " ξÑ0 2 1{3 ξ 1{3 Ψξ `1 p1 `op1qq ": f Ψ,0 pξqp1 `op1qq. (6.20)
It can be directly checked that arg max ξPR `fΨ,0 pξq " 1 2Ψ . We therefore get:

ξ ˚" ΨÑ`8 1 2Ψ `op 1 Ψ q " a ? Γn Ñ`8 4 Ã3 n Γ n 27 Bn a 2 `op Γ n a 2 q Ñ 0. (6.21)
From (6.11) and (6.20), we get:

min ρą1 P pλ n q ď a ? Γn Ñ n `8 ´32 2 4 Ãn a 2 2 1{3 ξ 1{3 Ψξ ˚`1 p1 `op1qq " (6.21) ´32 2 4 a 2 Ãn p 1 2 `1q Ψ ´1{3 p1 `op1qq ď (4.15) ´32 2 4 a 2 Ãn p 1 2 `1q p 2 3 Ã3 n Γ n 3 3 Bn a 2 q 1{3 p1 `op1qq " a ? Γn Ñ n `8 ´a4{3 Γ 1{3 n 2 2 B1{3 n p1 `op1qq (3.31)
" a ?

Γn Ñ n `8 ´a4{3 Γ 1{3 n 2}σ} 2{3 8 rϑs 2{3 1 p1 `op1qq.
From equations (4.9) and (6.13), the choice (6.21) yields:

ρ ´1 " ξΨ " 1 2 `op1q. (6.22)
Remark 15 (Controls of the parameters pλ, ρq for super Gaussian deviations)

We give here some useful estimates to control the remainder terms in the truncation procedure associated with unbounded innovations, see proof of Lemma 6. They specify the behaviour of the quantity ρλn Γn . In the regime of super Gaussian deviations, identities (4.10), (6.13) and the choice (6.21) (i.e. ξ ˚" 1 2Ψ ) yields:

ρ 2 λ 2 n Γ 2 n -CΨ ξ 2{3 ˚`p1 `a1 `ξ˚q 1{3 `p1 ´a1 `ξ˚q 1{3 ˘2 looooooooooooooooooooooooooomooooooooooooooooooooooooooon -2 1{3 Ψ 2{3 Ñ a ? Γn Ñ`8 `8.
Remark 16 Theorems 3 and 6 are actually a consequence of the more general following result which has a real importance for numerical applications. Indeed, for a given n P N, we have to control the non-asymptotic error in Theorems 3 and 6 . Furthermore, in Section 7 (see Remark 18) we will see that for θ P p 1 3 , 1s , for "reasonable" n (e.g. n " 5 ¨10 4 in the following Section 7) and a (« 1) we are already "out" of the Gaussian deviations regime, namely a ? Γn -1. This illustration justifies the interest of optimizing over ρ for Gaussian deviations and super Gaussian deviations.

Theorem 7 Let the assumptions of Theorem 3 be in force. For θ P p 1 3 , 1s, there exist explicit non-negative sequences pc n q ně1 and pC n q ně1 , respectively increasing and decreasing for n large enough, with lim n C n " lim n c n " 1 s.t. for all n ě 1 for all a ą 0,

P " | a Γ n ν n pAϕq| ě a ‰ ď 2 C n exp `cn P min pa, Γ n , ρq ˘,
where ρ ą 1 and

P min `a, Γ n , ρ ˘" ´pρ ´1q 1{3 ρ ? Γ n Φ n pa, ρq 8 p3a ´aΓ n pρ ´1q 1{3 Ãn Φ n pa, ρqq, with Φ n pa, ρq :" ´a ? Γ n Bn ``a 2 B2 n Γ n `pρ´1q
`2 Ãn 3 Bn

˘3˘1 2 ¯1 3 `´a ? Γ n Bn ´`a 2 B2 n Γ n `pρ´1q
`2 Ãn 3 Bn

˘3˘1 2 ¯1 3 (6.23) and r A n :" qνp|σ ˚∇ϕ| 2 q 2 `en and r B n " q 3 q 4 ´q}σ} 2 8 }∇ϑ} 2 8 2 `en ¯, (6.24)
where q :" qpnq ą 1, q :" qpnq ą 1, q :" qpnq with q, q q Ñ n 1 and e n is an explicit sequence going to 0.

The results of Theorem 3 explicitly follow taking (see for more details the proof of Lemma 7):

ρ ´1 " 1 2 B1{2 n a Ã3{2 n ? Γ n , for a ? Γ n Ñ n 0, ρ ´1 " 1 2 , for a ? Γ n Ñ `8.
Remark 17 It is natural to wonder if it is possible to get a sharper variance for super Gaussian deviations. Thereby it would be tempting to bootstrap Lemma 3. Such an iteration would lead to optimize polynomials of higher degrees. Recall from Lemma 4, that we already have to handle a polynomial of order 4 in the current setting. This illustrates that for very large deviations the highest term dominates and deteriorates the concentration. This is intrinsic to our approach. Such a phenomenon would even more pregnant when iterating the procedure the polynomial of higher degree yields, for a certain very large deviation regime, concentration bounds that become closer and closer to the exponential. However, bootstrapping might allow to improve the constants in the successive deteriorated concentration regimes. As indicated in Remark 16, this could be useful for some numerical purposes.

Numerical Results

Here, we have chosen to highlight the possible absence of non-degeneracy assumption for our results. To oversimplify, simulations are done with r " d " 1, X 0 and U 1 follow the standard normal distribution. Naturally, for a better convergence speed, we take θ « 1 3 , precisely θ " 1 3 `1 1000 . For this first example, we choose ϕ (solution of the Poisson equation). We take ϕ " σ " cos and for all x P R, bpxq " ´x 2 . By this pick, we compute numerically νp|σ ˚∇ϕ| 2 q « 0.1515 and νp|σ| 2 q « 0.4171 (that we provide here for comparison with the previous results in Chapter 3), with the same parameters (θ " 1 3 `1 1000 , n " 5 ¨10 4 and M C " 10 4 ).

Heed, for a non trivial test function ϕ, with our method, we cannot choose functions b and σ ‰ 0 canceling at the same point (0 here). Otherwise, the Poisson equation associated with the carré du champ source, Aϑ " |σ ˚∇ϕ| 2 ´νp|σ ˚∇ϕ| 2 q, would imply that ´νp|σ ˚∇ϕ| 2 q " 0, then ∇ϕ " 0, ν almost surely.

Let us now check that the Confluence Conditions (D p α ) are satisfied. For p P r1, 2q, we have for all

x P R d , ξ P R d A Dbpxq `Dbpxq 2 ξ, ξ E `1 2 r ÿ j"1 ´pp ´2q |xDσ ¨j pxqξ, ξy| 2 |ξ| 2 `|Dσ ¨j ξ| 2 " ´1 2 ξ 2 `1 2 sin 2 pxqξ 2 pp ´1q. (7.1)
So, for p " 3 2 , we directly obtain:

A Dbpxq `Dbpxq 2 ξ, ξ E `1 2 r ÿ j"1 ´p 3 2 ´2q |xDσ ¨j pxqξ, ξy| 2 |ξ| 2 `|Dσ ¨j ξ| 2 " ´1 2 ξ 2 `1 4 sin 2 pxqξ 2 ď ´1 4 ξ 2 ": ´αξ 2 . (7.2)
Note that, we have chosen a diffusion coefficient σ which degenerates on tkπ, k P Zu. However, thanks to the smoothness of the diffusion parameters, we can still here apply Lemma 6 in Chapter 3 which gives us a pointwise gradient bound of the solution of the Poisson problem in the current degenerate context. In other words:

rϑs 1 ď r|σ ˚∇ϕ| 2 s 1 α " 4rcos 2 sin 2 s 1 " 4 sup xPR pcosp2xq sinp2xqq " 2.
Hence, this inequality leads us to approximate rϑs 1 by 2. Pay attention that the control of the Lipschitz constant rϑs 1 is important for the super Gaussian deviations. Like illustrated in Remarks 16 and 18, this regime appears "sooner" than we might expect.

From Theorem 3, the function a Þ Ñ g n paq :" lnpPr ? Γ n |ν n pAϕq| ě asq is s.t. for a ą 0:

g n paq ď ´cn a 2 2νp|σ ˚∇ϕ| 2 q `lnp2C n q,
where pc n q ně1 and pC n q ně1 are sequences respectively increasing and decreasing for n large enough, with lim n C n " lim n c n " 1.

For Figure 4.1, the simulations have been performed for n " 5 ¨10 4 and the probability estimated by Monte Carlo simulation for M C " 10 4 realizations of the random variable ?

Γ n |ν n pAϕq|. The corresponding 95% confidence intervals have size at most of order 0.0016. We introduce the functions:

Spaq :" ´a2 2νp|σ ˚∇ϕ| 2 q , S sup paq :" ´a2 2}σ} 2 8 }∇ϕ} 2 8 .
Like in Theorem 7, we take

P min pa, Γ n , ρq " ´pρ ´1q 1{3 ρ ? Γ n Φ n pa, ρq 8 p3a ´pρ ´1q 1{3 Ãn Φ n pa, ρq a Γ n q,
where Ãn , Bn and Φ n pa, ρq are defined in (6.24). Through our numerical results, we take e n " 0.

We set also:

ρ 0 :" 1 `1 2 B1{2 n a Ā3{2 n ? Γ n , ρ 8 :" 3 2 .
We recall here that ρ and ρ 8 respectivly correspond to the optimal values of ρ in the Gaussian deviations and super Gaussian deviations (see Lemma 7). Eventually, we introduce:

P n,0,8 paq :" min ´Pmin pa, Γ n , ρ 0 q, P min pa, Γ n , ρ 8 q ¯, P n paq :" min ρą1 P min pa, Γ n , ρq.

Note that, the function P n,0,8 takes into account the multi-regime competition. We have estimated P n paq by a mesh method for ρ P p1, 2q and for a grid with 5 ¨10 5 steps.

From the above notations, we add the subscript σ to mean that we change νp|σ ˚∇ϕ| 2 q into }∇ϕ} 2 8 νp}σ} 2 q, i.e.

S σ paq " ´a2 2}∇ϕ} 2 8 νp}σ} 2 q , P n,σ paq :" min ρą1 P min,σ pa, Γ n , ρq, and we have changed Ãn into Ãσ,n :"

}∇ϕ} 2 8 νp}σ} 2 q 2 .
The quantities with subscript σ are those associated with the results in Chapter 3, recalled in the previous Theorem 2, where the variance is less sharp than the constants appearing in Theorems 3, 6 and 7. Thus, we can compare our main results with Remark 10 of Chapter 3 which is a weakened form of Theorem 7 where the carré du champ is changed into }∇ϕ} 2 8 νp}σ} 2 q like in Theorem 2. .1 reveals that the asymptotic curve S is much less sharp with respect to the realizations g n than our main estimations P n and P n,0,8 . In fact, these latter are very close to the realization g n . This claim enhances the significance of controlling finely, non-asymptotically, the deviation of the empirical measure.

In this plot, we can see that our pick of ρ for P n,0,8 , set in Lemma 7, is very close to the numerical optimization of P n over ρ. Nevertheless, observe that for a ą 0.5, P n,0,8 paq and P n paq slightly differ. It means that progressively the regime goes from Gaussian deviations (i.e. a ? Γn Ñ 0) to intermediate Gaussian deviations (i.e. a ? Γn " Op1q). Hence, the importance of optimizing globally the function ρ Þ Ñ P min pa, Γ n , ρq (appearing in (3.35)) in all regimes.

Remark 18 Remark that for the graphic 4.1, we chose n " 5 ¨10 4 , but for θ « 1 3 ,

?

Γ n « 37 and for θ « 1 2`0.5 , ?

Γ n « 26. In other words, for a « 1 we have intermediate Gaussian deviations as emphasized by the graphic. Hence the importance of the study of both regimes, Gaussian deviations ( a ? Γn Ñ 0) and super Gaussian deviations ( a ? Γn Ñ `8).

Appendix: Computation of asymptotic analysis

In this section, we perform asymptotic analysis for the map g 1 defined in (6.12) in proof of Lemma 4. We recall that for all ξ P R:

gpξq " ξ 1{3 ´p1`a1 `ξq 1{3 `p1´a1 `ξq 1{3 ¯´1´ξ 1{3 2 `p1`a1 `ξq 1{3 `p1´a1 `ξq 1{3 ˘¯. Lemma 8 g 1 pξq " ξÑ0 2 1{3 3ξ 2 3
`1 `op1q ˘, g 1 pξq "

ξÑ`8 8 3 5 ξ 2 `op 1 ξ 2 q.

Proof: Denote hpξq :" ξ 1{3 `p1 `?1 `ξq 1{3 `p1 ´?1 `ξq 1{3 ˘, so gpξq " hpξqp1´h pξq 2 q. Differentiating, we get:

h 1 pξq "
p1 ´?1 `ξq 1 3 `p1 `?1 `ξq

1 3 3 ξ 2 3 `ξ 1 3 p 1 6 ? 1 `ξ p1 `?1 `ξq 2 3 ´1 6 ? 1 `ξ p1 ´?1 `ξq 2 3 q " `1 ´?1 `ξ˘1 3 ``1 `?1 `ξ˘1 3 3 ξ 2 3 `1 6 ? 1 `ξ p1 ´?1 `ξq 2{3 ´p1 `?1 `ξq 2{3 ξ 2{3 . (a) For ξ Ñ 0, hpξq " ξÑ0 2 1{3 ξ 1{3 `opξ 1{3 q,
and

h 1 pξq " ξÑ0 2 1{3 3ξ 2 3 `ξ 1 3 p 1 6 ˆ22{3 ´22{3 ξ 2{3 q `op 1 ξ 2{3 q " 2 1{3 3ξ 2 3 `op 1 ξ 2{3 q,
which yields that

g 1 pξq " h 1 pξq `1 ´hpξq ˘" ξÑ0 2 1{3 3ξ 2 3
´`op 1 ξ 2{3 q.

(b) For ξ Ñ `8, In order to estimate g 1 we need to do a Taylor expansion up to the third order:

hpξq " ξÑ`8 ξ 1{3 p1 `ξq 1{6 `p1 `1 ? 1 `ξ q 1{3 ´p1 ´1 ? 1 `ξ q 1{3 " ξ 1{3 p1 `ξq 1{6 ´1 `1 3 ? 1 `ξ ´1 3 2 p1 `ξq `2 ˆ5 3 3 3!p1 `ξq 3{2 ´p1 ´1 3 ? 1 `ξ ´1 3 2 p1 `ξq ´2 ˆ5 3 3 3!p1 `ξq 3{2 q `op 1 ξ 3{2 q " ξ 1{2 `1 `1 6ξ `op 1 ξ q ˘`2 3 ? 1 `ξ `10 3 4 p1 `ξq 3{2 `op 1 ξ 3{2 q " ξ 1{2 `1 `1 6ξ `op 1 ξ q ˘`2 3 ? ξ ´1 3ξ 3{2 `10 3 4 p1 `ξq 3{2 `op 1 ξ 3{2 q " ξ 1{2 `2 3 ? ξ `1 3 2 ξ 3{2 ´1 3ξ 3{2 `10 3 4 p1 `ξq 3{2 `op 1 ξ 3{2 q " 2 3 ´8 81ξ `op 1 ξ q.
Differentiating the above expression, we get:

h 1 pξq " ξÑ`8 8 81ξ 2 `op 1 ξ 2 q, g 1 pξq " h 1 pξq `1 ´hpξq ˘" ξÑ0 `8 81ξ 2 `op 1 ξ 2 q ˘`1 ´2 3 `8 81ξ `op 1 ξ q " 16 3 5 ξ 2 ´8 3 5 ξ 2 `op 1 ξ 2 q " 8 3 5 ξ 2 `op 1 ξ 2 q.
Chapter 5

Extension to SDEs driven by a Poisson compound process

Abstract: In this article we approximate the invariant distribution ν of an ergodic Jump Diffusion driven by the sum of a Brownian motion and a Compound Poisson process with sub-Gaussian jumps. We first construct an Euler discretization scheme with decreasing time steps, particularly suitable in cases where the driving Lévy process is a Compound Poisson. This scheme is similar to those introduced by Lamberton and Pagès in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] for a Brownian diffusion and extended by Panloup in [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic differential equation driven by a lévy process[END_REF] to the Jump Diffusion with Lévy jumps. We obtain a non-asymptotic Gaussian concentration bound for the difference between the invariant distribution and the empirical distribution computed with the scheme of decreasing time step along a appropriate test functions f such that f ´νpf q is is a coboundary of the infinitesimal generator.

Introduction

Setting

Let pX t q tě0 be a d-dimensional càdlàg process solution of the stochastic differential equation:

dX t " bpX t qdt `σpX t qdW t `κpX t ´qdZ t .
(1.1)

where b : R d Ñ R d , σ : R d Ñ R d b R r and κ : R d Ñ R d b R r are Lipschitz continuous, pW t q tě0 is
a Wiener process of dimension r, and pZ t q tě0 is a R r -valued compound Poisson process (CPP), Z t " ř Nt k"1 Y k , where pY k q kPN are i.i.d. r -dimensional random vectors with common distribution π on BpR r q and pN t q tě0 is a Poisson process, independent of pY k q kPN . The processes pW t q tě0 and pZ t q tě0 are assumed to have the same dimension for the sake of simplicity. Moreover, pN t q tě0 , pY k q kPN and pW t q tě0 are independent and defined on a given filtered probability space pΩ, G, pG t q tě0 , Pq. We assume that b, σ, and κ satisfy a suitable Lyapunov condition (assumption pL V q in Section 1.3) which ensures the existence of an invariant distribution ν of pX t q tě0 (see [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic differential equation driven by a lévy process[END_REF]). For the sake of simplicity we also assume the uniqueness of the invariant distribution. We refer to [START_REF] Masuda | Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps[END_REF] under irreductibility and Lyapunov conditions for the existence and uniqueness of the invariant distribution for a diffusion driven by Lévy process.

The aim of this paper is to establish a non-asymptotic bound on the probability of the deviation ν n pf q ´νpf q, where ν n is an appropriate empirical measure such that lim nÑ8 ν n pf q " νpf q a.s. for all suitable test functions f . The algorithm that we define in this article is based on an Euler-like discretization scheme with decreasing time step pγ n q ně1 s.t. lim n γ n " 0. Lamberton and Pagès first introduced such a scheme in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] for a Brownian diffusion. They showed that the empirical measure of their scheme converges to the invariant measure of the diffusion and that it satisfies the Central Limit Theorem. The decreasing steps allows to the empirical measure to directly converge towards the invariant one. If we choose a constant time step γ k " h ą 0 in the scheme, the expected ergodic theorem is ν n pf q a.s.

ÝÑ n ν h pf q " ş R d f pxqν h pdxq, where ν h is the invariant distribution of the scheme which is supposed to converge toward the invariant measure of the diffusion (1.1) when h Ñ 0 (see e.g. For more details about this approach we refer to [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF], [START_REF] Talay | Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme[END_REF] and [START_REF] Malrieu | Concentration inequalities for Euler Schemes[END_REF]).

Next, Panloup in [START_REF] Panloup | Computation of the invariant measure of a levy driven SDE: Rate of convergence[END_REF] and [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic differential equation driven by a lévy process[END_REF] adapted the algorithm of [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] to the Jump Diffusion with Lévy jumps [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic differential equation driven by a lévy process[END_REF] and also showed the convergence and the Central Limit Theorem for the empirical measure in this case. In the same way as the questions of the convergence of the empirical measure ν n or of its limiting distribution, the natural question is that of the nature of the deviations ν n pf q´νpf q along appropriate test functions f . In the case of the Brownian diffusion this question was considered in Chapters 3 and 4. Note that in the Brownian diffusion case the innovations of the Euler scheme are designed in order to "mimic" Brownian increments, hence it is natural to assume that they satisfy some Gaussian Concentration property (assumption (GC) in Section 1.3). In particular this Gaussian Concentration property is satisfied by Gaussian or symmetric Bernoulli law. Taken as an assumption on the Brownian innovations of the scheme, it allows to show a non-asymptotic Gaussian Concentration bound for the probability of the deviations of ν n pf q from νpf q, see Chapters 3 and 4 with sharp constants. The deviation ν n pf q ´νpf q is evaluated along the functions f such that f ´νpf q is a coboundary of the infinitesimal generator of the diffusion.

When the diffusion contains Lévy jumps, it is not generally expected that these deviations will show a Gaussian behaviour. But such a behaviour seems natural if we suppose that the driving Levy process is a Compound Poisson process and the jump size vectors pY k q kPN satisfy a Gaussian Concentration property (GC). In this paper, we focus on this situation. Before giving its precise formulation we need to introduce some notations. First of all, we introduce our discretization scheme. In general, for a Euler scheme corresponding to a Jump Diffusion with Lévy jumps, one has to define a numerically computable jump vectors designed to "mimic" the increments of the driving Lévy process. In most cases, the increments of a Lévy process are not numerically computable, that is why it is important to propose different ways to approximate these increments according to the nature of the driving Lévy process. In this paper we introduce a scheme (S) particularly suitable in the case where a driven Lévy process is a Compound Poisson. Note that our scheme is close to the scheme (C) of [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic differential equation driven by a lévy process[END_REF]. Like in the previously mentioned articles, we denote time steps pγ k q kě1 , and for all n ě 0, we define:

X n`1 " X n `γn`1 bpX n q `?γ n`1 σpX n qU n`1 `κpX n qZ n`1 , (1.2)
where X 0 is an R d valued random variables such that X 0 P L 2 pΩ, F 0 , Pq, pU n q ně1 is an i.i.d. sequence of centered random variables matching the moments of the Gaussian law on R r up to order three, independent of X 0 . Furthermore, for all n ě 1 we put

Z n :" B n Y n , (1.3)
where pB n q ně1 are one-dimensional independent Bernoulli random variables, independent of X 0 , pU n q ně1 and pY n q ně1 , s.t. B n (law)

" Bernpµγ n q, where µ is an intensity of the Poisson process driving the CPP pZ t q tě0 . Without loss of generality we can suppose from now on that µ " 1. The choice (1.3) of the innovations Z n , n P N is motivated by the following heuristic reasoning: Z n has to "mimic" the increment of the CPP Z t " ř Nt k"1 Y k on the small time interval of the length γ n . The probability that the CPP does not jump on this interval is equal to expp´γ n q " 1 ´γn `opγ n q, and if the CPP jumps on this interval, it will most probably have only one jump. Hence we approximate the increment N γn of the CPP by a t0, 1u random variable with the probability of 1 equal to γ n .

We also introduce the empirical (random) measure of the scheme: for all A P BpR d q (where BpR d q is the Borel σ-field on R d ):

ν n pAq :" ν n pω, Aq :"

ř n k"1 γ k δ X k´1 pωq pAq ř n k"1 γ k . (1.4)
Obviously, to study long time behaviour, we have to consider steps pγ k q kě1 such that the current time of the scheme Γ n :"

ř n k"1 γ k Ñ n `8.
We recall as well that γ k Ó k 0. We suppose that both jump amplitudes pY n q ně1 and Brownian innovations pU n q ně1 satisfy a Gaussian concentration (see further the assumption (GC)). As we already mentioned, the aim of the paper is to show that this assumption implies a non-asymptotic Gaussian Concentration inequality for the probability of the deviations of ν n pf q from νpf q (see Theorem 2, Section 2). The main argument in the proof of Theorem 2 is the fact that the (GC) property of jumps sizes Y k , k P N, permits to show the similar Gaussian Concentration property for the jump innovations Z k , k P N. This result is given in Proposition 1. However the Gaussian Concentration property of jump innovations depends on the dimension of the jump heights. This dependence survives in the main Theorem 2 giving Gaussian Concentration of the deviation of ν n from ν.

The current chapter is organized as follows. In Section 1.2, we introduce some useful notations. The assumptions required for our main results are outlined in Section 1.3. In this part, we formulate a Gaussian concentration property of the jump innovation Z n , the proof is given in Section 2.4. We state in Section 1.4 some already known results connected with the approximation scheme. Our main results are in Section 2, and the demonstration is located in Section 2.3. Section 3 is dedicated to the analysis of the exponential integrability of Lyapunov function. Technical lemmas are stated in Section 2.2, but their proofs are postponed to Section 4. Eventually, we propose a numerical illustration of our main result in Section 5.

General notations

We set for any step sequence pγ n q ně1 : @ P p0, `8q, Γ p q n :"

n ÿ k"1 γ k , Γ n :" n ÿ k"1 γ k " Γ p1q n ,
where Γ n corresponds to the current time, hence Γ n Ñ nÑ`8

`8. For the sake of simplicity, from now on, the time step sequence will have the form: γ n -1 n θ with θ P p0, 1s, where for two sequences pu n q nPN , pv n q nPN the notation u n -v n means that Dn 0 P N, DC ě 1 s.t. @n ě n 0 , C ´1v n ď u n ď Cv n .

Henceforth, C will be a non negative constant, and pe n q ně1 , pR n q ně1 will be deterministic sequences s.t. e n Ñ n 0 and R n Ñ n 1, that may change from line to line. The constant C as well as the sequences pe n q ně1 , pR n q ně1 depend on known parameters appearing in the hypotheses set in Section 1.3 (which will be called (A) further). Other possible dependencies will be explicitly specified.

We denote by I m , m P td, ru the identity matrix of dimension m.

Through the article, for any smooth enough function f , for k P N we will denote D k f the tensor of the k th derivatives of f . Namely We define for pp, d, mq P N 3 , C p pR d , R m q the space of p-times continuously differentiable functions from R d to R m . Furthermore, for f P C p pR d , R m q, p P N, we set for β P p0, 1s the Hölder modulus:

D k f " pB i 1 . . . B i k f q 1ďi 1 ,...,i k ďd . Yet, for a multi-index α P N d 0 :" pN Y t0uq d , we set D α f " B α 1 x 1 . . . B α d x d f : R d Ñ R.
rf ppq s β :" sup x‰x 1 ,|α|"p |D α f pxq ´Dα f px 1 q| |x ´x1 | β ď `8,
where α P N d is a multi-index of length p, namely |α| :" ř d i"1 α i " p. In other words, in the above definition, the | ¨| in the numerator is the usual absolute value. We will also use the notation rrn, pss, pn, pq P pN 0 q 2 , n ď p, for the set of integers being between n and p.

Let us introduce for k P N 0 , β P p0, 1s and m P t1, d, d ˆru the Hölder spaces C k,β pR d , R m q :" tf P C k pR d , R m q: @α P N d , |α| P rr1, kss, sup

xPR d |D α f pxq| ă `8, rf pkq s β ă `8u, C k,β b pR d , R m q :" tf P C k,β pR d , R m q: }f } 8 ă `8u. (1.5)
In the above definition, we denote for all bounded mapping ζ : R d Ñ R m , m P t1, d, dˆru, the uniform norm }ζ} 8 :" sup xPR d }ζζ ˚pxq} with }ζpxq} " Tr pζζ ˚pxqq 1{2 , where for M P R m b R m , TrpM q is the trace of M . In particular, } ¨} is the Fröbenius norm * . Practically, with these notations, C k,β pR d , R m q stands for the subset of C k pR d , R m q whose elements have bounded derivatives up to order k and β-Hölder continuous k th derivatives. In particular, for k " 0, the space of Lipschitz continuous functions from R d to R m is denoted by C 0,1 pR d , R m q.

For a given Borel function f :

R d Ñ E, where E can be R, R d , R d b R r , R d b R d , we set for k P N 0 : f k :" f pX k q.
Moreover, for k P N 0 , we denote F k :" σ `X0 , pU j , Z j q jPrr1,kss ˘and r F k :" σ `X0 , pU j , Z j q jPrr1,kss , U k`1 ˘.

(1.6)

Eventually, we define the infinitesimal generator associated with the diffusion (1.1) which writes for all ϕ P C 2 pR d q and x P R d :

Aϕpxq " bpxq∇ϕpxq `1 2 T r `σσ ˚pxqD 2 ϕpxq ˘`ż R d pϕpx `κpxqyq ´ϕpxqq πpdyq ": r Aϕpxq `żR d pϕpx `κpxqyq ´ϕpxqq πpdyq, (1.7)
where π stands for the distribution of Y 1 , and r A is the infinitesimal generator of the continuous part of the diffusion.

Hypotheses

We assume the following set of hypothesis about the coefficients of the SDE (1.1) and the parameters of the scheme (1.2): (C1) The first value of the scheme X 0 is sub-Gaussian: there exists λ 0 P R ˚such that (GM) The sequences of random variables pU n q ně1 and pY n q ně1 are respectively i.i.d., such that E rU 1 s " E rY 1 s " 0; ErpU i 1 U j 1 q 1ďi,jďr s ": ErU b2 1 s " I r , ErY b2 1 s " I r ; ErpU i 1 U j 1 U k 1 q 1ďi,j;kďr s ": ErU b3 1 s " 0 † . Also, pU n q ně1 , pY n q ně1 and X 0 are independent.

@λ ă λ 0 , E rexppλ|X 0 | 2 qs ă `8. (
(GC) We say that r.v. G P L 1 satisfies Gaussian concentration property, if for every Lipschitz continuous function g : R r Ñ R and every λ ą 0:

E " exppλgpGqq ‰ ď exp ˆλE rgpGqs `λ2 rgs 2 1 2 ˙, (1.8)
We assume that U 1 and Y 1 satisfy the Gaussian concentration property.

(L V ) We assume the following Lyapunov like stability condition:

There exists a non-negative function V : R d ÝÑ rv ˚, `8q with v ˚ą 0 such that i) V is a C 2 continuous function s.t. }D 2 V } 8 ă `8, and lim |x|Ñ8 V pxq " `8.

ii) There is C V P p0, `8q such that for all x P R d :

|∇V pxq| 2 `|bpxq| 2 ď C V V pxq.
iii) There exist α V ą 0, β V P R `such that for all x P R d , AV pxq ď ´αV V pxq `βV .

(U) There is a unique invariant distribution ν to equation (1.1).

(S) We assume that the sequence pγ k q kě1 is small enough, namely for all k ě 1,

γ k ď 1 2 minp2, α V 4 `rbs 1 C V 8 `p ? C V rbs 1 2 `?C V }D 2 V }8 2 `CV 4 q ? C V ˘q,
where C V is given by the assumption (L V ). For β P p0, 1s, we introduce:

(T β ) We choose a test function ϕ such that i) ϕ P C 3,β pR d , Rq,
ii) x Þ Ñ x∇ϕpxq, bpxqy is Lipschitz continuous, we further assume that there exist C V,ϕ ą 0 s.t. for all x P R d : iii) |ϕpxq| ď C V,ϕ p1 `aV pxqq.

Remark 1 Under the assumption (C0) the equation (1.1) admits a unique non-explosive solution, cf [App09] (Theorem 6.2.9.).

Remark 2

The assumption (GC) is central for this paper. Note that the laws N p0, I r q and p 1 2 pδ 1 `δ´1 qq br , i.e. for Gaussian or symmetrized Bernoulli increments which are the most commonly used sequences for the sub-Gaussian innovations, satisfy (GC). Moreover, inequality (1.8) yields that for all r ě 0, Pr|U 1 | ě rs ď 2 expp´r 2 2 q (sub-Gaussian concentration of the innovation, see e.g. [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]).

Remark 3

The assumption (L V ) together with (C2) ensure, following [START_REF] Panloup | Computation of the invariant measure of a levy driven SDE: Rate of convergence[END_REF] (Proposition 1) the existence of at least one invariant distribution of the SDE (1.1). Note that this Lyapunov assumption (L V ) is equivalent to the similar Lyapunov assumption for the continuous part of the equation (1.1). Indeed, using second order Taylor expansion, the fact that πp¨q " 0 and πp| ¨|2 q " r ă 8, we get that

| ż R r pV px `κpxqyq ´V pxqqπpdyq| ď }κ} 2 8 r}D 2 V } 8 2 .
Hence the condition iii) of pL V q is equivalent that the generator of the diffusion without jumps satisfies AV pxq ď ´r α V V pxq `r β V , (1.9)

with αV " α V , βV " β V `}κ} 2 8 r}D 2 V }8 2
. For more information about Lyapunov function existence, see e.g. [START_REF] Gadat | Long time behaviour and stationary regime of memory gradient diffusions[END_REF].

Moreover, it is classic to see that this assumption constraints the drift coefficient b to be under a linear map. Indeed, this is the consequence of the fact that the Lyapunov function V has to be lower than the square norm, i.e. there exist constants K, c ą 0 such that for all |x| ě K, |V pxq| ď c|x| 2 and hence using ii) of (L V ) |bpxq| ď ? C V c|x|.

Remark 4

The assumption (T β ) allows to substantially simplify the proof of our results. The condition iii) is natural for ϕ Lipschitz continuous, which is obviously lower than the square root of a quadratic function (potentially V ). Whilst the condition ii) is a direct consequence if ϕ is the solution of the Poisson equation:

Aϕ " f, (1.10)
where f P C 1,β pR d , Rq s.t. νpf q " 0. If σ, κ P C 1,β b pR d , R dˆr q, b P C 1,β pR d , R d q and ϕ P C 3,β pR d , Rq, then both sides of the following identity:

x∇ϕ, by " f ´1 2 Tr `ΣD 2 ϕ ˘´ż R d
`ϕp¨`κp¨qyq ´ϕp¨q ˘πpdyq, are Lipschitz continuous.

From now on, we identify assumptions (C0), (C1), (C2), (GM), (GC), (L V ), (U), (S) and (T β ) for some β P p0, 1s to (A). Except when explicitly indicated, we assume throughout the paper that assumption (A) is in force.

We suppose that the step sequence pγ k q kě1 is taken such that γ k -k ´θ, θ P p0, 1s. This pick yields for any ě 0, Γ

p q n -n 1´ θ if θ ă 1, Γ p q n -lnpnq if θ " 1 and Γ p q n -1 if θ ą 1.
The corner stone of our analysis is the fact that the jumps innovations pZ n q nPN inherit the Gaussian concentration property of pY n q nPN : Proposition 1 (Gaussian concentration of the jumps innovation) Let g : R r Ñ R uniform Lipschitz continuous function, ε P p0, 1q and ρprq " ? rp3 `rq `1 8 `4 expp ? r `1 `r{2q.

(1.11)

Then for all 0 ă λ ă ε 6rgs 1 ρprq the following inequality holds for all n P N E exppλgpZ n qq ď exppλEgpZ n q `λ2 γ n p1 `r `εqrgs 2 1 2 q.

(1.12)

Remark 5 Let us point out that the concentration inequality is only valid for λ on a compact set. This constraint is due to the difficulty to approximate a compound Poisson process which has actually a sub-exponential tail (and not a sub-Gaussian one).

The proof of this proposition is given in Subsection 2.4.

Existing results

The natural next question concerns the rate of that convergence. In a Brownian diffusion framework, a Central Limit Theorem (CLT) was established by Lamberton and Pagès [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] for functions f of the form f ´νpf q " r Aϕ, namely f ´νpf q is a coboundary for r A, where r A denotes the continuous part of the infinitesimal generator (see (1.7) further). This choice of functions class comes from the characterization of the invariant distribution ν by a solution in the distribution sense of the stationary Fokker-Planck equation: r A ˚ν " 0 (where r A ˚stands for the adjoint of r A). In other words, for all functions ϕ P C 2 pR d , Rq, we have νp r

Aϕq " ş R d r Aϕpxqνpdxq " 0. In [START_REF] Panloup | Computation of the invariant measure of a levy driven SDE: Rate of convergence[END_REF], the author also provided the rate of convergence through a Central Limit Theorem (CLT) for the already mentioned general scheme:

Theorem 1 (CLT) Under (C2), (U) and (L V ), if E|Z t | 2p ă `8 for p ą 2, if ErU b3 1 s " 0, Er|U 1 | 2p s ă `8 and lim n Γ p2q n ?
Γn " 0 then for all function ϕ P C 3,1 pR d , Rq we have the following results (with pLq denoting the weak convergence):

a Γ n ν n pAϕq pLq ÝÑ N `0, σ 2 ϕ ˘, (1.13) with σ 2 ϕ :" ż R d
`|σ ˚∇ϕ| 2 pxq `żR d |ϕpx `κpxqyq ´ϕpxq| 2 πpdyq ˘νpdxq.

(1.14)

In the Brownian diffusion context (κ " 0), under some confluence and non-degeneracy or regularity assumptions, in Chapter 3, we established suitable derivatives controls for the Poisson problem (e.g. Schauder estimates). With a compound Poisson process, we think that a similar analysis may work. It will be a future research. Let us mention [START_REF] Priola | Pathwise uniqueness for singular sdes driven by stable processes[END_REF] for some Schauder estimates for Poisson equation, with a potential, associated with a SDE purely driven by stable processes but with a constant drift.

In [START_REF] Honoré | Non-Asymptotic Gaussian Estimates for the Recursive Approximation of the Invariant Measure of a Diffusion[END_REF], the authors have established a non-asymptotic Gaussian concentration with κ " 0 there are explicit sequences c n ď 1 ď C n converging to 1 such that for all n P N, for all a ą 0 and γ k -k ´θ, θ P p 1 3 , 1s,

Pr a Γ n ν n pAϕq ě as ď C n exp ˆ´c n a 2 2}σ} 2 8 }∇ϕ} 8 ˙, (1.15)
which is our goal for a diffusion with jump contributions. Remark that, in Chapter 4 a non-asymptotic Gaussian concentration was established with the asymptotically best constants for a particular large deviation called "Gaussian deviations" therein. In other words, for a " op ? Γ n q:

Pr a Γ n ν n pAϕq ě as ď C n exp ˆ´c n a 2 2νp|σ ˚∇ϕ| 2 q ˙.
(1.16)

In this present work, we aim to obtain a Gaussian deviations bound like (1.15) for the scheme (1.2). To do so, we will perform the so-called martingales increments method which was exploited successfully by Frikha and Menozzi [START_REF] Frikha | Concentration bounds for stochastic approximations[END_REF]. It was also the backbone of the analysis in Chapters 3 and 4. Here, we adapt these techniques for the stochastic differential equation (1.1) driven by the compound Poisson with Jump heigh sizes satisfying Gaussian concentration.

Main results

Result of non-asymptotic Gaussian concentration

Our main result is stated below.

Theorem 2 For θ P p 1 2`β , 1s, β P p0, 1s, assume that (A) is in force. For all positive sequence pχ n q ně1 with lim nÑ8 χ n " 0, there are two non-negative sequences pc n q ně1 and pC n q ně1 satisfying c n Õ 1 and C n OE 1 as n Ñ 8, such that for all n P N, a ą 0 , satisfying a ď χ n

?

Γn Γ p2q n , the following bound holds:

P " | a Γ n ν n pAϕq| ě a ‰ ď 2 C n exp `´c n a 2 2σ 2 8 ˘,

Strategy

For the analysis of ν n pAϕq, we will first perform an appropriate Taylor expansion (equation (2.3) below). An expansion of this kind is standard in this context, and analogous decompositions were already used in Chapter 3, Chapter 4 and in [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic differential equation driven by a lévy process[END_REF], [START_REF] Panloup | Computation of the invariant measure of a levy driven SDE: Rate of convergence[END_REF] with a jump component. It can be viewed as a kind of Itô formula for Euler scheme, because it permits to write the difference ϕpX n q ´ϕpX 0 q as a sum of a martingale, a term involving the generator and a remainder term. Recall that F k " σ `X0 , pU j , Z j q jPrr1,kss ˘, k P N ˚. Let us define the contributions of the decomposition of ν n pAϕq in the following lemma.

ψ ϕ k pX k´1 , U k q :" ? γ k σ k´1 U k ¨∇ϕpX k´1 `γk b k´1 q `γk ż 1 0 p1 ´tqTr ´D2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1 ´D2 ϕpX k´1 `γk b k´1 qΣ k´1 ¯dt, ∆ ϕ k pX k´1 , U k q :" ψ ϕ k pX k´1 , U k q ´E" ψ ϕ k pX k´1 , U k q|F k´1 ‰ , r ∆ ϕ k pX k´1 , Z k q :" ϕpX k´1 `κk´1 Z k q ´ϕpX k´1 q ´γk ż R r
" ϕpX k´1 `κk´1 yq ´ϕpX k´1 q ‰ πpdyq.

(2.1) Moreover, we define the remainder contributions in the decomposition of ν n pAϕq.

D k,ϕ 2,b pX k´1 q :" γ k ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt, D k,ϕ 2,Σ pX k´1 q :" γ k 2
Tr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σk´1 ¯, D k,ϕ j pX k´1 , U k , Z k q :" ϕpX k q ´ϕpX k´1 `γk b k´1 `?γ k σ k´1 U k q ´pϕpX k´1 `κk´1 Z k q ´ϕpX k´1 qq .

(2.2)

Lemma 1 (Local decomposition of the empirical measure) For all ϕ P C 2 pR d , Rq, k P N ˚the following decomposition holds:

ϕpX k q ´ϕpX k´1 q " γ k AϕpX k´1 q `∆ϕ k pX k´1 , U k q `r ∆ ϕ k pX k´1 , Z k q `Rϕ k pX k´1 , U k , Z k q, (2.3) where R ϕ k pX k´1 , U k , Z k q :" D k,ϕ 2,b pX k´1 q`D k,ϕ 2,Σ pX k´1 q`D k,ϕ j pX k´1 , U k , Z k q`E " ψ ϕ k pX k´1 , U k q|F k´1 ‰ .
(2.4) Furthermore, we have the following properties: ii) For all k P N ˚, ∆ ϕ k pX k´1 , U k q and r ∆ ϕ k pX k´1 , Z k q are martingale increments with respect to F k , namely:

i) For all k P N ˚, the functions u Þ Ñ ∆ ϕ k pX k´1 , uq and z Þ Ñ r ∆ ϕ k pX k´1 ,
E " ∆ ϕ k pX k´1 , U k q ˇˇF k´1 ‰ " 0, E " r ∆ ϕ k pX k´1 , Z k q ˇˇF k´1 ‰ " 0.
The proof of Lemma 1 is given in Section 4. Now we introduce the martingales associated to these martingale increments:

M ϕ n :" n ÿ k"1 ∆ ϕ k pX k´1 , U k q, Ă M ϕ n :" ÿ k"1 r ∆ ϕ k pX k´1 Z k q.
(2.5) Summing (2.3) over k we obtain the following global decomposition of the empirical measure:

ν n pAϕq " ´1 Γ n pM ϕ n `Ă M ϕ n `Rϕ n q, (2.6)
where we denoted

R ϕ n :" n ÿ k"1 R ϕ k pX k´1 , U k , Z k q ´pϕpX n q ´ϕpX 0 qq. (2.7)
Using the definition (2.2) we can write

R ϕ n " ´Lϕ n `Dϕ 2,b,n `Dϕ 2,Σ,n `Dϕ j,n `Ḡ ϕ n , with L ϕ n :" ϕpX n q ´ϕpX 0 q, D ϕ 2,b,n :" n ÿ k"1 D k,ϕ 2,b pX k´1 q, D ϕ 2,Σ,n :" n ÿ k"1 D k,ϕ 2,Σ pX k´1 q, D ϕ j,n :" n ÿ k"1 D k,ϕ j pX k´1 , U k , Z k q, Ḡϕ n :" n ÿ k"1 E rψ ϕ k pX k´1 , U k q|F k´1 s.
(2.8)

In the proof of Theorem 2, we need some key results stated below. The proofs of all these statements are postponed to Section 4. The main contribution in the decomposition (2.6) is given by the martingales M ϕ n and Ă M ϕ n . Their analysis is given with the help of the Gaussian Concentration inequality (1.8) and (1.12), trough the following lemma:

Lemma 2 (Concentration of the martingale increments) Let ∆ ϕ n and r ∆ ϕ n given by (2.1).

i) For all Λ ą 0 we have

E " exp ˆ´Λ Γ n ∆ ϕ n pX n´1 , U n q ˙ˇˇF n´1  ď exp ˆγn }σ} 2 8 }∇ϕ} 2 8 Λ 2 2Γ 2 n ˙.
ii) For all 0 ă ε ă 1, n P N ˚, for all Λ ą 0 s.t. Proposition 2 Under (A), there is a constant c V :" c V ppAqq ą 0 such that for all λ P r0, c V s:

I 1 2 V :" sup ně0 E " exp `λ? V pX n q ˘‰ ă `8.
(2.9)

Remark 7
In particular, we easily see that for all λ P r0, c V s and ξ P r0, 1 2 s:

I ξ V :" sup ně0 E " exp `λV pX n q ξ ˘‰ ă `8.
(2.10)

Note that for κ " 0 (purely continuous case), the integrability of exp `λV pX n q ξ ˘is available until ξ " 1 (see Proposition 1 in [START_REF] Honoré | Non-Asymptotic Gaussian Estimates for the Recursive Approximation of the Invariant Measure of a Diffusion[END_REF]). The lost of integrability is the consequence of the bound condition over λ in the Gaussian Concentration result of Proposition 1.

We have the following results for the initial term appearing in (2.3) which is handled thanks to the below result.

Lemma 3 (Initial term) For all Λ ą 0 s.t. Λ Γn ă c V 2C V,ϕ : E exp ´Λ |L ϕ n | Γ n ¯ď exp ´2C V,ϕ Λ Γ n ¯pI 1 2 V q 2C V,ϕ Λ c V Γn with c V , I 1 2
V given in Proposition 2.

Next the last remainders are controlled as following:

Lemma 4 (Remainders) For all Λ ą 0 s.t. Λ Γn ă 2c V `}∇ϕ}8rbs 1 `rx∇ϕ,bys 1 ˘?C V 1 Γ p2q n : E exp ´Λ Γ n ˇˇD ϕ 2,b,n ˇˇ¯ď pI 1{2 V q Λ `}∇ϕ}8rbs 1 `rx∇ϕ,bys 1 ˘?C V Γ p2q n 2c V Γn .
(2.11)

We also have, for all Λ ą 0 s.

t. Λ Γn ď c V }σ} 2 8 }D 3 ϕ}8 ? C V 1 Γ p2q n : E exp ´Λ Γ n ˇˇD ϕ 2,Σ,n ˇˇ¯ď pI 1{2 V q }σ} 2 8 }D 3 ϕ}8C 1 2 V ΛΓ p2q n 2c V Γn .
(2.12)

Lemma 5 (Bounds for the Conditional expectations) With the notations (2.8), and for θ P p 1 2`β , 1s, we have that

| Ḡϕ n | ? Γ n a.s. ď α n :" rϕ p3q s β › › σ › › p3`βq 8 E " |U 1 | 3`β ‰ p1 `βqp2 `βqp3 `βq Γ p 3`β 2 q n ? Γ n Ñ n 0.
Remark 8 The strongest condition over θ comes from this remainder term. Indeed, for

θ ă 2 3`β , Γ p 3`β 2 q n ? Γn -n 1´p2`βqθ 2
which goes to 0 if and only if θ ą 1 2`β . Whilst for the other remainders, for θ ă 1 2 , we need to have Γ p2q n ?

Γn -n 1´3θ 2
Ñ n 0 which is implied by θ ą 1 3 .

Now, let us deal with the remainder term D j,n due to the jump vector pZ k q kě1 . Lemma 6 (Remainder term due to the jumps) If 0 ă Λ Γn ă 1 12}κ}8}∇ϕ}8ρprq , then we have:

E " exp ˆΛ Γ n D ϕ j,n ˙ ď exp ˆp Λ ? Γ n `Λ2 Γ n qe n ˙, (2.13)
where we recall that e n " e n ppAqq, n ě 1, is a sequence such that e n Ñ n 0.

The proof of this lemma is one of the most intricate of this article, we the decided to postponed it to the end of Section 4.

Proof of our main result

Proof: Theorem 2 Through the following analysis, we deal with P "? Γ n ν n pAϕq ě a ‰ . The term Bhouou P "?

Γ n ν n pAϕq ď ´a‰ can be handled readily by symmetry. From notations introduced in (2.6), ν n pAϕq " ´1 Γn pR ϕ n `M ϕ n `Ă M ϕ n q. The idea is now to write for a, λ ą 0:

P " a Γ n ν n pAϕq ě a ‰ ď exp `´aλ ? Γ n ˘E" exp `´λ Γ n pR ϕ n `M ϕ n `Ă M ϕ n q ˘ı ď exp `´aλ ? Γ n ˘E" exp `´qλ Γ n pM ϕ n `Ă M ϕ n q ¯ı1{q E " exp `pλ Γ n |R ϕ n | ˘ı1{p , (2.14)
for 1 p `1 q " 1, p, q ą 1. We will choose later p " ppnq Ñ n `8 slowly enough, which implies that q " qpnq Ñ 1. Let λ ą 0. Recall that

R ϕ n " ´Lϕ n `Dϕ 2,b,n `Dϕ 2,Σ,n `Dϕ j,n `Ḡ ϕ n
By Cauchy-Schwarz inequality, we obtain:

E " exp `pλ Γ n |R ϕ n | ˘ı1{p ď ˆE exp ´2pλ Γ n ˇˇL ϕ n ˇˇ¯˙1 2p ˆE exp ´4pλ Γ n ˇˇḠ ϕ n ˇˇ¯˙1 4p ˆE exp ´8pλ Γ n ˇˇD ϕ 2,b,n ˇˇ¯˙1 8p ˆE exp ´16pλ Γ n ˇˇD ϕ 2,Σ,n ˇˇ¯˙1 16p ˆE exp ´16pλ Γ n ˇˇD ϕ j,n ˇˇ¯˙1 16p . (2.15)
We recall that all the long of our analysis, C ą 0 denotes a generic constant, pR n q ně1 and pe n q ně1 are generic non-negative sequences, depending on coefficients of assumption (A), which may change frome line to line, such that lim nÑ8 R n " 1, lim nÑ8 e n " 0. For the term associated with L n in (2.15), if 2pλ Γn ă c V 2C V,ϕ , by Lemma 3 we can write:

´E exp `2pλ |L ϕ n | Γ n ˘¯1 2p ď exp ´2C V,ϕ λ Γ n ¯pI 1 2 V q 2C V,ϕ λ c V Γn " exppC λ Γ n q.
(2.16)

From Lemma 5, with α n defined there and by Young inequality we obtain:

´E exp `4pλ Γ n | Ḡϕ n | ˘¯1 4p ď exp `λ ? Γ n α n ˘ď exp `λ2 Γ n 2p `α2 n p 2 ˘" R n exp `λ2 Γ n e n ˘.
(2.17)

In the last equality, R n " exppα 2 n p n {2q and e n " 1{2p n . Recall that α n Ñ n 0. We need to choose p " p n Ñ n `8. We choose p n such that p n α 2 n Ñ n 0. For the term involving

D ϕ 2,Σ,n from (2.15), if 16pλ Γn ď 2c V Γ p2q n }σ} 2 8 }D 3 ϕ}8 ? C V
, using Lemma 4, we can write

´E exp `16pλ Γ n ˇˇD ϕ 2,Σ,n ˇˇ˘¯1 16p ď pI 1{2 V q }σ} 2 8 }D 3 ϕ}8C 1 2 V λΓ p2q n c V Γn " exppC λΓ p2q n Γ n q " exppC λ ? Γ n e n q,
(2.18) where in the last equality we take e n " Γ p2q n ? Γn and recall that for all θ P p

ˇˇ˘¯1 4p ď pI 1{2 V q λ `}∇ϕ}8rbs 1 `rx∇ϕ,bys 1 ˘?C V Γ p2q n 2c V Γn " exppC λΓ p2q n Γ n q " exppC λ ?
Γ n e n q.

(2.19) Finally, Lemma 6 yields that if 0 ă 16pλ Γn ă 1 12}κ}8}∇ϕ}8ρprq , then:

E " exp ˆ16pλ Γ n D ϕ j,n ˙ ď exp ˆλ ? Γ n e n `λ2
Γ n e n ˙.

(2.20)

We gather (2.16), (2.17), (2.18), (2.19) and (2.20) into (2.15) and finally from (2.14) we obtain:

P " a Γ n ν n pAϕq ě a ‰ ď exp `´aλ ? Γ n ˘E" exp `´qλ Γ n pM ϕ n `Ă M ϕ n q ˘‰ 1 q exp `p λ ? Γ n `λ2 Γ n qe n ˘Rn .
(2.21) Now, let us control the martingale terms thanks to Lemma 2. Let 0 ă ε ă 1, and λ ą 0 s.t. qλ{Γ n ă Cε 6}κ}8}∇ϕ}8ρprq , we recall that ρprq is defined in (1.11). Thanks to Lemma 2 and the independence of Z n and U n conditionally to F n´1 we can write . Finally qλ n {Γ n ă Cε n is required to apply Lemma 2. We recall that we will choose p Ñ 8 and q Ñ 1 and finally ε Ñ 0.

E exp ´´qλ Γ n pM ϕ n `Ă M ϕ n q " E " exp `´qλ Γ n pM ϕ n´1 `Ă M ϕ n´1 q ˘E" exp `´qλ Γ n p∆ ϕ n pX n´1 , U n q `r ∆ ϕ n pX n´1 , Z n q ˘ˇˇF n´1 ı  ď E " exp `´qλ Γ n pM ϕ n´1 `Ă M ϕ n´1 q ˘exp ˆq2 λ 2 γ n 2Γ 2 n p}σ} 2
We recall also that from the statement of the theorem a " apnq can depends of n in such a way that a ? Γn ď χn

Γ p2q n Ñ n 0. But if q Ñ 1,

Proof of the Gaussian property of the jump innovation

Proof: Proposition 1 Suppose first that g : R r Ñ R is Lipschitz continuous with rgs 1 ď 1. The case of rgs 1 ą 1 follows by considering λ " λrgs 1 and g " g rgs 1 . We suppose w.l.o.g. that gp0q " 0. We recall that thank to the definition (1.3) the law of Z n is the same that the law of B n Y, where B n is a Bernoulli variable with parameter γ n , independent of the random vector Y with distribution π on BpR r q. We will establish first that for all ε P p0, 1q and 0 ă λ ă ε ρprq (see (1.11)) we have E exppλgpZ n qq ď exppλEgpZ n q `λ2 γ n p1 `pEgpY qq 2 `εq 2 q.

(2.25)

Denote for this proof m g :" EgpY q. Using (GC) property of Y we can write E exppλgpZ n qq " γ n E exppλgpY qq `p1 ´γn q ď γ n exppλm g `λ2 {2q `p1 ´γn q.

Denote

∆ exp n :" γ n exppλm g `λ2 2 q `p1 ´γn q ´exppλγ n m g `λ2 γ n p1 `m2 g `εq 2 q.

(2.26)

Here, the second exponential corresponds to the right hand side in (2.25). We will show that ∆ exp n ă 0. Indeed, let us develop the difference ∆ exp n by power series expansion:

∆ exp n " γ n pλm g `λ2 2 q ´pγ n λm g `λ2 γ n p1 `m2 g `εq 2 q `1 2 γ n pλm g `λ2 2 q 2 ´1 2 pγ n λm g `γn λ 2 p1 `m2 g `εq 2 q 2 `Qpλq " ´γn λ 2 2 pε `γn m 2 g q `γn λ 3 2 `mg p1 ´γn p1 `m2 g `εqq 1 8 γ n λ 4 `1 ´γn p1 `m2 g `εq 2 ˘`Qpλq,
where

Qpλq :" γ n ÿ kě3 1 k! pλm g `λ2 2 q k ´ÿ kě3 1 k! pγ n λm g `γn λ 2 p1 `m2 g `εq 2 q k .
In particular, using γ n ď 1 from (S), and ε ă 1, we can roughly estimate:

∆ exp n ď ´γn λ 2 2 ε `γn λ 3 2 `|m g |p3 `m2 g q ˘`1 8 γ n λ 4 `Qpλq, (2.27)
Because g is 1-Lipschitz continuous and from the assumption (GM) we obtain:

m 2 g " |EgpY n q| 2 ď E|gpY n q| 2 " E|gpY n q ´gp0q| 2 ď rgs 1 E}Y n } 2 " rgs 1 r ÿ k"1 |Y k | 2 ď r.
Using again γ n ď 1, λ ď 1, ε ď 1 we get

Qpλq ď γ n ÿ kě3 λ k k! p ? r `1 2 q k `γn ÿ kě3 λ k γ k´1 n k! p ? r `p2 `rq 2 q k ď 2γ n λ 3 expp ? r `1 `r{2q
Thus combined with (2.27) gives :

∆ exp n ď ´γn λ 2 2 ε `γn λ 3 2 `|m g |p3 `m2 g q ˘`1 8 γ n λ 4 `2γ n λ 3 expp ? r `1 `r{2q ď γ n λ 2 {2 ´ε `λ? rp3 `rq `1 8 λ 2 `4λ expp ? r `1 `r{2q ( ď γ n λ 2 {2 ´ε `λρprq ( .
which is negative if λρprq ă ε, with ρprq defined in (1.11). This proves the (2.25). Together with the inequality m 2 g ď r this proves the concentration inequality in the case rgs 1 ď 1.

Remark

In Chapter 3, the exponential moments of the Lyapunov function was used to control the remainder terms of the decomposition of the empirical measure. In this article, we also use the Lyapunov function for this purpose. But our framework yields more constraints over the analysis. Namely, we cannot directly use exppCV n q which is not a priori integrable. Indeed, let us consider the Compound Poisson process Zt :" ř Nt k"1 Ỹk where p Ỹk q kPN is an i.i.d. sequence of a standard normal variables independent of N t wich follows a Poisson law, which is the typical jump random variables that we aim to approximate. Conditionally to N t , Zt " N p0, N t q. So if we choose the Lyapunov function to be the standard quadratic map, i.e. for all x P R d , V pxq " |x| 2 `1. We obtain in fine:

ErexppλV p Zt qs " e λ E " Erexppλ Z2 t q|N t s ‰ ě e λ Er ż R d exppλ|y t | 2 q expp´| y t | 2 2N t q dy t p2πN t q 1{2 1 Ntě1 s, this is integrable if almost surely N t ă 1
2λ which is not true for λ ą 0.

Proof: Proposition 2 Preliminarily to the proof of this proposition, we write some useful controls thanks to assumption (L V ), for all x P R d ,

|∇ ? V pxq| " | ∇V pxq 2 ? V pxq | ď ? C V 2 , (3.1)
}D 2 ? V pxq} " } D 2 V pxq 2 ? V pxq ´∇V ∇V ˚pxq 4V 3{2 pxq } ď }D 2 V } 8 2 ? v ˚`C V 4 ? v ˚.
(3.2) ˘ı ": γ n A ? V pX n´1 q `V1 pX n´1 q `V2 pX n´1 q `V3 pX n´1 , U n q `V4 pX n´1 , U n , Z n q,

where for all x P R d , the first term is such that:

V 1 pxq " γ n ż 1 0 xbpxq, ∇V 2 ? V px `tγ n bpxqq ´∇V 2 ? V pxqydt " γ n ż 1 0 xbpxq ´bpx `tγ n bpxqq, ∇V 2 ? V px `tγ n bpxqqydt `γn ż 1 0 xb, ∇V 2 ? V ypx `tγ n bpxqq ´xb, ∇V 2 ? V ypxqdt ": V 1 1 pxq `V2 1 pxq. (3.6)
Because b is supposed to be Lipschitz continuous and thanks to (L V ) ii), we readily writes:

V 1 1 pxq ď γ 2 n rbs 1 4 |bpxq| ż 1 0 |∇V | 2 ? V px `tγ n bpxqqdt ď γ 2 n rbs 1 C V 8 ? V pxq. (3.7)
Whilst the next term is more subtle. Indeed, observe that thanks to (L V ) ii) the following term is bounded:

|∇xb, ∇V 2 ? V y| ď ˇˇDb ∇V 2 ? V | `ˇb D 2 V 2 ? V | `| p∇V qp∇V q ˚b 4V 3 2 ˇď ? C V rbs 1 2 `?C V }D 2 V } 8 2 `C3{2 V 4 ": C (3.8) , (3.8)
which directly yields again thanks to (L V ) ii) that

V 2 1 pxq ď γ 2 n C (3.8) ż 1 0 t|bpxq|dt ď γ 2 n C (3.8) ? C V ? V pxq 2 . (3.9)
Hence plugging (3.7) and (3.9) into (3.6) implies that:

V 1 pxq ď γ 2 n p rbs 1 C V 8 `C(3.8) a C V q ? V pxq. (3.10)
The second term is handled by (3.2):

V 2 pxq ď γ n 2 }σ} 2 8 p }D 2 V } 8 2 ? v ˚`C V 4 ? v ˚q.
(3.11)

The third term satisfies the following identity:

V 3 px, U n q " ? γ n σpxqU n ¨∇? V px `γn bpxqq `γn ż 1 0 p1 ´tqTr ´D2 ? V px `γn bpxq `t? γ n σpxqU n qσpxqU n b U n σpxq ˚¯dt (3.2) ď ? γ n σpxqU n ¨∇? V px `γn bpxqq `γn 2 p }D 2 V } 8 2 ? v ˚`C V 4 ? v ˚q}σ} 2 8 |U n | 2 ": V 1 3 px, U n q `V2 3 pU n q, (3.12)
and the last term is:

V 4 px, U n , Z n q " ?
V `x `γn bpxq `?γ n σpxqU n `κpxqZ n ˘´? V `x `γn bpxq `?γ n σpxqU n γn π `aV px `κpxq¨q ´aV pxq (3.1) ď ?

V px `γn bpxq `?γ n σpxqU n `κpxqZ n q ´?V px `γn bpxq `?γ n σpxqU n q `γn }κ} 8

? C V πp| ¨|q 2 ": V 1 4 px, U n , Z n q `γn }κ} 8 ? C V πp| ¨|q 2 . ( 3
? V pX n q ´?V pX n´1 q ď ´γn r α V 2 ? V pX n´1 q ´β 1 V s `γ2 n p rbs 1 C V 8 `C(3.8) a C V q ? V pX n´1 q `γn 2 }σ} 2 8 p }D 2 V } 8 2 ? v ˚`C V 4 ? v ˚q `V3 pX n´1 , U n q `γn }κ} 8 ? C V πp| ¨|q 2 `V1 4 pX n´1 , U n q ď ´γn α V 4 ? V pX n´1 q `γn βV `V1 3 pX n´1 , U n q `V2 3 pU n q `V1 4 pX n´1 , U n , Z n q, (3.14) for γ n ď α V 4p rbs 1 C V 8 `C(3.8) ? C V q " α V 4 `rbs 1 C V 8 `p ? C V rbs 1 2 `?C V }D 2 V }8 2 `CV 4 q ? C V which
corresponds to assumption (S) and

βV :" β1 V `1 2 }σ} 2 8 p }D 2 V } 8 2 ? v ˚`C V 4 ? v ˚q `}κ} 8 ? C V πp| ¨|q 2 .
We control the contribution of V 1 3 pX n´1 , U n q and V 2 3 pU n q (defined in (3.12)) in the exponential moment of a V pX n q by the Gaussian concentration hypothesis (GC) and V 1 4 pX n´1 , U n , Z n q (see (3.13)) thanks to Proposition 1. We define for all x P R d and λ ą 0:

I 1 pλ, xq :" E " exp `λV 1 3 px, U n q ˘ı, I 2 pλq :" E " exp `λV 2 3 pU n q ˘ı, I 3 pλ, xq :" E " exp `λV 1 4 px, U n , Z n q ˘‰ .
Indeed, by (GC), we first write:

I 1 pλ, xq ď exp ´λ2 γ n |σ ˚pxq∇ ? V px `γn bpxqq| 2 2 ¯(3.1) ď exp ´λ2 γ n C V }σ} 2 8 4 ¯. (3.15)
Next, it is well known that under (GC), for all c ă 1 2 , I c :" E rexppc|U n | 2 qs ă `8. So we have for all λ ă

2c ? v pC V {2`}D 2 V }8q}σ} 2
8 γ 1 , by Jensen's inequality:

I 2 pλq ď " E exp `c|U n | 2 ˘ıλγn 2c p }D 2 V }8 2 ? v ˚`C V 4 ? v ˚q}σ} 2 8 " exp ´γn lnpI c q λ 2c p }D 2 V } 8 2 ? v ˚`C V 4 ? v ˚q}σ} 2 8
¯.

(3.16) Now, let us deal with the third term I 3 pλ, xq. First of all, note that from definition in (3.13) and (3.1) the function z Þ Ñ V 1 4 pX n´1 , U n , zq is }κ} 8 ? C V -Lipschitz continuous. Furthermore, we have that

|ErV 1 4 px, U n , Z n q|U n s| " γ n |ErV 1 4 px, U n , Y n q|U n s| ď γ n a C V }κ} 8 πp| ¨|q.
Hence, by the Proposition 1, and for all 0 ă λ ă 1 6}κ}8 ?

C V ρprq (see (1.11)), for the corresponding notation of Proposition 1 we take ε " 1, and we get:

I 3 pλ, xq ď E " E " exp `λV 1 4 px, U n , Z n q ˘|U n ‰‰ ď E " exp ´λErV 1 4 px, U n , Z n q|U n s `p2 `rqγ n λ 2 rV 1 4 px, U n , ¨qs 2 1 2 ¯ı ď exp `λγ n }κ} 8 a C V πp| ¨|q `p2 `rq}κ} 2 8 C V λ 2 γ n 2 ˘.
(3.17)

From now on, we assume that for all

λ ă λ V :" min ´1, λ 0 2c , 2c ? v pC V {2 `}D 2 V } 8 q}σ} 2 8 γ 1 , 1 6}κ} 8 ? C V ρprq ¯.
Gathering identities (3.14), (3.15) and (3.16), and by the Cauchy-Schwarz inequality, we obtain that for all λ ă λ V ,

E exp `λ? V pX n q " E " exp `λ? V pX n´1 q ˘E" exp `λp ? V pX n q ´?V pX n´1 qq ˘ˇF n´1 s ı ď E " exp `λr ? V pX n´1 qp1 ´αV 4 γ n q `β V γ n s ˘I1 p2λ, X n´1 q 1{2 I 2 p4λq 1{4 I 3 p4λ, X n´1 q 1{4 ı ď exp `λγ n β1 V ˘E" exp `λp1 ´γn αV q ? V pX n´1 q ˘ı,
where we have defined:

β1 V :" βV `CV }σ} 2 8 2 `lnpI c q p }D 2 V }8 2 ? v ˚`C V 4 ? v ˚q}σ} 2 8 2c `}κ} 8 a C V 2πp| ¨|q `2p2 `rq}κ} 2 8 C V , and
αV :" min `1 γ 1 , α V 4 ˘P p0, 1 γ 1 s.
So p1 ´γn αV q P r0, 1q and we deduce by Jensen inequality:

E exp `λ? V pX n q ˘ď exp `λγ n β1 V ˘E" exp `λ? V pX n´1 q ˘ıp1´γn αV q . ( 3.18)
For all λ ą 0, we introduce C V,λ :" max `Ere λ ?

V pX 0 q s, e λ β1

V { αV ˘.

In particular, we have Ere λ ?

V pX 0 q s ď C V,λ . Let us check by induction that for all n P N:

Ere λ ? V pXnq s ď C V,λ .
We deduce from (3.18) and by induction assumption that:

E exp `λ? V pX n q ˘ď exp `λγ n β1 V ˘Cp1´γn αV q V,λ ď C V,λ .
We pick c V ă λ V and the proof is completed. Remark 10 Observe also that v ˚:" inf xPR d V pxq ą 0, we have that for all pn, ξq P N ˆr0, 1 2 s, λ ă λ V pv ˚q1´ξ :

E exppλV ξ n q " E exp ´λpv ˚qξ ´Vn v ˚¯ξ loomoon ě1 ¯ď E exp `λpv ˚qξ´1 V n ˘ď C V,λpv ˚qξ´1 ă `8.
Hence, ξ P r0, 1s, λ ă λ V pv ˚q1´ξ , sup nPN E exppλ ? V ξ n q ă `8.

Proof of the Technical Lemmas

Proof: Lemma 1

For k P rr1, nss, we first write:

ϕpX k q ´ϕpX k´1 q " pϕpX k q ´ϕpX k´1 `γk b k´1 `?γ k σ k´1 U k qq `pϕpX k´1 `γk b k´1 `?γ k σ k´1 U k q´ϕpX k´1 `γk b k´1 qq `pϕpX k´1 `γk b k´1 q´ϕpX k´1 qq ": T k´1,j pϕq `Tk´1,r pϕq `Tk´1,d pϕq, (

with

T k´1,j pϕq " pϕpX k q ´ϕpX k´1 `γk b k´1 `?γ k σ k´1 U k qq T k´1,d pϕq " γ k x∇ϕpX k´1 q, b k´1 y `γk ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt, T k´1,r pϕq " ? γ k σ k´1 U k ¨∇ϕpX k´1 `γk b k´1 q `γk ż 1 0 p1 ´tqTr ´D2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1 ¯dt.
Thanks to this splitting, we are able to isolate the deterministic, the sub-Gaussian random variable approximating Brownian increments and the jump contributions. Then we proceed by Taylor expansion up to the order 2 for the function ϕ in the two last terms of the r.h.s. of (4.1),

ϕpX k q ´ϕpX k´1 q " γ k AϕpX k´1 q `ϕpX k q ´ϕpX k´1 `γk b k´1 `?γ k σ k´1 U k q ´γk ż R r pϕpX k´1 `κk´1 yq ´ϕpX k´1 qq πpdyq `rγ k ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydts `r γ k 2 pD 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 qqs `r? γ k σ k´1 U k ¨∇ϕpX k´1 `γk b k´1 q `γk ż 1 0 p1 ´tqTr ´D2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1 ´D2 ϕpX k´1 `γk b k´1 qΣ k´1 ¯dts " γ k AϕpX k´1 q `Dk,ϕ 2,b pX k´1 q `Dk,ϕ 2,Σ `ψϕ k pX k´1 , U k q `ϕpX k q ´ϕpX k´1 `γk b k´1 `?γ k σ k´1 U k q ´γk ż R r
pϕpX k´1 `κk´1 yq ´ϕpX k´1 qq πpdyq, we then obtain:

ϕpX k q ´ϕpX k´1 q " γ k AϕpX k´1 q `Dk,ϕ 2,b pX k´1 q `Dk,ϕ 2,Σ `ψϕ k pX k´1 , U k q `rϕpX k´1 `κk´1 Z k q ´ϕpX k´1 q ´γk ż R r pϕpX k´1 `κk´1 yq ´ϕpX k´1 qq πpdyqs `rϕpX k q ´ϕpX k´1 `γk b k´1 `?γ k σ k´1 U k q ´pϕpX k´1 `κk´1 Z k q ´ϕpX k´1 qqs " γ k AϕpX k´1 q `Dk,ϕ 2,b pX k´1 q `Dk,ϕ 2,Σ pX k´1 q `ψϕ k pX k´1 , U k q `r ∆ ϕ k pX k´1 , Z k q `Dk,ϕ j pX k´1 , U k , Z k q. (4.2)
Note finally, that by definition of D k,ϕ j pX k´1 , U k , Z k q in the previous expansion (4.2):

ψ ϕ k pX k´1 , U k q " ϕpX k q´ϕpX k´1 q´γ k AϕpX k´1 q´r ∆ ϕ k pX k´1 , Z k q ´`D k,ϕ 2,b pX k´1 q`D k,ϕ 2,Σ pX k´1 q`D k,ϕ j pX k´1 , U k , Z k q " ϕpX k´1 `γk b k´1 `?γ k σ k´1 U k q`ϕpX k´1 `κk´1 Z k q´γ k AϕpX k´1 q´2ϕpX k´1 q ´r ∆ ϕ k pX k´1 , Z k q ´`D k,ϕ 2,b pX k´1 q `Dk,ϕ 2,Σ pX k´1 q ˘,
hence after differentiating, we see that u Þ Ñ ψ ϕ k pX k´1 , uq and hence u Þ Ñ ∆ ϕ k pX k´1 , uq are Lipschitz continuous with a modulus bounded by ? γ k´1 }σ k´1 }}∇ϕ} 8 ď ? γ k´1 }σ} 8 }∇ϕ} 8 .

Moreover, from the definition Er∆ ϕ k pX k´1 , U k q|F k´1 s " 0 and using the definition of Z n we get Er r ∆ ϕ k pX k´1 , Z k q|F k´1 s " (4.3)

E " ϕpX k´1 `κk´1 Z k q ´ϕpX k´1 q ˇˇF k´1 ‰ ´γk ż R r
" ϕpX k´1 `κk´1 yq ´ϕpX k´1 q ‰ πpdyq " 0.

Proof:

Lemma 2 We first prove the point ii). For all ε P p0, 1q and 0 ă Λ Γn ă ε 6r r ∆npX n´1 ,¨qs 1 ρprq (ρprq set in (1.11)), thanks to Proposition 1, we have for all n P N:

E " exp ˆ´Λ Γ n r ∆ ϕ n pX n´1 , Z n q ˙ˇˇF n´1  ď exp ˆ´Λ Γ n Er r ∆ ϕ n pX n´1 , Z n q|F n´1 s `γn Λ 2 2Γ 2 n r r ∆ ϕ n pX n´1 , ¨qs 2 1 p1 `r `εq ˙. (4.4)
By definition of r ∆ ϕ n pX n´1 , Z n q in (2.1), and from (4.3) we have:

Er Λ Γ n r ∆ ϕ n pX n´1 , Z n q ˇˇF n´1 s " 0, and r r ∆ ϕ n pX n´1 , ¨qs 2 1 ď }κ} 2 8 }∇ϕ} 2 8 .
The previous control with (4.4) directly yield:

E " exp ˆΛ Γ n r ∆ ϕ n pX n´1 , Z n q ˙ˇˇF n´1  ď exp ˆγn Λ 2 2Γ 2 n }κ} 2 8 }∇ϕ} 2 8 p1 `r `εq ˙, (4.5)
with the constraint 0 ă Λ Γn ă ε 6}κ}8}∇ϕ}8ρprq .

The demonstration of the point i), is a direct consequence of the previous analysis without using Proposition 1, which yields no restriction on λ.

Proof:

Lemma 3 By assumption (T β ), we know that there exists C V,ϕ ą 0 such that for all x P R d , |ϕpxq| ď C V,ϕ `1 `aV pxq ˘, so we obtain:

E exp ´Λ |ϕpX 0 q ´ϕpX n q| Γ n ¯ď E exp ´Λ C V,ϕ p2 `aV pX 0 q `aV pX n qq Γ n ď exp ´2C V,ϕ Λ Γ n ¯«E exp ´2C V,ϕ Λ a V pX 0 q Γ n ¯ff1 2 « E exp ´2C V,ϕ Λ a V pX n q Γ n ¯ff1 2 ď exp ´2C V,ϕ Λ Γ n ¯pI 1 2 V q 2C V,ϕ Λ c V Γn .
The last inequality is obtained by Jensen's inequality for Λ Γn ă c V 2C V,ϕ and by Proposition 2.

Proof:

Lemma 4 From the definition (2.2) we can write:

D k,ϕ 2,b " γ k ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q ´∇ϕpX k´1 q, b k´1 ydt " γ k " ż 1 0 x∇ϕpX k´1 `tγ k b k´1 q, b k´1 ´bpX k´1 `tγ k b k´1 qydt `ż 1 0 `x∇ϕ, bypX k´1 `tγ k b k´1 q ´x∇ϕ, bypX k´1 q ˘dt ı .
From the boundedness of ∇ϕ, Lipschitz property of the mapping x Þ Ñ bpxq (assumption (C 0 )) and Lipschitz property of the mapping x Þ Ñ x∇ϕpxq, bpxqy (assumption (T β )), using the assumption (L V ), ii) one derives that:

|D k,ϕ 2,b | ď γ 2 k ´}∇ϕ} 8 rbs 1 `rx∇ϕ, bys 1 ¯|b k´1 | 2 ď C (4.6) γ 2 k a V k´1 , (4.6)
for C (4.6) :" p}∇ϕ} 8 rbs 1 `rx∇ϕ, bys 1 q

? C V 2 . Hence |D ϕ 2,b,n | ď n ÿ k"1 C (4.6) γ 2 k a V k´1 .
Next, by the Jensen inequality (for the exponential function with 1 Γ p2q n ř k"1 γ 2 k δ k as a measure), we deduce that:

E exp ´Λ Γ n |D ϕ 2,b,n | ¯ď 1 Γ p2q n n´1 ÿ k"1 γ 2 k E " exp ´Γp2q n Λ Γ n C (4.6) a V k´1 ¯ı ď 1 Γ p2q n n´1 ÿ k"1 γ 2 k E " exp ´cV a V k´1 ¯ıC (4.6) ΛΓ p2q n c V Γn , for Λ Γ ď c V C (4.6) Γ p2q n " 2c V p}∇ϕ}8rbs 1 `rx∇ϕ,bys 1 q ? C V Γ p2q n
ă 1, and c V is introduced in Proposition 2 which readily yields:

E exp ´Λ Γ n |D ϕ 2,b,n | ¯ď 1 Γ p2q n n´1 ÿ k"1 γ 2 k pI 1{2 V q C (4.6) ΛΓ p2q n c V Γn " pI 1{2 V q C (4.6) ΛΓ p2q n c V Γn .
For the second inequality, we first use a Taylor expansion:

|D k,ϕ 2,Σ | " γ k 2 ˇˇTr ´`D 2 ϕpX k´1 `γk b k´1 q ´D2 ϕpX k´1 q ˘Σk´1 ¯¯ˇˇˇ( 4.7) ď 1 2 }σ} 2 8 }D 3 ϕ} 8 γ 2 k |b k´1 | ď 1 2 }σ} 2 8 }D 3 ϕ} 8 a C V γ 2 k |V k´1 | 1 2 So |D ϕ 2,Σ,n | ď 1 2 }σ} 2 8 }D 3 ϕ} 8 a C V n ÿ k"1 γ 2 k |V k´1 | 1 2 .
Hence, like previously, by Jensen inequality and Proposition 2 for 0 ă Λ statisfying

Λ Γn ď 2c V }σ} 2 8 }D 3 ϕ}8 ? C V Γ p2q n ă 1 we obtain E exp ´Λ Γ n |D ϕ 2,Σ,n | ¯ď E exp ´Λ 2Γ n }σ} 2 8 }D 3 ϕ} 8 a C V n ÿ k"1 γ 2 k |V k´1 | 1 2 ď 1 Γ p2q n n ÿ k"1 γ 2 k E exp ´ΛΓ p2q n 2Γ n }σ} 2 8 }D 3 ϕ} 8 a C V |V k´1 | 1 2 ď pI 1{2 V q ΛΓ p2q n }σ} 2 8 }D 3 ϕ}8 ? C V 2c V Γn . Proof:

Lemma 5

The proof is similar to the analysis of Lemma 3 in Chapter 3. By the definition (2.1), and because U k , k P rr1, nss, has the same moments as the standard Gaussian random variable up to order three (see (GM)) we have for all k P rr1, nss:

E rψ ϕ k pX k´1 , U k q|F k´1 s " γ k ż 1 0 p1 ´tqTr ´E" D 2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1 ´D2 ϕpX k´1 `γk b k´1 qΣ k´1 |F k´1 ‰ ¯dt, where E " Tr ´D2 ϕpX k´1 `γk b k´1 `t? γ k σ k´1 U k qσ k´1 U k b U k σ k´1 ´D2 ϕpX k´1 `γk b k´1 qΣ k´1 ¯ˇˇF k´1 ı " t ? γ k ż 1 0 E " Tr ´`rD 3 ϕpX k´1 `γk b k´1 `ut ? γ k σ k´1 U k q ´D3 ϕpX k´1 `γk b k´1 qsσ k´1 U k σk´1 U k b U k σ k´1 ˘¯ˇˇˇF k´1 ı du.
Then,

|E rψ ϕ k pX k´1 , U k q|F k´1 s| ď γ k ż 1 0 p1 ´tqt 1`β rϕ p3q s β E " γ 1`β 2 k }σ k´1 } 3`β |U k | 3`β ż 1 0 u β du ˇˇF k´1 ı dt " rϕ p3q s β γ 3`β 2 k }σ k´1 } 3`β E r|U k | 3`β s p1 `βqp2 `βqp3 `βq . (4.8)
We sum over k to get the result. Proof:

Lemma 6 Recall that we have denoted for n P N 0 , F n :" σ `X0 , pU j , Z j q jPrr1,nss ˘and r

F n " F n _ σpU n`1 q. E " exp `Λ Γ n n ÿ k"1 D k,ϕ j pX k´1 , U k , Z k q ˘ı (4.9) " E " exp `Λ Γ n n´1 ÿ k"1 D k,ϕ j pX k´1 , U k , Z k q ˘E" exp `Λ Γ n D ϕ j,n pX n´1 , U n , Z n q ˘ˇˇr F n´1
ıı .

The idea is to control the last conditional expectation using Proposition 2. Recall that

D ϕ j,n pX n´1 , U n , Z n q " ϕpX n´1 `γn b n´1 `?γ n σ n´1 U n `κn´1 Z n q ´ϕpX n´1 `γn b n´1 `?γ n σ n´1 U n q
´"ϕpX n´1 `κn´1 Z n q ´ϕpX n´1 q ‰ .

Moreover, we have for all z P R r :

|∇ z D ϕ j,n pX n´1 , U n , zq| " |κ n´1 `∇ϕpX n´1 `γn b n´1 `?γ n σ n´1 U n `κn´1 zq ´∇ϕpX n´1 `κn´1 zq ˘| ď 2}κ} 8 }∇ϕ} 8 . (4.10)
Hence for all X n´1 , U n fixed the function z Ñ D n,ϕ j,ϕ pX n´1 , U n , zq is Lipschitz continuous satisfying rD ϕ j,n pX n´1 , U n , zqs 1 ď 2}κ} 8 }∇ϕ} 8 .

This estimation is used to bound Λ for which we can apply the Proposition 1. However, we need a more subtle control of the last Lipschitz modulus. Namely, using Taylor expansion we can write

|∇ z D ϕ j,n pX n´1 , U n , zq| ď |κ n´1 `∇ϕpX n´1 `γn b n´1 `κn´1 zq ´∇ϕpX n´1 `κn´1 zq ˘| `|κ n´1 `∇ϕpX n´1 `γn b n´1 `?γ n σ n´1 U n `κn´1 zq ´∇ϕpX n´1 `γn b n´1 `κn´1 zq ˘| ď }κ} 8 `?2}∇ϕ} 1 2 8 |∇ϕpX n´1 `γn b n´1 `κn´1 zq´∇ϕpX n´1 `κn´1 zq| 1 2 `?γ n }D 2 ϕ} 8 }σ} 8 |U n | ď ? γ n }κ} 8 `?2C 1 4 V }∇ϕ} 1 2 8 }D 2 ϕ} 1 2 8 V 1 4 pX n´1 q `}D 2 ϕ} 8 }σ} 8 |U n | ˘. (4.11)
Now for all Λ satisying 0 ă Λ Γn ă 1 12}κ}8}∇ϕ}8ρprq , we get

E " exp ˆΛ Γ n D ϕ j,n pX n´1 , U n , Z n q ˙ˇˇr F n´1  (4.12) ď exp ´Λ Γ n ErD ϕ j,n pX n´1 , U n , Z n q| r F n´1 s `γn Λ 2 2Γ 2 n p1 `r `εqrD ϕ j,n pX n´1 , U n , ¨qs 2 1 ď exp ˆΛ Γ n γ n ErD ϕ j,n pX n´1 , U n , Y n q| r F n´1 s `γ2 n Λ 2 2Γ 2 n ´C1 ? V pX n´1 q `C2 |U n | 2 ¯˙,
where we have denoted

C 1 :" p1 `r `εq}κ} 2 8 4 a C V }∇ϕ} 8 }D 2 ϕ} 8 , C 2 :" 2p1 `r `εq}κ} 2 8 }D 2 ϕ} 2 8 }σ} 2 8 ,
and used the following identities:

D ϕ j,n pX n´1 , U n , 0q " 0, and ErD ϕ j,n pX n´1 , U n , Z n q| r F n´1 s " γ n ErD ϕ j,n pX n´1 , U n , Y n q| r F n´1 s,
which is a consequence of the definition of Z n in (1.3).

To control ErD ϕ j,n pX n´1 , U n , Y n q| r F n´1 s we introduce for all px, yq P pR d q 2 the function:

φpx, yq :" Erϕpx `κpyqY n qs ´ϕpxq, which readily implies that:

ErD ϕ j,n pX n´1 , U n , Y n q| r F n´1 s " φpX n´1 `γn b n´1 `?γ n σ n´1 U n , X n´1 q ´φpX n´1 , X n´1 q.
The idea in the following is to apply the expansion of Lemma 1 with κ " 0 to the function x Ñ φpx, yq , which also corresponds to the expansion of Lemma 1 in Chapter 3 for diffusion without jumps. If κ " 0, then X n " X n´1 `γn b n´1 `?γ n σ n´1 U n , we can write using (2.3) and the definition (2.1) of ∆ φ k with Ă M φ n " D φ j,n " 0:

φpX n , X n´1 q´φpX n´1 , X n´1 q " γ n r A φpX n´1 q`D φ 2,b,n pX n´1 q`D φ 2,Σ,n pX n´1 q`ψ φ n pX n´1 , U n q. (4.13)

All the terms in the right have obviously the same properties as the corresponding terms in the similar expansion of ϕ given by Lemma 1 with κ " 0. In particular, for all y P R d , the map φp¨, yq is Lipschitz continuous with }∇ φp¨, yq} 8 ď 2}∇ϕ} 8 , }D 2 φp¨, yq} 8 ď 2}D 2 ϕ} 8 , }D 3 φp¨, yq} 8 ď 2}D 3 ϕ} 8 . (4.14) Furthermore, D φ 2,b,n and D φ 2,Σ,n satisfy similar inequalities as (4.6) and (4.7) where ϕ is replaced by φ. We directly have thanks to the definitions (1.7), (2.2), identities (4.6), (4.7) and (4.14):

γ n | r A φpX n´1 q| " γ n |xb n´1 , ∇ φpX n´1 qy `1 2 T r `σσ ˚D2 φpX n´1 q ˘| ď γ n p2 a C V ? V pX n´1 q}∇ϕ} 8 `}σ} 2 8 }D 2 ϕ} 8 q ď γ n C 3 ? V pX n´1 q, |D φ 2,b,n pX n´1 q| " γ n | ż 1 0 x∇ φpX n´1 `tγ n b n´1 q ´∇ φpX n´1 q, b n´1 ydt| ď γ 2 n ´}∇ φ} 8 rbs 1 `rx∇ φ, bys 1 ¯|b n´1 | 2 ď γ 2 n 2C (4.6) ? V pX n´1 q, |D φ 2,Σ,n pX n´1 q| " ˇˇγ n 2 Tr ``D 2 φpX n´1 `γn b n´1 q ´D2 φpX n´1 q ˘Σn´1 ˘ď γ 2 n }σ} 2 8 }D 3 ϕ} 8 C 1 2 V ? V pX n´1 q,
where C 3 :" 2 ? C V }∇ϕ} 8 `}σ} 2 8 }D 2 ϕ} 8 pv˚q ´1 2 . Therefore, from the previous controls and using (4.12) and (4.13), we get that there is a constant C " CppAqq ą 0 such that:

E " exp ˆΛ Γ n D ϕ j,n pX n´1 , U n , Z n q ˙ˇˇr F n´1  (4.15) ď exp ´CΛ Γ n γ n `γn ? V pX n´1 q `ψ φ n pX n´1 , U n q ˘`CΛ 2 Γ 2 n γ 2 n `?V pX n´1 q `|U n | 2 ˘¯.
Next, the idea is to separate the unbounded contribution from the terms involving p ?

V pX k qq kPrr1,nss , p|U k | 2 q kPrr1,nss by a global Cauchy-Schwarz inequality:

E « exp ˜Λ Γ n n ÿ k"1 D k,ϕ j pX k´1 , U k , Z k q ¸ff (4.16) ď E « exp ˜2Λ Γ n n ÿ k"1 " D k,ϕ j pX k´1 , U k , Z k q ´Cγ 2 k p1 `2Λ Γ n q ? V pX k´1 q ´2CΛ Γ n γ 2 k |U k | 2  ¸ff1 2 ˆE" exp ˜2 n ÿ k"1 " Cγ 2 k ˆΛ Γ n `2Λ 2 Γ 2 n ˙?V pX k´1 q `2CΛ 2 Γ 2 n γ 2 k |U k | 2 ı ¸ı1 2 ": Υ 1 2 1 ˆΥ1 2 2 .
Again by the Cauchy-Schwarz inequality, we get :

Υ 1{2 2 ď E " exp ´4Cp Λ Γ n `2Λ 2 Γ 2 n q n ÿ k"1 γ 2 k ? V pX k´1 q ¯ı1 4 E " exp ´4CΛ 2 Γ 2 n n ÿ k"1 γ 2 k |U k | 2 ¯ı1{4 . (4.17)
We control the second expected value under condition

4CΛ 2 γ 2 1 Γ 2 n ă 1 using Jensen inequality: E " exp ´4CΛ 2 Γ 2 n n ÿ k"1 γ 2 k |U k | 2 ¯ı1{4 ď n ÿ k"1 γ 2 k Γ p2q n E " exp ´4CΛ 2 Γ p2q n Γ 2 n |U k | 2 ¯ı1{4 ď E " exp ´|U 1 | 2 4 ¯ı16CΛ 2 Γ p2q n Γ 2 n . ( 4

.18)

Because U 1 satisfies (GC), Erexpp |U 1 | 2 4 qs ă `8. We handle the first expectation in (4.17) by the same method, using Jensen inequality under condition

Λ Γn ă c V 2CΓ p2q n , Λ 2 Γ 2 n ă c V 16CΓ p2q n
and Proposition 2 and we obtain: Γn Ñ 0, we deduce that:

E " exp ´4C ˆΛ Γ n `2Λ 2 Γ 2 n ˙n ÿ k"1 γ 2 k ? V pX k´1 q ¯ı 1 4 ď 1 Γ p2q n n ÿ k"1 γ 2 k E " exp ´4CΓ p2q n `Λ Γ n `2Λ 2 Γ 2 n ˘?V pX k´1 q ¯ı 1 4 ď pI 1{2 V q CΓ p2q n `Λ c V Γn `2Λ 2 c V Γ 2 n q . ( 4
Υ 1{2 2 ď exp ´p Λ ? Γ n `Λ2 Γ n qe n ¯. (4.20)
The first term in (4.16), Υ 1 , is handled by identity (4.15).

Υ 1 ď E " exp ´2Λ Γ n γ n ψ φ n pX n´1 , U n q `2Λ Γ n n´1 ÿ k"1 " D k,ϕ j pX k´1 , U k , Z k q ´Cγ 2 k `1 `2Λ Γ n ˘?V pX k´1 q ´2CΛ 2 Γ 2 n γ 2 k |U k | 2 ‰ ¯ı " E " E " exp ´2Λ Γ n γ n ψ φ n pX n´1 , U n q|F n´1 ı ˆexp ´2Λ Γ n n´1 ÿ k"1 " D k,ϕ j pX k´1 , U k , Z k q ´Cγ 2 k `1 `2Λ Γ n ˘?V pX k´1 q ´2CΛ 2 Γ 2 n γ 2 k |U k | 2 ‰ ¯ı pGCq ď exp ´2Λ Γ n γ 1`3 `β 2 n 2rϕ p3q s β › › σ › › p3`βq 8 E " |U 1 | 3`β ‰ p1 `βqp2 `βqp3 `βq `4Λ 2 Γ 2 n 4γ 3 n }σ} 2 8 }∇ϕ} 2 8 `E" exp ´2Λ Γ n n´1 ÿ k"1 " D k,ϕ j pX k´1 , U k , Z k q ´Cγ 2 k `1 `2Λ Γ n ˘?V pX k´1 q ´C2Λ 2 Γ 2 n γ 2 k |U k | 2 ‰ ¯ı.
The last inequality is a consequence of the bound (4.8) in the proof of Lemma 5 and the Lipschitz modulus control of ψ φ n pX n´1 , ¨q in Lemma 1. Hence, we iterate this procedure and with some positive constants C1 , C2 not depending on Λ neither n but only on the assumptions we get :

Υ 1 ď exp ´C 1 ΛΓ p 5`β 2 q n Γ n `C 2 Λ 2 Γ p3q n Γ 2 n ¯" exp ˆp Λ ? Γ n `Λ2 Γ n qe n ˙, (4.21)
where, using Γ p2q n ?

Γn Ñ 0 for θ ą 1 3 . Eventually, inequalities (4.20) and (4.21) yields that:

E " exp `Λ Γ n n ÿ k"1 D k j pX k´1 , U k , Z k q ˘ı ď exp ˆp Λ ? Γ n `Λ2 Γ n qe n ˙.
5

Numerical Results

This section is a numerical illustration of the deviations results of the empirical measure ν n from Theorem 2. We consider the mono-dimensional case, d " r " 1. The innovations pU i q iě1 and X 0 are Gaussian variables. Also, a difficulty is to approximate the jump part of generator Aϕ, namely πpϕpx `κpxq¨q ´ϕpxqq for x P R. To avoid this problem, we choose pY k q kě0 to be Bernoulli variables, hence we directly get πpϕpx `κpxq¨q ´ϕpxqq " 1 2 `ϕpx `κpxqq `ϕpx ´κpxqq ˘´ϕpxq. We consider for the coefficients and the test function bpxq " ´x 2 , and σpxq " κpxq " ϕpxq " cospxq in (1.1). Note, in particular, that we have picked a degenerate framework. Thanks to Theorem 2, for pγ k q kě1 " pk ´θq kě1 , θ P r1{3, 1s (corresponding to β " 1 therein) the function a P R `Þ Ñ g n paq :" log `Pr| a Γ n ν n pAϕq| ě as ȋs such that for g n paq ď ´cn a 2 2}σ} 2 8 }∇ϕ} 2 8 `4}κ} 2 8 }∇ϕ} 2 8 `logp2C n q ": c n S ν paq `logp2C n q,

In Figure 5.1, we plot the the curves of g n for θ " 1 3 `10 ´3 We perform the simulations for n " 5 ˆ10 4 in Figure 5.1, the probability is estimated by Monte Carlo simulation with M C " 10 4 realizations of the random variable | ? Γ n ν n pAϕq| in the unbiased case. The Figure 5.1 enhance the fact that g n paq is indeed under a quadratic form in a. Nevertheless, we see s that the result of Theorem 2 is not sharp, to obtain such a result we have to avoid the dimension dependency and a sharp inequality of Proposition 1. Sharp Schauder Estimates for some Degenerate Kolmogorov Equations

Abstract : We provide here some sharp Schauder estimates for degenerate PDEs of Kolmogorov type when the coefficients lie in some suitable anisotropic Hölder spaces and the first order term is non-linear and unbounded. We proceed through a perturbative approach based on forward parametrix expansions. Due to the low regularizing properties of the degenerate variables, for the procedure to work, we heavily exploit duality results between appropriate Besov spaces.

Our method can be seen as constructive and provides, even in the non-degenerate case, an alternative approach to Schauder estimates.

Introduction and Main Results

We aim at proving, for a fixed time horizon T ą 0 and given integers n, d P N, Schauder estimates for degenerate scalar valued Kolmogorov PDEs of the form:

# B t upt, xq `xFpt, xq, Dupt, xqy `1 2 Tr `D2
x 1 upt, xqapt, xq ˘" ´f pt, xq, t P r0, T q, upT, xq " gpxq,

(1.1) for x " px 1 , ¨¨¨, x n q P R nd where for all i P rr1, nss, x i P R d . The source f and the terminal condition g are scalar. The diffusion matrix a is R d b R d -valued. Also, we denote by Fpt, xq :" pF 1 pt, xq, ¨¨¨, F n pt, xqq the vector of the R d valued mappings F i which have, for i P rr2, nss, the following structure:

@pt, xq P r0, T s ˆRnd , F i pt, xq :" F i pt, x i´1:n q, x i´1:n :" px i´1 , ¨¨¨, x n q.

(1.2)

In (1.1), the notation D " pD x 1 , ¨¨¨, D xn q stands for the full spatial gradient and D x i denotes the partial gradient w.r.t. to x i .

Our goal is to prove that, under suitable non-degeneracy conditions on the diffusion matrix a, which only acts on the first d components of the underlying space, and a weak Hörmander like condition on the first order non-linear term F, there exists a unique weak solution to (1.1) which satisfies some appropriate Schauder estimates when the source f , the terminal condition g and the coefficients a, F lie in appropriate Hölder spaces.

For notational convenience we will denote the spatial operator in (1.1) by pL t q tPr0,T s , i.e. for all ϕ P C 2 0 pR nd , Rq (space of twice continuously differentiable functions with compact support):

L t ϕpxq " xFpt, xq, Dϕpxqy `1 2 Tr `D2 x 1 ϕpxqapt, xq ˘. (1.3)
Under suitable non-degeneracy and regularity assumptions on a, F it can be shown that the martingale problem associated with (1.3) is well posed, see e.g. [START_REF] Menozzi | Parametrix techniques and martingale problems for some degenerate Kolmogorov equations[END_REF], [START_REF] Menozzi | Martingale problems for some degenerate Kolmogorov equations[END_REF], [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF]. In that case, there exists a unique weak solution to the stochastic differential equation

dX 1 t " F 1 pt, X 1 t , . . . , X n t qdt `σpt, X 1 t , . . . , X n t qdW t , dX 2 t " F 2 pt, X 1 t , . . . , X n t qdt, dX 3
t " F 3 pt, X 2 t , . . . , X n t qdt, . . .

dX n t " F n pt, X n´1 t , X n t qdt, t ě 0, (1.4)
where pW t q tě0 is a Brownian motion on some filtered probability space pΩ, pF t q tě0 , Pq. The operator pL t q tě0 then corresponds to the generator of the process in (1.4) where σ is a square root of a. The well posedness of the martingale problem in particular implies that (1.1) admits a solution in the mild sense on a suitable function space (see e.g. Kolokoltsov [START_REF] Kolokoltsov | Markov Processes, Semigroups and Generators[END_REF]).

The deterministic system (1.1) and its corresponding stochastic counterpart (1.4) appear in many applicative fields from physics to finance (see e.g. [START_REF] Talay | Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme[END_REF], [START_REF] Rey-Bellet | Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators[END_REF] or [START_REF] Barucci | Some results on partial differential equations and asian options[END_REF]). Another issue concerns the regularizing properties of the Brownian motion in the SDE (1.4). In a Hölder framework for the coefficients F, σ, some minimal thresholds that guarantee weak uniqueness holds where established in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF]. In the current work, we specifically quantify how adding to these thresholds some suitable regularity, depending on the stage i P rr1, nss of the chain and the variable j P rrpi ´1q _ 1, nss, we get a global similar smoothing effect.

Through the Schauder estimates of Theorem 1, we indeed precisely quantify the parabolic bootstrap associated with pL t q tě0 emphasizing its intrinsic regularization properties. It is known from the seminal work of Lunardi [START_REF] Lunardi | Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in R n[END_REF] on the topic that the Schauder estimates for degenerate Kolmogorov equations differ from those in the usual non-degenerate setting (see e.g. [START_REF] Krylov | Lectures on elliptic and parabolic equations in Hölder spaces[END_REF] or [START_REF] Friedman | Partial differential equations of parabolic type[END_REF]). They reflect in some sense the multiple scales in the systems (1.1) and (1.4) (see Section 1.1 below) and are stated in terms of anisotropic Hölder spaces. In particular, those spaces emphasize that the higher is the index of the considered variable in rr1, nss, the weaker is the associated regularity gain.

Let us now shortly describe some particular cases of dynamics of type (1.1) for which some Schauder estimates have already been proved. We again mention the work by Lunardi [START_REF] Lunardi | Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in R n[END_REF], who considered the special case of a homogeneous linear drift Fpxq " Ax satisfying the condition (1.2). Precisely, the matrix A writes in this case:

A " ¨a1,1 ¨¨¨¨¨¨¨¨¨a 1,n a 2,1 ¨¨¨¨¨¨¨¨¨a 2,n 0 d,d a 3,2 ¨¨¨¨¨¨a 3,n . . . 0 d,d . . . . . . 0 d,d ¨¨¨0 d,d a n,n´1 a n,n ‹ ‹ ‹ ‹ ‹ ' ,
where the entries pa i,j q ijPrr1,nss 2 are in R d bR d s.t. `ai,i´1 ˘iPrr2,nss are non-degenerate elements of R d b R d (which expresses the weak Hörmander condition) * . Also, the homogeneous diffusion coefficient a belongs to an appropriate anisotropic Hölder space and asymptotically converges when |x| Ñ 8 to a non-degenerate constant matrix of R d bR d . The assumptions on the asymptotic behavior on the diffusion coefficient have then been relaxed by Lorenzi [START_REF] Lorenzi | Schauder estimates for degenerate elliptic and parabolic problems with unbounded coefficients in n[END_REF], in the kinetic framework, i.e. n " 2 with the notations of (1.4), up to additional regularity assumptions on a which could also be unbounded. Priola established later in [START_REF] Priola | Global Schauder estimates for a class of degenerate Kolmogorov equations[END_REF] Schauder estimates, without dimensional constraints for time homogeneous drifts of the form

Fpxq " Ax `ˆF 1 pxq 0 pn´1qd,d ˙, (1.5)
for a non-linear drift F1 acting on the non-degenerate variable in the expected anisotropic Hölder space. The underlying technique consisted in establishing bounds on the derivatives of the semi-group of the perturbed degenerate Ornstein-Uhlenbeck process (i.e. with F1 ) from the usual unperturbed one (with pAxq 1 only) through the Girsanov theorem assuming first that F1 is smooth. This initial smoothness of F1 is required in order to compute the associated tangent flows. Through the continuity approach, the author then managed to obtain the estimates for a bounded variable diffusion coefficient lying in the natural expected Hölder space similar to the one of [START_REF] Lunardi | Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in R n[END_REF] with the same asymptotic conditions. The smoothness of F1 is then relaxed through an approximation procedure viewing the difference between the Hölder drift and its mollification as a source term and exploiting the estimates established for the smooth drift. Let us also mention the work of Di Francesco and Polidoro [START_REF] Francesco | Schauder estimates, harnack inequality and gaussian lower bound for kolmogorov-type operators in non-divergence form[END_REF] who derived Schauder estimates for a linear drift of the previous type using an alternative notion of continuity regarding the diffusion coefficient a, which somehow involves the unbounded transport associated with the drift.

Put it differently, in the current framework of degenerate Kolmogorov equations, focusing on the drift, Schauder estimates hold, to the best of our knowledge, for either linear drifts or Hölder perturbations on the non-degenerate variable of a linear drift.

In the non-degenerate setting, Schauder estimates for unbounded non-linear drift coefficients were obtained under mild smoothness assumptions by Krylov and Priola [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF]. Their idea was to use the flow associated with the first order vector field in L t , i.e. 9 θ t pxq " Fpt, θ t pxqq, to precisely get rid of the unbounded terms.

In the current framework, we will face both difficulties: the degeneracy as well as the non-linearity and unboundedness of the drift. We will prove, in the framework of Hölder spaces for the source f , the terminal condition g and the coefficients a, F, Schauder estimates similar to those of the previously quoted works. The diffusion coefficient a and the source term f will have, as in the non-degenerate case, the same regularity. We mention that, in contrast with the non-degenerate case, this is will not be the case for the drift F for which some additional smoothness on the degenerate entries pF i q iPrr2,nss is needed to guarantee the well posedness of (1.4). In particular, the Hölder indexes of F will be above the minimal thresholds appearing in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF]. The flow associated with the drift term will again play a key role in our setting.

We will here proceed through a perturbative approach. The idea is to perform a first order parametrix expansion (or Duhamel expansion) of a solution of (1.1) with mollified coefficients around a suitable linearized Ornstein-Uhlenbeck type semi-group. Since we are interested in controlling derivatives, at least along the non-degenerate variable x 1 , we will use the so-called forward parametrix approach, which had previously been successfully used by Friedman [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] or Il'in et al. [START_REF] Il'in | Second-order linear equations of parabolic type[END_REF] in the non-degenerate setting to obtain pointwise bounds on the fundamental solution and its derivatives for the corresponding heat-type equation or in [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF] to derive strong uniqueness for kinetic SDEs of type (1.4) (i.e. n " 2 with the previous notations). This approach actually allows to exploit cancellation techniques which are crucial when derivatives come in.

On the other hand, for the well posedness of the corresponding martingale problem or density estimates of the fundamental solution of (1.1), the backward approach introduced by McKean and Singer [MS67] might be easier. It was extended in the current degenerate setting, which involves unbounded coefficients, and successfully exploited for the two previously indicated applications in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] and [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF]. Unfortunately, this approach does not allow to benefit from the previously mentioned cancellation techniques. This becomes a hard issue in the parabolic setting involving a non-trivial terminal condition.

The perturbative approach is not usual to establish Schauder type estimates. The standard way is to proceed through a priori estimates to establish for a given solution of the PDE in a suitable function space, the expected bound. Existence and uniqueness issues, in the considered function space, for the solution of the equation are addressed in a second time. We can refer to [START_REF] Krylov | Lectures on elliptic and parabolic equations in Hölder spaces[END_REF] for a clear presentation of this approach and to [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF] for an extension of this method to non-degenerate operators with unbounded drift coefficients. We will here obtain that the solutions of (1.1) with mollified coefficients satisfy, uniformly w.r.t. the mollification parameter, a Schauder type estimate (see Sections 3 to 5 below). From the well posedness of the martingale problem established in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] under our current assumptions, we will then derive that the martingale solution to (1.1) actually itself satisfies the Schauder controls. Since we want to be in the sharpest possible Hölder setting for the coefficients, source and terminal functions, we will need to establish some subtle controls (in particular we have no true derivatives of the coefficients) which will heavily rely on duality results for Besov spaces (see Section 4.2 below and e.g. Chapter 3 in Lemarié-Rieusset [START_REF] Lemarie-Rieusset | Recent developments in the Navier-Stokes problem[END_REF]).

Let us emphasize that the perturbative approach developed here provides, even in the non-degenerate case, a new alternative to establish Schauder estimates. It can be seen as a constructive one in the sense that, from a sequence of smooth solutions, that uniformly satisfy the expected control, we will extract through convergence in law arguments a limit solution which also satisfies the bound. Uniqueness of the solution in the considered class then again follows from uniqueness in law of the underlying limit process.

The drawback of our approach is that, for the parabolic problem (1.1), we first have to establish our estimates in small time. This is intuitively clear since the perturbative methods (expansions along an ad hoc proxy) are precisely designed for small times. To obtain the result for an arbitrary given time, we then have to iterate the estimate. We are therefore far from the optimal constants for the Schauder estimates established in the non-degenerate setting for time dependent coefficients by Krylov and Priola [START_REF] Krylov | Poisson stochastic process and basic Schauder and Sobolev estimates in the theory of parabolic equations[END_REF]. However, to the best of our knowledge, our approach is the only one allowing to derive Schauder estimates for a parabolic degenerate Kolmogorov equation with fully non-linear drift in Hölder space. Also, we think the strategy developed in the current work should apply for the elliptic degenerate Kolmogorov equation with good potential term, i.e. the negative sign of the potential would allow to integrate on an infinite time horizon (getting therefore rid of the small time constraint) or for unbounded sources f that would need in that cases to be somehow controlled by an associated potential as in [START_REF] Krylov | Elliptic and parabolic second-order PDEs with growing coefficients[END_REF].

In connection with the strong regularizing properties of Brownian motion, let us mention that the perturbative approach we develop here allows as well to address the problem of strong well posedness for the SDE (1.4) in a Hölder framework for the coefficients. This was done by Chaudru de Raynal in [START_REF] Chaudru | Weak regularization by stochastic drift: result and counter example[END_REF] for n " 2. Let us mention as well in this kinetic case the works of Fedrizzi et al. [START_REF] Fedrizzi | Regularity of stochastic kinetic equations[END_REF] and Zhang [START_REF] Zhang | Stochastic hamiltonian flows with singular coefficients[END_REF] who derived strong uniqueness for L p drifts on the non-degenerate component and a linear degenerate dynamics in (1.4). We establish strong uniqueness for the full chain for some suitable related Hölder thresholds for the drift in Chapter 7. Let us mention that, from a PDE viewpoint, strong uniqueness for the associated SDE is heavily related to controls of the derivatives of the solution of the PDE in all the directions (including the degenerate ones). In Chapter 7, under some slightly stronger Hölder regularity assumptions on the coefficients (which for strong uniqueness issues then in turn become the source term in the PDE with the Zvonkin approach), we derive some pointwise bounds for the derivatives of the PDEs. Some related issues were also considered under additional smoothness conditions by Lorenzi [Lor05a] in the case of a linear drift.

Before stating our main results, we recall some properties associated with the system (1.1). We first describe in Section 1.1 how the intrinsic multi-scales of the degenerate Kolmogorov like equations appear. We then introduce the appropriate setting of Hölder spaces to consider in Section 1.2. We eventually conclude the introduction stating in Section 1.3 our main results concerning Schauder estimates associated with (1.1).

Intrinsic scales of the system and associated distance

Let us now briefly expose how the system typically behaves. To do so, consider the following operator:

L 0 :" B t `xA 0 x, Dy `1 2 ∆ x 1 , A 0 " ¨0d,d
L 0 v " 0 ùñ L 0 pv ˝δλ q " 0. (1.7)
This hence lead us to introduce the homogeneous pseudo-norm corresponding to the dilation operator δ λ . Precisely, setting for all 0 ď s ď t ă `8, px, yq P pR nd q 2 : d P `pt, xq, ps, yq ˘" ps ´tq

1 2 `d ÿ i"1 |y i ´xi | 1 2i´1 , (1.8)
we indeed have d P `δλ `pt, xq ˘, δ λ `ps, yq ˘˘" λd P `pt, xq, ps, yq ˘. In our current setting we will mainly use the spatial part deriving from the parabolic homogeneous pseudo-distance d P in (1.8). We set accordingly

d `x, y ˘" d ÿ i"1 |y i ´xi | 1 2i´1 .
(1.9)

From a technical point of view, these distances express the spatial homogeneity associated with the intrinsic time scales of the variances for the Gaussian process with generator L 0 . From a more probabilistic viewpoint the exponents in (1.9) can be related to the characteristic time-scales of the iterated integrals of the Brownian motion (see e.g. [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF], [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF]). The non-degeneracy and boundedness assumption on a as well as the Hörmander condition on the pF i q iPrr2,nss that we assume for (1.1)-(1.4) (see assumptions (UE) and (H) below) will allow us to consider for our analysis the previous pseudo-distances associated with the simplest yet typical equation in the class described by (1.1)-(1.4).

Associated Hölder spaces

We first recall some useful notations and spaces. We denote for k P N, β P p0, 1q by } ¨}C k`β pR m ,R q , m P t1, d, ndu, P t1, d, d 2 , ndu the usual homogeneous Hölder space, see e.g. Krylov [START_REF] Krylov | Lectures on elliptic and parabolic equations in Hölder spaces[END_REF]. Precisely, for ψ P C k`β pR m , R q, denoting by α " pα 1 , ¨¨¨, α m q P N m a generic multi-index and |α| " ř m i"1 α i , we define the semi-norm:

}ψ} C k`β pR m ,R q :" k ÿ i"1 sup |α|"i }D α ψ} L 8 pR m ,R q `sup |α|"k rD α ψs β , rD α ψs β :" sup px,yqPpR m q 2 ,x‰y |D α ψpxq ´Dα ψpyq| |x ´y| β , (1.10)
where | ¨| denotes the Euclidean norm on the considered space. We will also need to consider the associated subspace with bounded elements. Namely, we set:

C k`β b pR m , R q :" tψ P C k`β pR m , R q : }ψ} L 8 pR m ,R q ă `8u.
We define correspondingly the Hölder norm:

}ψ} C k`β b pR m ,R q :" }ψ} C k`β pR m ,R q `}ψ} L 8 pR m ,R q .
(1.11)

We are now in position to define our inhomogeneous Hölder spaces with multi-index of regularity. Let ψ : R nd Ñ R be a smooth function. We first introduce, for i P rr1, nss, x P R d the perturbation operator that writes:

@z P R nd , Π x i pψqpzq :" ψpz 1 , ¨¨¨, z i `x, ¨¨¨, z n q.
(1.12)

We then define for all i P rr1, nss, the mapping pz, xq P R nd ˆRd Þ ÝÑ ψ i pz, xq :" Π x i pψqpzq.

(1.13)

Let us introduce the following inhomogeneous Hölder space in d-metric: given a parameter γ P p0, 1q, and k P N, we say that ψ is in

C k`γ d pR nd , R q, if }ψ} C k`γ d pR nd ,R q :" n ÿ i"1 sup zPR nd }ψ i pz, ¨q} C k`γ 2i´1 pR d ,R q ă `8. (1.14)
For the sake of simplicity, we will write:

}ψ} L 8 :" }ψ} L 8 pR nd ,R q , and }ψ} C k`γ d :" }ψ} C k`γ d pR nd ,R q .
The subscript d stands here to indicate the dependence of the Hölder exponents appearing in the r.h.s. on the underlying pseudo-norm d reflecting the scale invariance of the system (see equation (1.9) and the comments above for details). Note in particular that, for k " 0, there exists C :" Cpn, dq ě 1 such that:

C ´1rψs γ d ď }ψ} C k`γ d pR nd ,R q ď Crψs γ d , rψs γ d :" sup x‰x 1 ,px,x 1 qPpR nd q 2 |ψpxq ´ψpx 1 q| d γ px, x 1 q , (1.15)
see also e.g. Lunardi [START_REF] Lunardi | Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in R n[END_REF]. Also, from (1.11) and (1.14), we write that

ψ P C k`γ b,d pR nd , R q if }ψ} C k`γ b,d pR nd ,R q :" n ÿ i"1 sup zPR nd }ψ i pz, ¨q} C k`γ 2i´1 b pR d ,R q ă `8.
From now on, we will denote:

}ψ} C γ b,d :" }ψ} C γ b,d pR nd ,R q .
Finally, through the article, we use the following notation for all functions

ϕ 1 P L 8 `r0, T s, C k`γ b,d pR m , R q ˘and ϕ 2 P L 8 `r0, T s, C k`γ d pR m , R q ˘: }ϕ 1 } L 8 pC k`γ b,d q :" sup tPr0,T s }ϕ 1 pt, ¨q} C k`γ b,d pR m ,R q , and }ϕ 2 } L 8 pC k`γ d q :" sup tPr0,T s }ϕ 2 pt, ¨q} C k`γ d pR m ,R q .

Assumptions and main result

With these notations at hand we can now state our assumptions and main results. In the following, we will assume: (H) Weak Hörmander like condition. For all i P rr2, nss, there exists a closed convex subset E i´1 Ă GL d pRq (set of invertible d ˆd matrices) s.t., for all t ě 0 and px i´1 , . . . , x n q P R pn´i`2qd , D x i´1 F i pt, x i´1 , . . . , x n q P E i´1 . For example, E i´1 may be a closed ball included in GL d pRq, which is an open set.

(S) Smoothness of the Coefficients. Fix γ P p0, 1q. We suppose the following conditions hold.

(i) Smoothness of the diffusion coefficient. We assume that a is measurable in time and that a P L 8 `r0, T s,

C γ b,d pR nd , R d b R d q ˘.
(ii) Smoothness of the drift in time. We only assume here that the drift is measurable in time and bounded at the origin, i.e. the measurable mapping t Þ Ñ Fpt, 0q is bounded.

(iii) Smoothness of the drift in space. We now state, for each level i P rr1, nss the smoothness assumptions on the drift component F i (see the remark below for more explanations):

F i P L 8 pr0, T s, C p2i´3q_0`γ d pR ppn´i`2q^nqd , R d q.
(1.16) For a fixed parameter γ P p0, 1q, we will say that (A) is in force as soon as (UE), (H), (S) hold.

Remark 1 Let us come back to assumption (S)-(iii), which may seem difficult to understand at first sight. Namely, we here explain a little bit how the particular thresholds appearing in this assumption come from as well as the precise regularity imposed on each component of the drift F w.r.t. any space variables. ' Note first that for i " 1 assumption (S)-(iii) readily says, with the previous notations for Hölder spaces, that F 1 P L 8 pr0, T s, C γ d pR nd , R d qq. ' For each level i P rr2, nss of the chain in (1.2), we shall consider different types of assumptions on F i depending on the variables x i´1 and x i:n " px i , ¨¨¨, x n q respectively. Let us now fix i P rr2, nss.

The component x i´1 is hence the one which transmits the noise. Coherently with the usual Hörmander setting, we need some differentiability of F i w.r.t. x i´1 . In order to have a global coherence, in terms of time-space homogeneity, for all the considered variables, the specific smoothness to be considered for that variable is that F i pt, ¨, x i:n q is in C 1`γ 2pi´1q´1 pR d , R d q. Recalling now the previous definition of d and of the associated Hölder spaces, we have

C 1`γ 2pi´1q´1 pR d , R d q " C 2i´3`γ 2pi´1q´1 pR d , R d q " C 2i´3`γ d pR d , R d q.
Now, at level i, the components x i:n are above the current characteristic time-scale, i.e. the vector of their associated time rescaling, which writes according to the homogenous quasi-metric d P in (1.8) as pt i´1 2 , t pi`1q´1 2 , ¨¨¨, t n´1 2 q, has in small time entries that are actually smaller or equal than the time rescaling of the current variable x i in t i´1{2 . We recall as well that, in order to have the well posedness of the martingale problem associated with the operator pL t q tě0 in (1.3), some natural minimal thresholds of Hölder continuity appear for these variables. Precisely, at level i, F i must be Hölder continuous in x j , j P rri, nss, with index strictly greater than 2i´3 2j´1 (see [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] for details). Here, still to have a global coherence, in terms of time-space homogeneity, for all the considered variables, we assume that F i is 2i´3 2j´1 `γ 2j´1 Hölder continuous in its j th variable. This precisely corresponds to the minimal threshold required to which we add the intrinsic γ-Hölder regularity w.r.t to the associated scale appearing in d for the considered entry. Thus, with a slight abuse of notations,

z Þ Ñ F i pt, x i´1 , x i , ¨¨¨, x j´1 , z, x j`1 , ¨¨¨, x n q is supposed to be in C 2i´3 2j´1 `γ 2j´1 pR d , R d q " C 2i´3`γ d pR d , R d q.
' "Gathering" the regularity conditions assumed on each variable for each component of F hence gives assumption (S)-(iii).

Remark 2 Concerning the time regularity in the previous assumptions we can refer to the earlier work of Kruzhkov et al. [START_REF] Kružkov | Schauder type estimates, and theorems on the existence of the solution of fundamental problems for linear and nonlinear parabolic equations[END_REF] who first consider this type of regularity to establish Schauder estimates in the classical non degenerate framework with bounded coefficients. We can also mention Lorenzi [START_REF] Lorenzi | Optimal Hölder regularity for nonautonomous Kolmogorov equations[END_REF] for extensions to unbounded coefficients.

We are now in position to state our main result.

Theorem 1 (Schauder Estimates for degenerate Kolmogorov Equations.)

Let γ P p0, 1q be given. Suppose that (A) is in force and that the terminal condition g and source term f of the Cauchy problem (1.1) satisfy: g P C 2`γ b,d pR nd , Rq and f P L 8 `r0, T s, C γ b,d pR nd , Rq ˘.

Then, there exists a unique weak solution u in C 2`γ b,d pR nd , Rq to (1.1). Furthermore, there exists a constant C (1) :" C (1) `pAq, T ˘s.t.

}u} L 8 pC 2`γ b,d q ď C (1) `}g} C 2`γ b,d `}f } L 8 pC γ b,d q ˘.
(1.17)

Section 2 below is dedicated to the presentation and description of the various steps that we perform to obtain our main result of Theorem 1. From now on we will denote by C a generic constant that may change from line to line but only depends on known parameters in (A) and the considered fixed final time T , i.e. C :" CppAq, T q. We reserve the notation c for generic constants that may also change from line to line, depend on (A) but are also independent of T , i.e. c :" cppAqq.

Detailed Guide to the proof

We describe in this section the approach we follow to derive our main results. The point is to emphasize the various difficulties arising and to introduce the adapted tools that can be used to circumvent them.

The first step of our strategy is to mollify equation (1.1) in order to get a well-posed Cauchy problem in the classical sense. Precisely, for ϕ P C 2 0 pR nd , Rq, m P N and t P r0, T s we define the operator:

L m t ϕpxq :" xF m pt, xq, Dϕpxqy `1 2 Tr `D2 x 1 ϕpxqa m pt, xq ˘, (2.1)
where F m , a m are mollified versions in space of the initial coefficients F, a in (1.3), i.e. F m pt, xq " Fpt, ¨q ‹ φ m pxq, a m pt, xq " apt, ¨q ‹ φ m pxq, where for all z P R nd , φ m pzq :" m nd φpzmq for a smooth, i.e. C 8 , non-negative function φ : R nd Ñ R `s.t. ş R nd φpzqdz " 1 and the previous convolutions are to be understood componentwise. We introduce correspondingly the stochastic differential equation with generator pL m t q tě0 . Namely, for fixed pt, xq P r0, T s ˆRnd and s ě t,

X m,t,x s " x `ż s t F m pu, X m,t,x u qdu `ż s t Bσ m pu, X m,t,x u qdW u , (2.2)
where σ m is a square root of a m . The dynamics in (2.2) is similar to the one in (1.4) up to the mollification of the coefficients. It can be deduced from the well-posedness of the martingale problem, which holds under our current assumptions from [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF], that pX m t q tPr0,T s ñ m pX t q tPr0,T s (convergence in law on the path space) where pX t q tPr0,T s is the unique weak solution of (1.4) (see also [START_REF] Stroock | Multidimensional diffusion processes[END_REF]).

Consider now mollified versions f m , g m of the source f and the final condition g in (1.1). It is then rather direct to derive through stochastic flows techniques, see e.g. Kunita [START_REF] Kunita | Stochastic Flows and Stochastic Differential Equations[END_REF], that

u m pt, xq :" Erg m pX m,t,x T qs `ż T t Erf m ps, X m,t,x s qsds, (2.3)
belongs for any given m to C 8 b pR nd , Rq (space of infinitely differentiable functions with bounded derivatives) and precisely solves for all x P R nd :

$ ' & ' % B t u m pt, xq `xF m pt, xq, Du m pt, xqy `1 2 Tr `D2
x 1 u m pt, xqa m pt, xq ˘" ´fm pt, xq, t P r0, T q, ˆRnd , u m pT, xq " g m pxq, x P R nd .

(2.4) The idea is now to obtain controls of the norms }u m } L 8 pC 2`γ b,d q which are uniform w.r.t. the mollifying parameter m. To this end, we will use a perturbative method by expanding u m around a suitable Ornstein-Uhlenbeck like Gaussian proxy corresponding to an appropriate linearization of the dynamics in (2.2). Consider first the deterministic dynamics deriving from (2.2) obtained setting σ m to 0, i.e.

9 θ m v,τ pξq " F m pv, θ m v,τ pξqq, v P r0, T s, θ m τ,τ pξq " ξ, (2.5)
where pτ, ξq P r0, T s ˆRnd are freezing parameters, respectively in time and space to be specified. Fix 0 ď t ă s ď T and x P R nd . The typical linearization of (2.2) on the time interval rt, ss around pθ m v,τ pξqq vPrt,ss writes:

d Xm,pτ,ξq v " rF m pv, θ m v,τ pξqq`DF m pv, θ v,τ pξqqp Xm,pτ,ξq v ´θm v,τ pξqqsdv`Bσ m pv, θ m v,τ pξqqdW v , @v P rt, ss, Xm,pτ,ξq t " x, (2.6)
where for all z P R nd , DF m pv, zq :"

¨0d,d ¨¨¨¨¨¨¨¨¨0 d,d D z 1 F m,2 pv, zq 0 d,d ¨¨¨¨¨¨0 d,d 0 d,d D z 2 F m,3 pv, z 2:n q 0 d,d 0 d,d . . . . . . 0 d,d . . . . . . 0 d,d ¨¨¨0 d,d D z n´1 F m,n pv, z n´1 , z n q 0 d,d ‹ ‹ ‹ ‹ ‹ '
(2.7) denotes the subdiagonal of the Jacobian matrix D z F m pv, ¨q at point z. From our previous assumptions (non-degeneracy of σ and Hörmander like condition), the Gaussian process with dynamics (2.6) admits a well controlled multi-scale density pm,pτ,ξq pt, s, x, ¨q (see e.g. Section 3.1 below and for instance [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF], [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF]). Namely, there exists C :" CppAq, T q ě 1 s.t. for j P t1, 2u and all pi, kq P rr1, nss 2 , for all 0 ď t ă s ď T, px, yq P pR nd q 2 : and for all u ą 0, T u is the intrinsic scale matrix:

|D x k D j x i pm,
T u " ¨uI d,d 0 d,d ¨¨¨0 d,d 0 d,d u 2 I d,d 0 d,d . . . . . . . . . . . . . . . 0 d,d ¨¨¨0 d,d u n I d,d ‹ ‹ ‹ ' , (2.9)
that is, the i th diagonal entry of u ´1 2 T u reflects the time order of the variances of the pi ´1q th iterated integral of the standard Brownian motion at time u. Observe as well that the time singularities in (2.8) precisely reflect the typical scale of the associated variable, i.e. differentiating in x k yields an additional time singularity in ps ´tq ´k`1 2 where ps ´tq k´1 2 is exactly the order of the standard deviation of the pk ´1q th iterated integral of the Brownian motion.

Denoting by p Lm,pτ,ξq v q vPrt,T s the generator of (2.6), it also holds that:

`Bs ´p Lm,pτ,ξq s q ˚˘p m,pτ,ξq pt, s, x, yq " 0, pm,pτ,ξq pt, s, ¨, yq Ñ sÓt δ y p¨q, `Bt `L m,pτ,ξq t ˘p m,pτ,ξq pt, s, x, yq " 0, pm,pτ,ξq pt, s, x, ¨q Ñ tÒs δ x p¨q.

The above equations are respectively the forward and backward Kolmogorov equations. In the first one, the operator p Lm,pτ,ξq s q ˚acts on the forward variable y whereas in the second one, Lm,pτ,ξq t acts in the backward variable x. We will use the notation P m,pτ,ξq T,t for the corresponding semi-group, i.e. for the associated Green kernel (with fixed final time T ą 0). For fixed pt, xq P r0, T sˆR nd and the above Gaussian proxy, for which pτ, ξq still remain to be specified, we recall that Duhamel's formula (first order parametrix expansion) yields that:

u m pt, xq " P m,pτ,ξq T,t g m pxq `G m,pτ,ξq f m pt, xq `ż T t ds ż R nd
pm,pτ,ξq pt, s, x, yqpL m s ´L m,pτ,ξq s qu m ps, yqdy.

(2.11)

Recall from the statement of our main Theorem 1 that we have to give bounds on }u m } L 8 pC 2`γ b,d q . For this introduction to the proof, we will focus on the contributions D 2 x 1 u m , that already exhibits all the difficulties and for which we want to establish a control in time-space supremum norm and for the γ-Hölder modulus associated with the distance d.

Differentiating in D 2 x 1 equation (2.11) gives:

D 2 x 1 u m pt, xq " D 2 x 1 P m,pτ,ξq

¯,

we get that the terms of last contribution in the above r.h.s. can precisely be absorbed by the exponential off-diagonal bound in (2.8).

We therefore eventually derive for the non-degenerate contribution with the notation of (2.8):

ˇˇż T t ds ż R d D 2
x 1 pm,pτ,ξq pt, s, x, yq∆ 1,Fm,σm pτ, s, θ m s,t pξq, y, u m qdy ˇˇˇˇˇˇp τ,ξq"pt,xq

ď ż T t ds ps ´tq 1´γ 2 ż R nd C ppτ,ξq C ´1 pt, s, x, yq ´}D y 1 u m ps, ¨q} L 8 `}D 2 y 1 u m ps, ¨q} L 8 ¯dy ď 2C γ pT ´tq γ 2 `}D y 1 u m } L 8 `}D 2 y 1 u m } L 8 ď 2C γ pT ´tq γ 2 }u m } L 8 pC 2`γ b,d q ,
(2.14) yielding precisely a time smoothing effect corresponding exactly to the Hölder continuity exponent γ of the coefficients. The previous choice of pτ, ξq is known as the forward parametrix and seems adapted as soon as one is led to estimate derivatives of the solution. This was for instance already the choice performed in the non-degenerate case by Il'in et al [START_REF] Il'in | Second-order linear equations of parabolic type[END_REF] or Friedman [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] in relation with Schauder estimates or by Di Francesco and Polidoro [START_REF] Francesco | Schauder estimates, harnack inequality and gaussian lower bound for kolmogorov-type operators in non-divergence form[END_REF] in the current degenerate Kolmogorov setting with a linear drift.

Let us mention that, as far as one is concerned with density estimates, which formally amounts to replace u m ps, xq, u m ps, yq in (2.11) with p m pt, T, x, zq, p m ps, T, y, zq (density at some fixed point z P R nd of X m T starting from x at time t), or with the well-posedness of the martingale problem, another choice, consisting in freezing in pτ, ξq " ps, yq in the above equation, could also be considered. Note that the freezing parameters would here depend on the time and spatial integration variables. This backward approach was first introduced by McKean and Singer [START_REF] Mckean | Curvature and the eigenvalues of the Laplacian[END_REF] and led successfully to density estimates and well-posedness of the martingale problem for the current model (1.4) in the respective works [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF], [START_REF] Menozzi | Martingale problems for some degenerate Kolmogorov equations[END_REF], [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF].

However, when dealing with derivatives, the forward perturbative approach appears more flexible since it allows to exploit cancellation techniques whereas this is much trickier in the backward case for which pm,ps,yq pt, s, x, yq is not a density w.r.t. y. Some associated errors associated with this approach are thoroughly discussed in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF].

Let us now turn to the contributions associated with the degenerate variables in the difference pL m s ´L Under the current assumptions, we do not expect to have uniform controls w.r.t. to the smoothing parameter m for the derivatives pD y i u m q iPrr2,nss in the degenerate directions. Our strategy will first consist for those terms in performing an integration by parts leading to: where the notation "b" between two tensors means the usual tensor product. In particular, `D2

ˇˇn ÿ i"2 ż T t ds ż R nd dyD 2 x 1 pm,
x 1 pm,pτ,ξq pt, s, x, yq b ∆ i,Fm pτ, s, θ m s,t pξq, yq ˘is a tensor lying in pR d q b3 . Furthermore, D y i ¨refers to an extended form of the divergence over the i th variable (y i P R d ). Precisely, from (2.16), we rewrite for all i P rr2, nss, ps, yq P rt, T sR nd

D y i ¨`Θ m,α i,pt,xq ps, yq ˘" d ÿ j"1 B y j i ´D2
x 1 pm,pτ,ξq pt, s, x, yq `∆i,Fm pτ, s, θ m s,t pξq, yq ˘j¯, with y i " py 1 i , ¨¨¨, y n i q. In other words, this "enhanced" divergence form decreases by one the order of the input tensor. As a particular case, if d " 1, ∆ i,Fm pτ, s, θ m s,t pξq, yq is a scalar and the divergence form corresponds to the standard differentiation, i.e. D y i ¨" B y i .

Here α " p2, 0, ¨¨¨, 0q is the multi-index which reflects that the above contribution involves the second order derivatives of the frozen heat-kernel w.r.t. to its non-degenerate components.

In view of our main estimates in Theorem 1, we will use the duality between suitable Besov spaces to derive bounds for the spatial integrals in (2.16). Introduce, for all fixed i P rr2, nss and any py 1 , ¨¨¨, y i´1 , y i`1 , ¨¨¨, y n q ": py 1:i´1 , y i`1:n q P R pn´1qd the mappings u i,ps,y 1:i´1 ,y i`1:n q m : y i Þ Ñ u m ps, y 1:i´1 , y i , y i`1:n q, Ψ i,pt,xq,ps,y 1:i´1 ,y i`1:n q m : y

i Þ Ñ D y i ¨`Θ m,α
i,pt,xq ps, yq ˘.

(2.17)

The underlying idea is that we actually want, for the i th variable, to control uniformly in m the Hölder modulus ru i,ps,y 1:i´1 ,y i`1:n q m s 2`γ 2i´1 uniformly in ps, y 1:i´1 , y i`1:n q P rt, T s ˆRpn´1qd .

To prove this property we recall that, setting αi :" 2`γ 2i´1 , C αi b pR d , Rq " B αi 8,8 pR d , Rq with the usual notations for Besov spaces (see e.g. Triebel [START_REF] Triebel | Theory of function spaces[END_REF]).

Let us now recall some definitions/characterizations from Section 2.6.4 of Triebel [START_REF] Triebel | Theory of function spaces[END_REF]. For α P R, q P p0, `8s, p P p0, 8s, B α p,q pR d q :" tf P S 1 pR d q : }f } H α p,q ă `8u where SpR d q stands for the Schwarz class and

}f } H α p,q :" }ϕpDqf } L p pR d q `´ż 1 0 v pm´α 2 qq }B m v h v ‹ f } q L p pR d q dv v ¯1 q , (2.18)
with ϕ P C 8 0 pR d q (smooth function with compact support) is s.t. ϕp0q ‰ 0, ϕpDqf :" pϕ f q _ where f and pϕ f q _ respectively denote the Fourier transform of f and the inverse Fourier transform of ϕ f . The parameter m is an integer s.t. m ą α 2 and for v ą 0,

z P R d , h v pzq :" 1 p2πvq d 2 exp `´|z| 2
2v ˘is the usual heat kernel of R d . We point out that the quantities in (2.18) are well defined for q ă 8. The modifications for q " `8 are obvious and can be written passing to the limit.

Observe that the quantity }f } H α p,q , where the subscript H stands to indicate the dependence on the heat-kernel, depends on the considered function ϕ and the chosen m P N. It also defines a quasi-norm on B s p,q pR d q. The previous definition of B α p,q pR d q is known as the thermic characterization of Besov spaces and is particularly well adapted to our current framework. By abuse of notation we will write as soon as this quantity is finite }f } H α p,q ": }f } B α p,q . As indicated above, it is easily seen from ( 2 ). We will therefore write from (2.16) and with the notations of (2.17)

n ÿ i"2 ż T t ds ˇˇż R nd dyD y i ¨`Θ m,α
i,pt,xq ps, yq ˘um ps, yq ˇˇˇˇˇˇp τ,ξq"pt,xq

ď n ÿ i"2 ż T t ds ż R pn´1qd
dpy 1:i´1 , y i`1:n q}Ψ i,pt,xq,ps,y 1:i´1 ,y i`1:n q m } B ´α i

1,1 }u i,ps,y 1:i´1 ,y i`1:n q m } B αi 8,8 ď }u m } L 8 pC 2`γ b,d q n ÿ i"2 ż T t ds ż R pn´1qd
dpy 1:i´1 , y i`1:n q}Ψ i,pt,xq,ps,y 1:i´1 ,y i`1:n q m } B ´α i

1,1 . (2.19)
Exploiting the thermic characterization of Besov spaces (see again (2.18) and Section 4.2), It will be shown in Lemma 5 below that there exists C :" CppAqq s.t. for all i P rr2, nss and all m P N:

ż R pn´1qd
dpy 1:i´1 , y i`1:n q}Ψ i,pt,xq,ps,y 1:i´1 ,y i`1:n q m } B ´α i

1,1 ď C ps ´tq 1´γ 2 .
(2.20) Therefore: (2.22) which is precisely homogeneous to the bound obtained for the non-degenerate variables in (2.14). In both cases, the contribution pT ´tq γ 2 derives from the assumed smoothness of the coefficients a, F w.r.t. d which exactly leads to the same global control for the a priori most singular part of expansion (2.12).

n ÿ i"2 ż T t ds ż R pn´1qd dpy 1:i´1 , y i`1:n q}Ψ i,pt,xq,ps,y 1:i´1 ,y i`1:n q m } B ´α i 1,1 ď CpT ´tq γ 2 , ( 2
From the previous bounds and (2.12) we thus obtain:

|D 2 x 1 u m pt, xq| ď ´|D 2 x 1 P m,pτ,ξq T,t g m pxq|`|D 2 x 1 Gm,pτ,ξq f m pt, xq| ¯ˇˇp τ,ξq"pt,xq `CpT ´tq γ 2 }u m } L 8 pC 2`γ b,d q . (2.23) Since P m,pτ,ξq T,t
is a true semi-group, and Gm,pτ,ξq the associated Green kernel (precisely because we used a forward perturbative expansion), it will be derived in Lemma 6 (thanks to cancellation techniques) that there exists C :" CppAqq s.t. for all pt, xq P r0, T s ˆRnd :

´|D 2 x 1 P m,pτ,ξq T,t g m pxq| `|D 2 x 1 Gm,pτ,ξq f m pt, xq| ¯ˇˇp τ,ξq"pt,xq ď Cp}g m } C 2`γ b,d `pT ´tq γ 2 }f m } L 8 pC γ b,d q q ď Cp}g} C 2`γ b,d `pT ´tq γ 2 }f } L 8 pC γ b,d q q.
(2.24) Equation (2.24) eventually leads to the following estimate on |D 2 x 1 u m pt, xq|:

|D 2 x 1 u m pt, xq| ď C ´}g} C 2`γ b,d `pT ´tq γ 2 p}f } L 8 pC γ b,d q `}u m } L 8 pC 2`γ b,d q q ¯.
(2.25)

For T small enough, this equation would be compatible with the estimates of Theorem 1. Equation (2.25) might even seem too strong since it also exhibits, additionally to the control of the term associated with the perturbation, a small contribution (in pT ´tq γ 2

for a small enough T ) w.r.t. to the source f m . This is precisely because }D 2

x 1 u m pt, ¨q} L 8 is not one of the critical terms in the Hölder norm }u m pt, ¨q} C 2`γ b,d , i.e. the regularity of the coefficients still gives that it can be viewed as a remainder at first sight. Now, a typical critical term of the Hölder norm, for which we precisely exploit totally the spatial regularity of the coefficients, is rD 2

x 1 u m pt, ¨qs γ d (see assumption (S) and (1.15)). Let us now detail how we can handle it and in which sense it can be viewed as critical.

Of course, if g " 0, for t P r0, T s and given spatial points px, x 1 q P pR nd q 2 we can assume w.l.o.g. that dpx, x 1 q ď pT ´tq 1 2 , i.e. the spatial points are close w.r.t. the characteristic time scale pT ´tq 1 2 for the homogeneous metric d. Indeed, if dpx, x 1 q ą pT ´tq 1 2 , equation (2.25) readily gives:

|D 2 x 1 u m pt, xq ´D2 x 1 u m pt, x 1 q| ď |D 2 x 1 u m pt, xq| `|D 2 x 1 u m pt, x 1 q| ď 2CpT ´tq γ 2 p}f } L 8 pC γ b,d q `}u m } L 8 pC 2`γ b,d q q ď 2Cd γ px, x 1 qp}f } L 8 pC γ b,d q `}u m } L 8 pC 2`γ b,d q q. (2.26)
Let us now focus, as above, on the Hölder control associated with the perturbative contribution in (2.12) when dpx, x 1 q ď pT ´tq 1 2 . Namely,

D 2 x 1 ∆ τ,ξ,ξ 1 m pt, T, x, x 1 q :" ż T t ds ż R nd D 2
x 1 pm,pτ,ξq pt, s, x, yqpL m s ´L m,pτ,ξq s qu m ps, yqdy

´ż T t ds ż R nd D 2
x 1 pm,pτ,ξ 1 q pt, s, x 1 , yqpL m s ´L m,pτ,ξ 1 q s qu m ps, yqdy,

(2.27)

where we recall that a priori the spatial freezing points pξ, ξ 1 q in (2.27) (see also (2.12)) should be different for x and x 1 and depend on the position of dpx, x 1 q w.r.t. the current characteristic time scale in the time integral. Following the terminology of heat kernels, we will say that at time s P rt, T s the points x, x 1 are in the diagonal regime if c 0 d 2 px, x 1 q ď s´t, i.e. their homogeneous distance is small w.r.t. the characteristic time for a parameter c 0 to be specified later on. We insist again that we have the usual equivalence between time and space, i.e. time has to be compared with the square of the spatial metric. Similarly, we will say that the off-diagonal regime holds when c 0 d 2 px, x 1 q ą ps ´tq. Observing that in the diagonal case s ě t `c0 d 2 px, x 1 q (and in the off-diagonal one s ă t `c0 d 2 px, x 1 q) we split the time integral in (2.27) as: ∆ τ,ξ,ξ 1 m pt, T, x, x 1 q :" ∆ τ,ξ,ξ 1 m,diag pt, T, x, x 1 q `∆τ,ξ,ξ 1 m,off-diag pt, x, x 1 q, with ∆ τ,ξ,ξ 1 m,off-diag pt, x, x 1 q :"

ż t`c 0 d 2 px,x 1 q t ds ż R nd
pm,pτ,ξq pt, s, x, yqpL m s ´L m,pτ,ξq s qu m ps, yqdy,

´ż t`c 0 d 2 px,x 1 q t ds ż R nd
pm,pτ,ξ 1 q pt, s, x 1 , yqpL m s ´L m,pτ,ξ 1 q s qu m ps, yqdy,

∆ τ,ξ,ξ 1 m,diag pt, T, x, x 1 q :" ż T t`c 0 d 2 px,x 1 q ds ż R nd
pm,pτ,ξq pt, s, x, yqpL m s ´L m,pτ,ξq s qu m ps, yqdy

´ż T t`c 0 d 2 px,x 1 q ds ż R nd
pm,pτ,ξ 1 q pt, s, x 1 , yqpL m s ´L m,pτ,ξ 1 q s qu m ps, yqdy.

(2.28)

Intuitively, for the term D 2 x 1 ∆ τ,ξ,ξ 1 m,off-diag pt, x, x 1 q, since x, x 1 are far at the characteristic time scale ps ´tq 1 2 , there is no expectable gain in expanding D 2

x 1 pm,pτ,ξ 1 q pt, s, x 1 , yq D2

x 1 pm,pτ,ξq pt, s, x, yq. One therefore writes:

|D 2 x 1 ∆ τ,ξ,ξ 1 m,off-diag pt, x, x 1 q| ď | ż t`c 0 d 2 px,x 1 q t ds ż R nd D 2 x 1 pm,pτ,ξq pt, s, x, yqpL m s ´L m,pτ,ξq s qu m ps, yqdy| `| ż t`c 0 d 2 px,x 1 q t ds ż R nd D 2
x 1 pm,pτ,ξ 1 q pt, s, x 1 , yqpL m s ´L m,pτ,ξ 1 q s qu m ps, yqdy|. Now, provided ξ " x, ξ 1 " x 1 one derives from the previous equation, similarly to (2.19), (2.20), that

|D 2 x 1 ∆ τ,ξ,ξ 1 m,off-diag pt, x, x 1 q| ď C}u m } L 8 pC 2`γ b,d q ż t`c 0 d 2 px,x 1 q t ds ps ´tq 1´γ 2 ď C}u m } L 8 pC 2`γ b,d q c γ 2 0 d γ px, x 1 q. (2.29)
For c 0 small enough, we obtain again an estimate that would be compatible with the global bound on }u m } L 8 pC 2`γ b,d q stated in Theorem 1. Turning now to D 2

x 1 ∆ τ,ξ,ξ 1 m,diag pt, T, x, x 1 q one would therefore be tempted to carry on the analysis with the previous freezing points ξ " x, ξ 1 " x 1 . Intuitively, in the diagonal regime this should not have too much impact. This is only partly true, since if we proceed so we will be led to investigate the difference of operators at different freezing spatial points and this leads to compare quantities like θ m s,t pxq ´θm s,t px 1 q for which we want a uniform control w.r.t. m. Since the initial (unmollified) coefficients a, F are only Hölder continuous in space, this quantity is typically controlled (see Lemma 3) as:

d `θm s,t pxq, θ m s,t px 1 q ˘ď C `dpx, x 1 q `ps ´tq 1 2 ˘, (2.30)
where the time contribution precisely reflects the roughness of the coefficients. Unfortunately, this approach would lead to a final control of order `dpx, x 1 q `ps tq 1 2 ˘γ ď C `dγ px, x 1 q `ps ´tq γ 2 ˘which is not enough on the considered integration set.

Recall indeed that, in the diagonal regime c 0 d 2 px, x 1 q ď ps ´tq and the term ps ´tq γ 2 in the previous r.h.s. is too big. This means that for ∆ τ,ξ,ξ 1 m,diag pt, T, x, x 1 q, it would be more appropriate to consider the same spatial freezing point. In that case, taking ξ " ξ 1 " x and expanding the difference of the derivatives of the frozen Gaussian densities yields:

∆ τ,ξ,ξ 1 m,diag pt, T, x, x 1 q " ´n ÿ j"1 ż T t`c 0 d 2 px,x 1 q ds ż R nd dy ż 1 0 dµD x j D 2
x 1 pm,pτ,ξq pt, s, x `µpx 1 ´xq, yq ¨px 1 ´xq j pL m s ´L m,pτ,ξq s qu m ps, yq

" ´n ÿ j"1 ż T t`c 0 d 2 px,x 1 q ds ż R nd dy ż 1 0 dµD x j D 2
x 1 pm,pτ,ξq pt, s, x `µpx 1 ´xq, yq ¨px 1 ´xq j ´∆1,Fm,σm pt, s, θ m s,t pxq, y, u m q `n ÿ i"2 @ ∆ i,Fm,σm pt, s, θ m s,t pxq, yqD y i u m ps, yq D ¯, (2.31) using the notations introduced in (2.12) and (2.15) for the last equality. In the previous identities (2.31) and from now on, the symbol "¨" between two tensors means the usual tensor contraction. In particular D x j D 2

x 1 pm,pτ,ξq pt, s, x `µpx 1 ´xq, yq ¨px 1 ´xq j is a d ˆd matrix.

In the current diagonal regime, it can be shown from (2.8) and the homogeneity of the distance d that there is C ą 1 such that for ξ " x: |T ´1 s´t px ´x1 q|. Since c 0 d 2 px, x 1 q ď s ´t ðñ c 0 d 2 `ps ´tq 1 2 T ´1 s´t x, ps ´tq 1 2 T ´1 s´t x 1 ˘ď 1, we readily derive from the definition of d in (1.9) that ps ´tq|T ´1 s´t pm m,pt,ξq s,t qpx ´x1 q| 2 ď C. These points are thoroughly discussed in Section 3.

|D x j D 2 x 1 pm,
From (2.31), (2.32) reproducing the previously described analysis, we finally derive:

D 2 x 1 ∆ τ,ξ,ξ 1 m,diag pt, T, x, x 1 q ď C}u m } L 8 pC 2`γ b,d q n ÿ j"1 ż T t`c 0 d 2 px,x 1 q ds ps ´tq 1`pj´1 2 q´γ 2 |px ´x1 q j | ď C}u m } L 8 pC 2`γ b,d q n ÿ j"1 |px ´x1 q j | pc 0 d 2 px, x 1 qq j´1 2 ´γ 2 ď C c n´1 2 ´γ 2 0 }u m } L 8 pC 2`γ b,d q d γ px, x 1 q, (2.33)
using again the definition of d in (1.9) for the last inequality. We have again globally gained, thanks to the smoothness of the coefficients, a power γ 2 in the time singularities of equation (2.32).

From the previous discussion we now have to specify how to modify the freezing parameter depending on the position of the current time variable w.r.t. to the homogeneous spatial distance between the considered points. This can actually been done from the Duhamel formulation up to an additional discontinuity term. Restarting from (2.11) we can indeed rewrite for given pt, x 1 q P r0, T s ˆRnd and any r P pt, T s, ξ 1 P R nd :

u m pt, x 1 q " P m,pτ,ξ 1 q r,t u m pr, x 1 q `G m,pτ,ξ 1 q r,t f m pt, x 1 q `ż r t ds ż R nd dyp m,pτ,ξ 1 q pt, s, x 1 , yqpL m s ´L m,pτ,ξ 1 q s qu m ps, yq, @0 ď v ă r ď T, Gm,pτ,ξ 1 q r,v f m pt, xq " ż r v ds ż R nd
dyp m,pτ,ξ 1 q pt, s, x 1 , yqf m ps, yq.

(2.34)

Differentiating the above expression in r P pt, T s yields for any ξ 1 P R nd :

0 " B r P m,pτ,ξ 1 q r,t u m pr, x 1 q `żR nd dyp m,pτ,ξ 1 q pt, r, x 1 , yqf m pr, yq `żR nd dyp m,pτ,ξ 1 q pt, r, x 1 , yqpL m r ´L m,pτ,ξ 1 q r qu m pr, yqdy.

(2.35)

Integrating (2.35) between t and t 0 P pt, T s for a first given ξ 1 and between t 0 and T with a possibly different ξ1 yields:

0 " P m,pτ,ξ 1 q t 0 ,t u m pt 0 , x 1 q ´um pt, x 1 q `ż t 0 t ds ż R nd dyp m,pτ,ξ 1 q pt, s, x 1 , yqf m ps, yq `ż t 0 t ds ż R nd dyp m,pτ,ξ 1 q pt, s, x 1 , yqpL m s ´L m,pτ,ξ 1 q s qu m ps, yq `P m,pτ, ξ1 q T,t u m pT, x 1 q ´P m,pτ, ξ1 q t 0 ,t u m pt 0 , x 1 q `ż T t 0 ds ż R nd dyp m,pτ, ξ1 q pt, s, x 1 , yqf m ps, yq `ż T t 0 ds ż R nd
dyp m,pτ, ξ1 q pt, s, x 1 , yqpL m s ´L m,pτ, ξ1 q s qu m ps, yq.

Recalling that u m pT, x 1 q " g m px 1 q (terminal condition), and with the notations of (2.34) the above equation rewrites:

u m pt, x 1 q " P m,pτ, ξ1 q T,t g m px 1 q `G m,pτ,ξ 1 q t 0 ,t f m pt, x 1 q `G m,pτ, ξ1 q T,t 0 f m pt, x 1 q `P m,pτ,ξ 1 q t 0 ,t u m pt 0 , x 1 q ´P m,pτ, ξ1 q t 0 ,t u m pt 0 , x 1 q `ż T t ds ż R nd
dy ´Isďt 0 pm,pτ,ξ 1 q pt, s, x 1 , yqpL m s ´L m,pτ,ξ 1 q s q `Isąt 0 pm,pτ, ξ1 q pt, s, x 1 , yqpL m s ´L m,pτ, ξ1 q s q ¯um ps, yq.

(2.36)

We see that for ξ 1 ‰ ξ1 we have an additional discontinuity term deriving from the change of freezing point along the time variable. Of course expression (2.36) can be differentiated in space and taking then t 0 " t `c0 d 2 px, x 1 q, (2.37)

i.e. t 0 precisely corresponds to the critical time at which a change of regime occurs, and ξ 1 " x 1 , ξ1 " x precisely allows, when expanding D 2 x 1 u m pt, xq ´D2

x 1 u m pt, x 1 q using (2.12) for the first term and (2.36) for the second one, to exploit the previous analysis that led to (2.29) and (2.33) and which precisely relied on the suitable choice of freezing point. We again insist on the fact that, in the analysis, t 0 is an additional freezing parameter, which is a posteriori chosen according to (2.37) as a function of pt, x, x 1 q. In particular the parameter t 0 does not intervene in the various possible differentiations of the considered perturbative expansions.

This approach eventually leads to:

|D 2 x 1 u m pt, xq ´D2 x 1 u m pt, x 1 q| ď d γ px, x 1 q " C 1 `}g} C 2`γ b,d `}f } L 8 pC γ b,d q ˘`C `c´n`1 2 `γ 2 0 `cγ 2 0 ˘}u m } L 8 pC 2`γ b,d q ı `ˇˇ`D 2 x 1 P m,pτ,ξ 1 q t 0 ,t u m pt 0 , x 1 q ´D2 x 1 P m,pτ, ξ1 q t 0 ,t u m pt 0 , x 1 q ˘ˇt 0 "t`c 0 d 2 px,x 1 q ˇˇˇˇˇ.
(2.38)

The last contribution can be controlled through cancellation techniques and the key estimate (2.30) on the difference of the flows. The specific choice of t 0 " t `c0 d 2 px, x 1 q then precisely provides the required order leading to:

|D 2 x 1 u m pt, xq ´D2 x 1 u m pt, x 1 q| ď d γ px, x 1 q " C 1 `}g} C 2`γ b,d `}f } L 8 pC γ b,d q ˘`C `c´n`1 2 `γ 2 0 `cγ 2 0 `c0 ˘}u m } L 8 pC 2`γ b,d q ı .
We refer to Lemma 10 for results associated with the discontinuity term in (2.38).

We have detailed up to now what happens with the second order derivatives w.r.t. the non-degenerate variables. The previous procedure can be applied as well to control the Hölder moduli w.r.t. the degenerate ones. We therefore end up with the following kind of estimate:

}u m } L 8 pC 2`γ b,d q ď C 1 `}g} C 2`γ b,d `}f } L 8 pC γ b,d q ˘`}u m } L 8 pC 2`γ b,d q " C `c´n`1 2 `γ 2 0 `c γ 2 0 ˘`T γ 2 ı .
(2.39) Equation (2.39) would provide exactly the expected control if T and C are small enough. On the one hand, the final parameter T can always be chosen small enough. Indeed, the further extension of the Schauder estimates to an arbitrary fixed final time horizon can then be obtained by iterative applications (chaining) of the main estimates on a sufficiently small time interval. On the other hand, it will appear from the proofs that the constant C in (2.39) actually depends on the Hölder norms of the considered coefficients. If these quantities are small, i.e. the coefficients do not vary much and the components that transmit the noise are almost linear, then C will be small. For

" C `c´n`1 2 `γ 2 0 `c γ 2 0 ˘`T γ 2 ı ď k 0 ă 1, we eventually derive: }u m } L 8 pC 2`γ b,d q ď C 1 1 ´k0 `}g} C 2`γ b,d `}f } L 8 pC γ b,d q ˘,
which is precisely the expected control. The general case, is proved through a scaling argument which also allows to balance the opposite effects of c 0 (meant to be small) in the above bounds. This last point will be discussed in Section 6.

The remaining part of this article is organized as follows. We prove in Section 3 various properties for the density of the linearized Gaussian proxy: precise pointwise estimates for the density itself and its derivatives (see equation (2.8)) and some useful controls allowing cancellation arguments in our perturbative analysis. We also give therein some properties associated with the rescaled proxy. Section 4 is then devoted to the control of the supremum norms of the non-degenerate derivatives, corresponding to the previous equation (2.25). Section 5 addresses the issues of Hölder controls. Section 6 is concerned with the above mentioned scaling issues and we also conclude there the final proof of Theorem 1. Eventually, some auxiliary, but crucial, technical results are proved in Appendix 7 for the regularity of the flow and the mean, in Appendix 8 for the regularity of the resolvent and the covariance, and in Appendix 9 for a scaling analysis.

Gaussian proxy and associated controls

We first aim here at proving the control of equation (2.8). We recall that our point is to control the density of p Xm,pτ,ξq s q sPpt,T s satisfying (2.6).

WARNING: for notational simplicity, for the rest of the document we drop the sub and superscripts in m associated with the regularizations. We rewrite, with some notational abuse, for fixed pτ, ξq P r0, T s ˆRnd , the dynamics in (2.6) as:

d Xpτ,ξq v " rFpv, θ v,τ pξqq `DFpv, θ v,τ pξqqp Xpτ,ξq v ´θv,τ pξqqsdv `Bσpv, θ v,τ pξqqdW v , @v P rt, ss, Xpτ,ξq t " x, (3.1)
keeping in mind that F, θ, σ in (3.1) are smooth coefficients. We will give in the next subsection some key-controls to investigate the terms appearing the perturbative expansions (2.11) and (2.12).

@ζ P R nd , C ´1 (3.5) pv ´tq ´1|T v´t ζ| 2 ď x Kpτ,ξq v,t ζ, ζy ď C (3.5) pv ´tq ´1|T v´t ζ| 2 , (3.5)
where for all u ą 0, we denote by T u the intrinsic scale matrix introduced in (2.9). Namely: as in Proposition 1. Also, there exists C :" CppAq, T q ą 0 s.t. for all multiindex α " pα 1 , ¨¨¨, α n q P N n , |α| ď 3 and denoting by D α

T u " ¨uI d,d 0 d,d ¨¨¨0 d,d 0 d,d u 2 I d,
x :" D α 1 x 1 ¨¨¨D αn xn , we have:

|D α x ppτ,ξq pt, s, x, yq| ď C ps ´tq ř n i"1 α i pi´1 2 q`n 2 d 2
exp ´´C ´1ps ´tq ˇˇT ´1 s´t `mpτ,ξq s,t pxq ´y˘ˇˇ2 ": C ps ´tq

ř n i"1 α i pi´1 2 q ppτ,ξq C ´1 pt, s, x, yq, (3.7
)

with ş R nd dyp pτ,ξq
C ´1 pt, s, x, yq, up to a modification of the constants in (3.7).

Remark 3 (A slight abuse of notation)

To ease the reading we denote, when there is no possible ambiguity, pC ´1 pt, s, x, yq :" ppt,xq C ´1 pt, s, x, yq.

Remark 4 (Regularizing effect of the quasi-distance) From equation (3.7), we derive from the definition of d in (1.9) that for any given β ą 0, there exists C β s. pC ´1 pt, s, x, yq, up to a modification of C, which gives the statement for one partial derivative. The controls on the higher order derivatives are obtained similarly (see e.g. the proof of Lemma 5.5 of [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF] for the bounds on D 2

x 1 ppτ,ξq pt, s, x, yq). We now give some useful controls involving the previous Gaussian kernel which will be used in our perturbative analysis. The main interest of the estimates below is that they precisely allow to exploit cancellation techniques.

Proposition 3 For all 0 ď t ď s ď T , px, ξq P R nd ˆRnd , the following identities hold: (3.17)

ż R nd ppτ,
Where, in (3.13), we define for all pi, jq P rr1, nss 2 and M P pR nd q b2 , rM s i,j is the d ˆd block matrix coresponding to the entry of M on the i th line and the j th column.

Proof: First of all remark that equation (3.13) simply follows from a direct covariance computation.

Observe now that from Proposition 2, we have ş R nd ppτ,ξq pt, s, x, yqpy ´mpτ,ξq s,t pxqq 1 dy " 0 d . Differentiating twice this expression w.r.t. x 1 and using the Leibniz formula (recalling as well the identity (3.4) (3.16). Eventually, (3.17) can be derived again from derivation or observing that the sum of the length of the multi-derivation index, here 3, and the power integrated, here 2, is an odd number.

We conclude this section with a key lemma which will be useful to perform our sensitivity analysis of Section 5.

Lemma 1 (Sensitivities of the covariance) There exists C :" CppAqq s.t. for given pξ, ξ 1 q P pR nd q 2 and 0 ď t ă s ď T, px, x 1 q P pR nd q 2 : |r Kpτ,ξq s,t s 1,1 ´r Kpτ,ξ 1 q s,t s 1,1 | ď Cps ´tq `dγ pξ, ξ 1 q `ps ´tq

γ 2 ˘. (3.18)
The proof of Lemma 1 is postponed to Appendix 7 for the sake of clarity. We importantly point out that, in Lemma 1, the constant C mainly depends on the Hölder norms of the coefficients and is small provided the coefficients do not vary much. Precisely, it can be shown that C writes:

C :" C`} a} L 8 pC γ d q `n ÿ i"2 }F i } L 8 pC 2i´3`γ d q ˘(3.19)
for some universal constant C, where, in the above equation, we write with the notation of Section 1.2:

}F i } L 8 pC 2i´3`γ d q :" sup pt,zqPr0,T sˆR pn´i`2qd }pD x i´1 F i q i´1 pt, z, ¨q} C γ 2pi´1q´1 pR d ,R d bR d q `n ÿ j"i sup pt,zqPr0,T sˆR pn´i`2qd }pF i q j pt, z, ¨q} C 2i´3`γ 2j´1 pR d ,R d q . (3.20)
Namely, the quantity }F i } L 8 pC 2i´3`γ d q gathers the Hölder moduli of F i at the intrinsic associated scales according to the distance d in the variables j P rri, nss as well as the Hölder norm of the gradient w.r.t. the component which transmits the noise (but importantly not its supremum norm). Said differently, the L 8 pC 2i´3`γ d q norm of F i gathers the Hölder norms of the fractional parts of `Dt 2i´3`γ 2j´1 u x j F i ˘jPrri´1,nss in the j th variable with corresponding Hölder index 2i´3`γ 2j´1 ´t 2i´3`γ 2j´1 u, where t¨u stands for the integer part. We again refer to Appendix 7 for a precise statement and proof of this assertion (3.19) (See also Lemma 12 below and its proof for similar properties).

Additional controls associated with the (mollified) flow and its linearization

We now state three important estimates associated with the linearization or sensitivity w.r.t. the initial point for the frozen (mollified) differential system (2.5). For the sake of simplicity their proof is also postponed to Appendix 7. We have:

Lemma 2 There exists C :" CppAqq s.t. for all 0 ď t ď s ď T , px, x 1 q P R nd ˆRnd and px, x 1 q P pR nd q 2 : dpθ s,t pxq, θ s,t px 1 qq ď C ´dpx, x 1 q `ps ´tq

1 2 ¯.
The proof of Lemma 2 is postponed to Appendix 7.1.

The second important result concerns the impact of the freezing point in the linearization procedure. Namely, Lemma 3 (Sensitivity of the linearized flow w.r.t. the freezing parameter) Bhououou There exists C :" CppAqq s.t. for all τ " t, px, x 1 q P pR nd q 2 at the change of regime time t 0 defined in (2.37):

d

`mpτ,xq t 0 ,t px 1 q, m pτ,x 1 q t 0 ,t px 1 q ˘" d `mpτ,xq t 0 ,t px 1 q, θ t 0 ,t px 1 q ˘ď Cc 1 2n´1 0 dpx, x 1 q.

Again, the proof of Lemma 3 is postponed to Appendix 7.1.

Control of the supremum of the derivatives in the non-degenerate variables

WARNING: for notational simplicity, we drop from now on the sub and superscripts in τ associated with the linearization, since this parameter is chosen to be equal to t. For example, ppτ,ξq , m pτ,ξq s,t , θ u,τ become respectively pξ , m ξ s,t , θ u,t .

The point is now to control }u m } C 2`α b,d

. From (2.3) it is direct to derive that for all pt, xq P r0, T s ˆRnd :

|upt, xq| ď }g} L 8 `T }f } L 8 .
Let us now start from (2.11) to control pointwise the second order derivatives of u m in the non-degenerate variables, i.e. }D 2

x 1 u} L 8 . The first ones can be controlled similarly and more directly. Write for all pt, xq P r0, T s ˆRnd :

|D 2 x 1 upt, xq| ď }D 2 x 1 P ξ T,t g} 8 `}D 2 x 1 Gξ f } 8 `ˇˇż T t ds ż R nd D 2 x 1 pξ pt, s, x, yq∆ 1,F,σ pt, s, θ s,t pξq, y, uqdy ˇň ÿ i"2 ż T t ds ż R nd D y i ¨´`D 2 x 1 pξ pt, s, x, yq b ∆ i,F pt, s, θ s,

Control of the non-degenerate part of the perturbative term

The aim of this section is to prove identity (2.14) appearing in the user's guide to the proof.

To this end, we provide a general differentiation result, which will be useful as well in Section 5 to deal with the Hölder norms. Under the current assumptions on a, F, the following lemma holds.

Lemma 4 (First Derivative Control Lemma for the non-degenerate variable)

For all multi-index α " pα 1 , . . . , α n q P N d : ˇˇż

R nd D α
x pξ pt, s, x, yq∆ 1,F,σ pt, s, θ s,t pξq, y, uqdy ˇˇˇˇˇˇξ "x ď C}u} L 8 pC 2`γ b,d q ps ´tq ´řn j"1 α j pj´1 2 q`γ 2 .

(4.3)

Partie III: Degenerate Kolmogorov chains

Proof: Lemma 4

We first recall the control (2.12)

|∆ 1,F,σ pt, s, y, θ s,t pξq, uq|ď ´rF 1 s d,γ }D x 1 ups, ¨q} L 8 `1 2 raps, ¨qs d,γ }D 2 x 1 ups, ¨q} L 8 ¯dγ py, θ s,t pξqq.

From this control and Proposition 2, we directly obtain:

ˇˇż R nd D α
x pξ pt, s, x, yq∆ 1,F,σ pt, s, θ s,t pξq, y, uqdy ˇˇˇˇˇˇξ

"x ď ´żR nd |D α x pξ pt, s, x, yq|d γ `y, θ s,t pξq ˘`}F 1 } L 8 pC γ d q }D x 1 u} L 8 `}a} L 8 pC γ d q }D 2 x 1 u} L 8 ˘dy ¯ˇˇξ "x ď C}u} L 8 pC 2`γ b,d q ps ´tq ´řn j"1 α j pj´1 2 q ż R nd pC ´1 ps, t, x, yqd γ pθ s,t pxq, yqdy ď C}u} L 8 pC 2`γ b,d q ps ´tq ´řn j"1 α j pj´1 2 q`γ 2 ,
with the notations of Remark 3 for the last but one inequality. ˝Equation (2.14) readily follows from Lemma 4 taking α " p2, 0, . . . , 0q. Namely:

ˇˇż T t ds ż R nd D α x pξ pt, s, x, yq∆ 1,F,σ pt, s, θ s,t pξq, y, uqdy ˇˇˇˇξ "x ď C}u} L 8 pC 2`γ b,d q ż T t ds ps ´tq 1´γ 2 ď C}u} L 8 pC 2`γ b,d q pT ´tq γ 2 . (4.4)

Control of the degenerate part of the perturbative term

The point is here to control the terms

n ÿ i"2 ż T t ds ż R nd D 2
x 1 pξ pt, s, x, yq @ ∆ i,F pt, s, θ s,t pξq, yq, D y i ups, yq D dy appearing in equation (2.16) of the user's guide to the proof. We precisely want to derive equation (2.22). The bound will actually follow from the more general following result, which will again be useful for the Hölder norm in Section 5.

Lemma 5 (First Besov Control Lemma) For all multi-index α " pα 1 , . . . , α n q P N d :

n ÿ i"2 ˇˇż R nd D α x pξ pt, s, x, yq @ ∆ i,F pt, s, θ s,t pξq, yq, D y i ups, yq D dy ˇˇˇˇˇˇξ "x ď C}u} L 8 pC 2`γ b,d q ps ´tq ´řn j"1 α j pj´1 2 q`γ 2 . (4.5)
Proof: Lemma 5 Similarly to (2.16), we define for all i P rr2, nss, ϑ α i,pt,xq ps, yq :"

D y i ¨`D α x pξ pt, s, x, yq b ∆ i,F pt, s, θ s,t pξq, yq ˘": D y i ¨`Θ α i,pt,xq ps, yq ˘. (4.6)
The contribution of the l.h.s. in (4.5) then rewrites:

n ÿ i"2 ˇˇż R nd D α
x ppt,ξq pt, s, x, yq @ ∆ i,F pt, s, θ s,t pξq, yq, D y i ups, yq D dy ˇˇˇˇˇˇξ

"x " n ÿ i"2 ˇˇż R nd ϑ α i,pt,xq ps, yqups, yqdy ˇˇ. (4.7)
The point now is to observe that for any fixed i P rr2, nss and z " pz 1 , ¨¨¨, z i´1 , z i`1 , ¨¨¨, z n q P R pn´1qd the mapping

y i Þ Ñ ups, z 1:i´1 , y i , z i`1:n q is in C 2`γ 2i´1 b pR d q " B 2`γ 2i´1
8,8 pR d q using the Besov space terminology, see e.g. Triebel [START_REF] Triebel | Theory of function spaces[END_REF], uniformly in s P r0, T s. The point is now to put in duality the mappings y i Þ Ñ u m ps, y 1:i´1 , y i , y i`1:n q and Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q :

y i Þ Ñ ϑ α i,pt,xq ps, yq, (4.8)
see e.g. Proposition 3.6 in [START_REF] Lemarie-Rieusset | Recent developments in the Navier-Stokes problem[END_REF]. This precisely means that we have to prove that Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q lies in the suitable Besov space, Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q P B ´2`γ 2i´1 1,1 pR d q and to control the associated norm We will actually prove that those norms provide an integrable quantity w.r.t. y 1:i´1 , y i`1:n as well as an integrable time singularity. This will be done through the thermic characterization of Besov spaces, see e.g. Section 2.6.4 in [START_REF] Triebel | Theory of function spaces[END_REF] as well as (2.18) above. Precisely, we recall that for a function ψ : R d Ñ R in B ´α i 1,1 pR d q, αi :" 2`γ 2i´1 a quasi-norm is given by:

}ψ} B ´α i 1,1 pR d q :" }ϕpDqψ} L 1 pR d ,Rq `ż 1 0 dv v v αi 2 }h v ‹ ψ} L 1 pR d ,Rq , (4.9)
with

@z P R d , h v pzq :" 1 p2πvq d 2 exp `´|z| 2 v ˘,
being the usual heat kernel of R d , ‹ standing for the usual convolution on R d for ϕ P C 8 0 pR d , Rq s.t. ϕp0q ‰ 0. Taking ψ " Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q in the above characterization and from definition (4.6), the main advantage of using (4.9) consists in rebalancing the derivative appearing in the definition (4.6) to the heat kernel or to the smooth compactly supported function ϕ. Precisely, focusing on the heat kernel part of (4.9), the other contribution being handled more directly, we get:

}h v ‹ Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq " ż R d ˇˇż R d h v pz ´yi qϑ α i,pt,xq ps, yqdy i ˇˇdz " (4.6) ż R d ˇˇż R d h v pz ´yi qD y i ¨`Θ α i,pt,xq ps, yq ˘dy i ˇˇdz " ż R d ˇˇż R d D α
x pξ pt, s, x, yq @ ∆ i,F pt, s, θ s,t pξq, yq, D z h v pz ´yi q D dy i ˇˇdz. (4.10)

We now want to estimate the }Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q } B ´α i 1,1

. To this end, we split the time integral in (4.9) into two parts writing:

ż 1 0 dv v v αi 2 }h v ‹ Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq " ż ps´tq β i 0 dv v v αi 2 }h v ‹ Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq `ż 1 ps´tq β i dv v v αi 2 }h v ‹ Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq (4.11)
for a parameter β i ą 0 to be specified. Precisely, in order to have a similar smoothing effect in time than for the terms appearing in (4.4), we now want to calibrate β i to obtain:

ż 1 ps´tq β i dv v v αi 2 }h v ‹ Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq ď C ps ´tq ř j"1 α j pj´1 2 q´γ 2 qczi pt, s, x, py 1:i´1 , y i`1:n qq, (4.12)
where introducing:

qc pt, s, x, yq :" n ź j"1 N cps´tq 2j´1 `pθ s,t pxq ´yq j ˘" pc ´1 pt, s, x, yq,
where for a ą 0, z P R d , N a pzq " 1

p2πaq d 2
exp `´|z| 2 2a ˘stands for the standard Gaussian density of R d with covariance matrix aI d , we introduce: qczi pt, s, x, py 1:i´1 , y i`1:n qq " ź jPrr1,nss,j‰i N cps´tq 2j´1 `pθ s,t pxq ´yq j ˘.

(4.13)

To choose properly the parameter β i leading to (4.12), we now write:

ż 1 ps´tq β i dv v v αi 2 }h v ‹ Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq ď ż 1 ps´tq β i dv v v αi 2 ż R d dz ˇˇż R d D α x pξ pt, s, x, yq @ ∆ i,F pt, s, θ s,t pξq, yq, D z h v pz ´yi q D dy i ˇˇˇˇˇˇξ "x ď C ż 1 ps´tq β i dv v v αi 2 ż R d dz ż R d dy i h cv pz ´yi q v 1 2 qc pt, s, x, yq ps ´tq ř n j"1 α j pj´1 2 q d 2i´3`γ pθ s,t pxq, yq ď C ż 1 ps´tq β i dv v v αi 2 ż R d dz ż R d dy i h cv pz ´yi q v 1 2 qc pt, s, x, yq ps ´tq ř n j"1 α j pj´1 2 q ps ´tq 2i´3`γ 2 ď C qczi pt, s, x, py 1:i´1 , y i`1:n qq ż 1 ps´tq β i dvv ´3 2 `α i 2 ps ´tq ´řn j"1 α j pj´1 2 q`2 i´3`γ 2 ď C qczi pt, s, x, py 1:i´1 , y i`1:n qqps ´tq r´1 2 `α i 2 sβ i ´řn j"1 α j pj´1 2 q`2 i´3`γ
using Proposition 2 for the third inequality (see also Remark 4, equation (3.8)) recalling as well that 0 ď t ă s ď T is small.

To obtain (4.12), we then take:

r´1 2 `α i 2 sβ i ´n ÿ j"1 α j pj ´1 2 q`2 i ´3 `γ 2 " ´n ÿ j"1 α j pj ´1 2 q`γ 2 ðñ β i " p2i ´3qp2i ´1q 2i ´3 ´γ . (4.14)
The key point is now to check that the previous choice of β i also yields a bound similar to (4.12) for the contribution in (4.11) associated with v P r0, ps ´tq β i s.

To this end, we restart from identity (4.10), which allows to exploit partial cancellations w.r.t. the integration variable y i . Namely, write:

ż R d h v pz ´yi qD y i ¨Θα i,pt,xq ps, yqdy i " ż R d h v pz ´yi qD y i ¨´Θ α i,pt,xq ps, yq ´Θα i,pt,xq ps, y 1:i´1 , z, y i`1:n q ¯dy i " ż R d D α x pξ pt, s, x, yq @ F i ps, yq ´Fi ps, y 1:i´1 , z, y i`1:n q, D z h v pz ´yi q D dy i `żR d `Dα
x pξ pt, s, x, yq ´Dα

x pξ pt, s, x, y 1:i´1 , z, y i`1:n q @F i ps, y 1:i´1 , z, y i`1:n q ´Fi ps, θ s,t pξqq ´Dx i´1 F i ps, θ s,t pξqqpy ´θs,t pξqq i´1 , D z h v pz ´yi q D dy i ": ´T1 `T2 ¯`v, t, s, x, py 1:i´1 , z, y i`1:n q ˘, (4.15) using the definition in (4.6) and (4.10) for the last decomposition.

Write now:

|T 1 `v, t, s, x, py 1:i´1 , z, y i`1:n q ˘| ď C ż R d h cv pz ´yi q v 1 2 qc pt, s, x, yq ps ´tq ř n j"1 α j pj´1 2 q |z ´yi | 2i´3`γ 2i´1 dy i ď C ż R d h cv pz ´yi q v 2´γ 4i´2
qc pt, s, x, yq ps ´tq

ř n j"1 α j pj´1 2 q dy i . (4.16)
We thus derive from (4.16):

|T 1 `v, t, s, x, py 1:i´1 , z, y i`1:n q ˘| (4.17)

ď C v 2´γ
4i´2 ps ´tq ř n j"1 α j pj´1 2 q qczi pt, s, x, py 1:i´1 , y i`1:n qqN cv`ps´tq 2i´1 `z ´θs,t pxq i ˘.

Write now from (4.15) and the Taylor formula applied to the i th variable for the difference of the derivatives of the densities:

|T 2 `v, t, s, x, py 1:i´1 , z, y i`1:n q ˘| ď C ż R d dy i h cv pz ´yi q v 1 2
ż 1 0 dµ qc pt, s, x, y 1:i´1 , z `µpy i ´zq, y i`1:n q ps ´tq ř n j"1 α j pj´1 2 q`2 i´1 2 ˆ|y i ´z| ´ˇˇF i ps, y 1:i´1 , z, y i`1:n q ´Fi ps, y 1:i´1 , θ s,t pxq i:n q ˇF i ps, y 1:i´1 , θ s,t pxq i:n q ´Fi ps, θ s,t pξqq ´Dx i´1 F i ps, θ s,t pxqqpy ´θs,t pxqq i´1 ˇˇď

C ż R d dy i h cv pz ´yi q ż 1 0 dµ qc pt, s, x, y 1:i´1 , z `µpy i ´zq, y i`1:n q ps ´tq ř n j"1 α j pj´1 2 q`2 i´1 2 ˆ´|z ´θs,t pxq i | 2i´3`γ 2i´1 `|pθ s,t pxq ´yq i´1 | 1`γ 2pi´1q´1 `n ÿ k"i`1 |pθ s,t pxq ´yq k | 2i´3`γ 2k´1 ¯.
Writing, for any µ P r0, 1s,

|z ´θs,t pxq i | ď µ|z ´yi | `|z `µpy i ´zq ´pθ s,t pxqq i |,
we thus derive

|T 2 `v, t, s, x, py 1:i´1 , z, y i`1:n q ˘| ď C ż R d dy i h cv pz ´yi q ż 1 0 dµ qc pt, s, x, y 1:i´1 , z `µpy i ´zq, y i`1:n q ps ´tq ř n j"1 α j pj´1 2 q`2 i´1 2 ˆ´|y i ´z| 2i´3`γ 2i´1
`d2i´3`γ ´θs,t pxq, py 1:i´1 , z `µpy i ´zq,

y i`1:n q ¯ď C ż R d dy i h cv pz ´yi q ż 1 0 dµq c pt, s, x, y 1:i´1 , z `µpy i ´zq, y i`1:n q ˆ´v 2i´3`γ 2p2i´1q
ps ´tq

ř n j"1 α j pj´1 2 q`2 i´1 2 `1 ps ´tq ř n j"1 α j pj´1 2 q`1´γ 2 ď C qczi pt, s, x, y 1:i´1 , y i`1:n q ż 1 0 dµ ż R d h cv pz ´yi qN cps´tq 2i´1 pz `µpy i ´zq´pθ s,t pxqq i qdy i ˆ´v 2i´3`γ 2p2i´1q
ps ´tq

ř n j"1 α j pj´1 2 q`2 i´1 2 `1 ps ´tq ř n j"1 α j pj´1 2 q`1´γ 2 ¯, (4.18)
using again (3.8) for the second inequality. From (4.15), (4.17) and (4.18) we derive, with the notation introduced in (4.13):

}h v ‹ Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1 q } L 1 pR d ,Rq ď ´1 v 2´γ 4i´2 ps ´tq ř n j"1 α j pj´1 2 q `v 2i´3`γ 2p2i´1q ps ´tq ř n j"1 α j pj´1 2 q`2 i´1 2 `1 ps ´tq ř n j"1 α j pj´1 2 q`1´γ 2 C qczi pt, s, x, py 1:i´1 , y i`1:n qq ˆż 1 0 dµ ż R d dz ż R d dy i h cv pz ´yi qN cps´tq 2i´1 pz `µpy i ´zq ´pθ s,t pxqq i q ď C qczi pt, s, x, py 1:i´1 , y i`1:n qq ˆ´1 v 2´γ 4i´2 ps ´tq ř n j"1 α j pj´1 2 q `v 2i´3`γ 2p2i´1q
ps ´tq

ř n j"1 α j pj´1 2 q`2 i´1 2 `1 ps ´tq ř n j"1 α j pj´1 2 q`1´γ 2 ¯,
using the change of variable pw 1 , w 2 q " pz ´yi , z `µpy i ´zq ´pθ s,t pxqq i q for the last inequality. From the above computations and with the notations of (4.11), we derive:

ż ps´tq β i 0 dvv αi 2 ´1}h v ‹ Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1 q } L 1 pR d ,Rq ď C qczi pt, s, x, py 1:i´1 , y i`1:n qq ż ps´tq β i 0 dv v v αi 2 ˆ´1 v 2´γ 4i´2 ps ´tq ř n j"1 α j pj´1 2 q `v 2i´3`γ 2p2i´1q ps ´tq ř n j"1 α j pj´1 2 q`2 i´1 2 `1 ps ´tq ř n j"1 α j pj´1 2 q`1´γ 2 ":
C qczi pt, s, x, py 1:i´1 , y i`1:n qqB α,β i pt, sq.

Let us now prove that for β i " p2i´3qp2i´1q 2i´3´γ . defined in (4.14), we have:

B α,β i pt, sq ď C ps ´tq ř n j"1 α j pj´1 2 q´γ 2 . ( 4.19)
To prove (4.19), we now write:

B α,β i pt, sq ď C " v αi 2 ´2´γ 4i´2 ps ´tq ř n j"1 α j pj´1 2 q `v αi 2 `2i´3`γ 2p2i´1q ps ´tq ř n j"1 α j pj´1 2 q`2 i´1 2 `v αi 2 ps ´tq ř n j"1 α j pj´1 2 q`1´γ 2 ı v"ps´tq β i v"0 ď C " ps ´tq β i p αi 2 ´2´γ 4i´2 q´ř n j"1 α j pj´1 2 q `ps ´tq β i p αi 2 `2i´3`γ 2p2i´1q q´p ř n j"1 α j pj´1 2 q`2 i´1 2 q `ps ´tq β i αi 2 ´řn j"1 α j pj´1 2 q´1`γ 2 ı .
From the above equation, (4.19) holds as soon as β i can be chosen so that the three following conditions hold:

β i ˆα i 2 ´2 ´γ 4i ´2 ˙´γ 2 ě 0, β i ˆα i 2 `2i ´3 `γ 2p2i ´1q ˙´2i ´1 2 ´γ 2 ě 0, β i αi 2 ´1 ě 0.
Recalling that αi 2 "

1`γ 2 2i´1 and for the previous choice of β i , the above conditions rewrite:

ˆp2i ´3qp2i ´1q 2i ´3 ´γ ˙ˆ1 `γ 2 2i ´1 ´1 ´γ 2 2i ´1 ˙´γ 2 ě 0 ðñ p2i ´3q 2i ´3 ´γ γ ´γ 2 ě 0, ˆp2i ´3qp2i ´1q 2i ´3 ´γ ˙ˆ2 `γ 2p2i ´1q `2i ´3 `γ 2p2i ´1q ˙´2i ´1 2 ´γ 2 ě 0 ðñ ˆ2i ´3 2i ´3 ´γ ˙p2i ´1 `2γq ´p2i ´1 `γq ě 0, ˆp2i ´3qp2i ´1q 2i ´3 ´γ ˙1 `γ 2 2i ´1 ´1 ě 0 ðñ ˆ2i ´3 2i ´3 ´γ ˙p1 `γ 2 q ´1 ě 0.
All the above conditions are true for i P rr2, nss, γ P p0, 1s. Note the chosen β i seems to be rather sharp in the sense that letting γ go to 0 the above constraints become equalities. This proves (4.19). We finally get:

ż 1 0 dv v v αi 2 }h v ‹Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq ď C ps ´tq ř n j"1 α j pj´1 2 q´γ 2 qczi pt, s, x, py 1:i´1 , y i`1:n qq.
(4.20) Reproducing the previous computations we also write for a C 8 compactly supported function ϕ:

}ϕpDqΨ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq ď ż R d ˇˇż R d D y i p ϕpz ´yi q ¨`D α
x pξ pt, s, x, yq b ∆ i,F pt, s, θ s,t pξq, yqydy i ˇˇdz ˇˇˇξ

"x ď C ps ´tq ř n j"1 α j pj´1 2 q ż R d qc pt, s, x, yqd 2i´3`γ pθ s,t pxq, yqdy i ď C
ps ´tq ř n j"1 α j pj´1 2 q´pi´3 2 `γ 2 q qczi pt, s, x, py 1:i´1 , y i`1:n qq.

From (4.9) and (4.20), we finally obtain:

}Ψ α i,pt,xq,ps,y 1:i´1 ,y i`1:n q } B ´α i 1,1 ď C ps ´tq ř n j"1 α j pj´1 2 q´γ 2 qczi pt, s, x, py 1:i´1 , y i`1:n qq, (4.21)
which together with (4.7) and (4.8) gives the result. Equation (2.23) now follows from Lemma 5 taking α " p2, 0, . . . , 0q. Namely,

ˇˇn ÿ i"2 ż T t ds ż R nd D 2 x 1 pξ pt, s, x, yq @ ∆ i,F pt, s, θ s,t pξq, yq, D y i ups, yq D ¯ˇˇˇˇˇˇξ "x ď CpT ´tq γ 2 }u} L 8 pC 2`γ b,d q .
(4.22)

Non-degenerate derivatives for the frozen semi-group : terminal condition and source

The main result of this section is the following lemma.

Lemma 6 (Derivatives of frozen semi-group and Green kernel) There exists a constant C :" CppAq s.t. for all pt, xq P r0, T s ˆRnd ,

|D 2 x 1 P ξ T,t gpxq| ˇˇξ "x ď C}D 2 x 1 g} L 8 ď C}g} C 2`γ b,d , |D 2 x 1 Gξ f pt, xq| ˇˇξ "x ď CpT ´tq γ 2 }f } L 8 pC γ b,d q q.
Proof: Lemma 6 Note first that, since pξ is a density one has the following cancellation identity:

ˇˇD 2 x 1 P ξ T,t gpxq ˇˇˇˇξ "x " ˇˇż R nd D 2 x 1 pξ pt, T, x, yqrgpyq ´gpm ξ T,t pxqqsdy ˇˇˇˇˇˇξ "x ď ˇˇż R nd D 2 x 1 pξ pt, T, x, yqrgpyq ´gpy 1 , m ξ T,t pxq 2:n qsdy ˇˇˇˇˇˇξ "x `ˇˇż R nd D 2 x 1
pξ pt, T, x, yqrgpy 1 , m ξ T,t pxq 2:n q ´gpm ξ T,t pxqqsdy ˇˇˇˇˇˇξ

"x . (4.23)
The first term in the r.h.s. of the previous identity is readily controlled thanks to Proposition 2 ˇˇż

R nd D 2
x 1 pξ pt, T, x, yqrgpyq ´gpy 1 , m ξ T,t pxq 2:n qsdy ˇˇˇˇˇˇξ

"x ď C}g} C 2`γ b,d ż R nd
pT ´tq ´1 pC ´1 pt, T, x, yqd 2`γ pm ξ T,t pxq, yqdy ˇˇξ

"x ď CpT ´tq γ 2 }g} C 2`γ b,d . (4.24)
The second term of (4.23) is more subtle. We need to expand gpy 1 , m ξ T,t pxq 2:n q in its non-degenerate variable to take advantage of the corresponding regularity of g. Namely,:

ˇˇż R nd D 2 x 1 pξ pt, T, x, yqrgpy 1 , m ξ T,t pxq 2:n q ´gpm ξ T,t pxqqsdy ˇˇˇˇˇˇξ "x " ˇˇˇż R nd D 2 x 1 pξ pt, T, x, yq ˆ@D x 1 gpm ξ T,t pxqq, py ´mξ T,t pxqq 1 D `ż 1 0 dµp1´µqTr ´D2 x 1 g `mξ T,t pxq 1 `µpy´m ξ T,t pxqq 1 , m ξ T,t pxq 2:n ˘`y´m ξ T,t pxq ˘b2 1 ¯dy ˇˇˇˇˇˇˇˇξ "x ď C}D 2 x 1 g} L 8 ż R nd pC ´1 pt, T, x, yq T ´t |py ´mξ T,t pxqq 1 | 2 dy ˇˇˇξ "x ď C}g} C 2`γ b,d , (4.25)
recalling from Proposition 3 for the last but one inequality that

ż R d dyD 2 x 1 pm,ξ pt, T, x, yq @ D x 1 gpm ξ T,t pxqq, py ´mξ T,t pxqq 1 D ˇˇξ "x " 0 d,d .
Gathering identities (4.24), (4.25) into (4.23), we obtain the stated control for the term

|D 2 x 1 P ξ T,t gpxq| ˇˇξ "x
.

Let us now turn to the Green kernel. We directly get from Proposition 2:

|D 2 x 1 Gξ f pt, xq| ˇˇξ "x ď ˇˇż T t ds ż R nd D 2
x 1 pξ pt, s, x, yqrf ps, yq ´f ps, θ s,t pξqqsdy ˇˇˇˇξ

"x ď Crf s L 8 pC γ q ż T t ds ż R nd 1 s ´t pC ´1 ps, t, x, yqdpθ s,t pxq, yq γ dy ď Crf s L 8 pC γ q pT ´tq γ 2 ,
which gives the result.

Similarly to Lemma 6, we prove from Proposition 2 the following result:

Lemma 7 For γ P p0, 1q, under (A), there exists C :" CppAq, T q, s.t. for all function ψ P C γ b,d pR nd , Rq, and any given multi-index α, |α| P rr1, 3ss, for all 0 ď t ă s ď T,

x P R nd : |D α x P ξ s,t ψpxq| ˇˇξ"x ď C}ψ} C γ b,d ps ´tq ´řn i"1 |α i |pi´1 2 q`γ 2 . (4.26)
Note carefully that in the above lemma |α| ě 1. Indeed, if |α| " 0 we cannot benefit from any regularizing effects which are precisely due to cancellation techniques.

Hölder controls

The purpose of the section is to give suitable Hölder controls of the terms rD 2 x 1 upt, ¨qs γ and sup zPR d rupt, z, ¨qs 2`γ in order to derive our main Schauder estimate of Theorem 1. For a fixed time t P r0, T s, the point is to consider for fixed x P R nd the perturbative expansion (2.11) and for another spatial point x 1 P R nd , the possibly more refined version provided by equation (2.36) which precisely allows to take into account the various regimes depending on dpx, x 1 q and ps ´tq 1{2 , s P rt, T s detailed in the previous user's guide to the proof.

We will address separately two cases. For a constant c 0 to be specified later on (but formally meant to be small ), we consider:

-The globally off-diagonal regime T ´t ă c 0 d 2 px, x 1 q. In that case, the spatial points

x, x 1 are globally far for the corresponding homogeneous distance d over the time horizon s P rt, T s. Hence, there is no specific need to exploit (2.36). Expanding the quantities D 2

x 1 upt, xq ´D2

x 1 upt, x 1 q and upt, z, x 2:n q ´upt, z, x 1 2:n q with (2.11) is enough to get the result.

-The mixed regime T ´t ě c 0 d 2 px, x 1 q. In that case, up to the transition time t 0 defined in (2.37), s ´t ă c 0 d 2 px, x 1 q, i.e. the off-diagonal regime holds for the times s close to t. It is then crucial to exploit the more refined perturbative expansion (2.36) to derive suitable bounds for D 2 x 1 upt, xq´D 2 x 1 upt, x 1 q and u `t, pz, x 2:n q ˘´u `t, pz, x 1 2:n q ˘. In particular, this leads to handle carefully the additional terms appearing from the change of freezing parameter.

Precisely, in the globally off-diagonal regime we first write from (2.11) for all px, x 1 q P pR nd q 2 s.t. pT ´tq ă c 0 d 2 px, x 1 q, D 2

x 1 upt, xq ´D2

x 1 upt, x 1 q " ˜D2

x 1 P ξ T,t gpxq ´D2

x

1 1 P ξ 1 T,t gpx 1 q `D2 x 1 Gξ f pt, xq ´D2 x 1 Gξ 1 f pt, x 1 q `ż T t ds ż R nd ´D2 x 1 pξ pt, s, x, yqpL s ´L ξ s qups, yq ´D2 x 1 pξ 1 pt, s, x 1 , yqpL s ´L ξ 1 s qups, yq ¯dy ¸ˇˇˇˇp ξ,ξ 1 q"px,x 1 q , ( 5.1)
and similarly, for all z P R d , u `t, pz, x 2:n q ˘´u `t, pz, x 1 2:n q

" ˜P ξ T,t gpz, x 2:n q ´P ξ 1 T,t gpz, x 1 2:n q `G ξ f `t, pz, x 2:n q ˘´G ξ 1 f `t, pz, x 1 2:n q ż T t ds ż R nd
´p ξ pt, s, pz, x 2:n q, yqpL s ´L ξ s qups, yq ´p ξ 1 pt, s, pz, x 1 2:n q, yqpL s ´L ξ 1 s qups, yq ¯dy ¸ˇˇˇˇp ξ,ξ 1 q" `pz,x 2:n q,pz,x 1 2:n q . (5.2)

In this case, we derive from Lemmas 4, 5, 6 and equation (4.1):

|D 2 x 1 upt, xq ´D2 x 1 upt, x 1 q| ď C " |D 2 x 1 P ξ T,t gpxq ´D2 x 1 1 P ξ 1 T,t gpx 1 q| ``}f } L 8 pC γ b,d q `}u} L 8 pC 2`γ b,d q ˘pT ´tq γ 2 ı ď C " |D 2 x 1 P ξ T,t gpxq ´D2 x 1 1 P ξ 1 T,t gpx 1 q| ``}f } L 8 pC γ b,d q `}u} L 8 pC 2`γ b,d q ˘cγ 2 0 d γ px, x 1 q ı , |u `t, pz, x 2:n q ˘´u `t, pz, x 1 2:n q ˘| ď C ˜| P ξ T,t gpz, x 2:n q ´P ξ 1 T,t gpz, x 1 2:n q| `| Gξ f `t, pz, x 2:n q ˘´G ξ 1 f `t, pz, x 1 2:n q ˘| `}u} L 8 pC 2`γ b,d q c γ 2 0 d γ px, x 1 q ¸.
Therefore, these equations give the expected controls up to appropriate estimates for the Hölder moduli of the frozen semigroup and Green kernel which are obtained below, for the so-called mixed regime, on which we can focus.

In the mixed regime, we will start from identity (2.36) for the expansion of upt, x 1 q, where we have chosen ξ 1 " x 1 and ξ1 " x. Namely, upt, x 1 q " P ξ1 T,t gpx 1 q `G ξ 1 t 0 ,t f pt, x 1 q `G ξ1 T,t 0 f pt, x 1 q `P ξ 1 t 0 ,t upt 0 , x 1 q ´P ξ1 t 0 ,t upt 0 ,

x 1 q `ż T t ds ż R nd dy ´Isďt 0 pξ 1 pt, s, x 1 , yqpL s ´L ξ 1 s q `Ĩ sąt 0 p ξ1 pt, s, x 1 , yqpL s ´L ξ1 s q ¯ups, yq.
Again, t 0 must be here seen as a frozen parameter, which is a posteriori, i.e. after possible differentiation, chosen as in (2.37).

The terms to control then write for the Hölder norm of the derivatives w.r.t. the non-degenerate variables:

D 2 x 1 upt, xq ´D2 x 1 upt, x 1 q " ˜´D 2 x 1 P ξ T,t gpxq ´D2 x 1 1 P ξ T,t gpx 1 q ¯`´D 2 x 1 Gξ t 0 ,t f pt, xq ´D2 x 1 Gξ 1 t 0 ,t f pt, x 1 q D2 x 1 Gξ T,t 0 f pt, xq ´D2 x 1 Gξ t 0 ,t f pt, x 1 q ¯`´D 2 x 1 P ξ 1 t 0 ,t upt 0 , x 1 q ´D2 x 1 P ξ t 0 ,t upt 0 , x 1 q ż T t ds ż R nd dy ´p ξ pt, s, x, yqpL s ´L ξ s q
´"I sďt 0 pξ 1 pt, s, x 1 , yqpL s ´L ξ 1 s q `Ĩ sąt 0 p ξ pt, s, x 1 , yqpL s ´L ξ s q ı¯u ps, yq ¸ˇˇˇˇp ξ,ξ 1 q"px,x 1 q . (5.3) Equation (5.2) would be adapted accordingly.

As a consequence of the previous equations, in order to derive the expected bounds we will devote a subsection to the Hölder controls for the frozen semi-group, for the frozen Green kernel, for the discontinuity term coming from the change of freezing point and for the perturbative contribution (remainder).

Hölder norms for the frozen semi-group

We precisely want to establish the following result.

Lemma 8 There exists C :" CppAqq s.t. for all pt, x, x 1 q P r0, T s ˆRnd ˆRnd , taking pξ, ξ 1 q " # px, x 1 q, if pT ´tq 1{2 ă c 0 dpx, x 1 q, px, xq, if pT ´tq 1{2 ě c 0 dpx, x 1 q, one has:

ˇˇD 2 x 1 P ξ T,t pxq ´D2 x 1 P ξ 1 T,t px 1 q ˇˇď C}g} C 2`γ b,d d γ px, x 1 q, ˇˇP ξ T,t pxq ´P ξ 1 T,t px 1 q ˇˇď C}g} C 2`γ b,d d 2`γ px, x 1 q, for x 1 " x 1 1 .
Hölder norms of the derivatives w.r.

t. the non-degenerate variables

Let us deal with the first inequality of Lemma 8, i.e. the norms over the non-degenerate variables x 1 . For the frozen semi-group, we say that the off-diagonal regime (resp. diagonal regime) holds when T ´t ď c 0 d 2 px, x 1 q (resp. T ´t ě c 0 d 2 px, x 1 q). ' Off-diagonal regime. If T ´t ď c 0 d 2 px, x 1 q, like in (4.23), we write:

D 2 x 1 P ξ T,t gpxq ´D2 x 1 P ξ 1 T,t gpx 1 q " " ż R nd D 2 x 1 pξ pt, T, x, yqrgpyq ´gpy 1 , m ξ T,t pxq 2:n qsdy ´żR nd D 2 x 1 pξ 1 pt, T, x 1 , yqrgpyq ´gpy 1 , m ξ 1 T,t px 1 q 2:n qsdy ‰ `" ż R nd D 2 x 1 pξ pt, T, x, yqrgpy 1 , m ξ T,t pxq 2:n q ´gpm ξ T,t pxqqsdy ´żR nd D 2 x 1 pξ 1 pt, T, x 1 , yqrgpy 1 , m ξ 1 T,t px 1 q 2:n q ´gpm ξ 1 T,t px 1 qqsdy ‰ ": ∆ t,T,ξ,ξ 1 D 2 x 1 P1 gpx, x 1 q `∆t,T,ξ,ξ 1 D 2 x 1 P2 gpx, x 1 q. ( 5.4)
The first term, which is associated with the degenerate variables, is controlled directly thanks to (4.24), which again readily follows from Proposition 2 (see as well Remark 3.8), for pξ, ξ 1 q " px, x 1 q. One readily gets:

ˇˇ∆ t,T,ξ,ξ 1 D 2 x 1 P1 gpx, x 1 q ˇˇˇˇξ "x ď 2CpT ´tq γ 2 }g} C 2`γ b,d ď 2Cc γ 2 0 }g} C 2`γ b,d d γ px, x 1 q. ( 5.5)
The second term is more delicate, we proceed like in (4.25):

∆ t,T,ξ,ξ 1 D 2 x 1 P2 gpx, x 1 q " " ż R nd D 2 x 1 pξ pt, T, x, yq ż 1 0 dµp1 ´µq Tr ´"D 2 x 1 g `mξ T,t pxq 1 `µpy ´mξ T,t pxqq 1 , m ξ T,t pxq 2:n ˘´D 2 x 1 g `mξ T,t pxq ˘‰ `y ´mξ T,t pxq ˘b2 1 ¯dy ´żR nd D 2 x 1 pξ 1 pt, T, x 1 , yq ż 1 0 dµp1 ´µq Tr ´"D 2 x 1 g `mξ 1 T,t px 1 q 1 `µpy ´mξ 1 T,t px 1 qq 1 , m ξ 1 T,t px 1 q 2:n ˘´D 2 x 1 g `mξ 1 T,t px 1 q ˘‰ `y ´mξ 1 T,t px 1 q ˘b2 1 ¯dy  `" 1 2 ż R nd D 2 x 1 pξ pt, T, x, yqTr ´D2 x 1 gpm ξ T,t pxqqpy ´mξ T,t pxqq b2 1 ¯dy ´1 2 ż R nd D 2 x 1 pξ 1 pt, T, x 1 , yqTr ´D2 x 1 gpm ξ 1 T,t px 1 qqpy ´mξ 1 T,t px 1 qq b2 1 dy  ": ∆ t,T,ξ,ξ 1 D 2 x 1 P21 gpx, x 1 q `∆t,T,ξ,ξ 1 D 2 x 1 P22 gpx, x 1 q. ( 5.6)
The first contribution of the previous identity is handled exploiting the smoothness of D 2

x 1 g and Proposition 2. Namely,

|∆ t,T,ξ,ξ 1 D 2 x 1 P21 gpx, x 1 q| ď CrD 2 x 1 gs γ d ż R nd dy pT ´tq ´p ξ C ´1 pt, T, x, yq|py ´mξ T,t pxqq 1 | 2`γ `p ξ 1 C ´1 pt, T, x 1 , yq|py ´mξ 1 T,t px 1 qq 1 | 2`γ ¯ˇˇp ξ,ξ 1 q"px,x 1 q .
Hence,

ˇˇ∆ t,T,ξ,ξ 1 D 2 x 1 P21 gpx, x 1 q ˇˇp ξ,ξ 1 q"px,x 1 q ď 2CpT ´tq γ 2 }g} C 2`γ b,d ď 2Cc γ 2 0 }g} C 2`γ b,d d γ px, x 1 q. (5.7)
Let us now decompose the last contribution of (5.6):

ˇˇ∆ t,T,ξ,ξ 1 D 2 x 1 P22 gpx, x 1 q ˇˇp ξ,ξ 1 q"px,x 1 q ď 1 2 ż R nd dy ps ´tq pξ C ´1 pt, T, x, yq|D 2 x 1 gpm ξ T,t pxqq ´D2 x 1 gpm ξ 1 T,t px 1 qq||py ´mξ T,t pxqq 1 | 2 `1 2 ˇˇż R nd D 2 x 1 pξ pt, T, x, yqTr ´D2 x 1 gpm ξ 1 T,t px 1 qqpy ´mξ T,t pxqq b2 1 D2 x 1 pξ 1 pt, T, x 1 , yqTr ´xD 2 x 1 gpm ξ 1 T,t px 1 qqpy ´mξ 1 T,t px 1 qq b2 1 ¯dy ˇď C|D 2 x 1 gpm ξ T,t pxqq ´D2 x 1 gpm ξ 1 T,t px 1 qq| " C|D 2 x 1 gpθ T,t pxqq ´D2 x 1 gpθ T,t px 1 qq|, (5.8)
exploiting Proposition 2 and equation (3.16) in Proposition 3 to observe that the second contribution of the first inequality above vanishes and recalling as well (2.13) for the last equality to identify the linearized flows, respectively frozen in ξ " x, ξ 1 " x 1 , with the initial non-linear ones.

From Lemma 2, we derive that for T ´t ď c 0 d 2 px, x 1 q:

ˇˇ∆ t,T,ξ,ξ 1 D 2 x 1 P22 gpx, x 1 q ˇˇp ξ,ξ 1 q"px,x 1 q ď C}g} C 2`γ b,d `dγ px, x 1 q `pT ´tq γ 2 ˘.
(5.9) Plugging (5.5), (5.7), (5.9) into (5.4) yields the result.

' Diagonal regime. If T ´t ą c 0 d 2 px, x 1 q, we directly write:

|D 2 x 1 P ξ T,t gpxq ´D2 x 1 P ξ T,t gpx 1 q| ď ˇˇż R nd rD 2 x 1 pξ pt, T, x, yq ´D2 x 1 pξ pt, T, x 1 , yqsgpyqdy ˇď n ÿ k"1 ˇˇż R nd D x k D 2 x 1
pξ pt, T, x 1 `µpx ´x1 q, yq ¨px ´x1 q k gpyqdy ˇ":

∆ t,T,ξ,ξ 1 DD 2 x 1 P3 gpx, x 1 q, (5.10)
recalling the notation D " pD x 1 , ¨¨¨, D xn q introduced after equation (1.1). This contribution is dealt through the cancellation tools of Proposition 3 (see equations (3.15), (3.17)). We get:

∆ t,T,ξ,ξ 1 DD 2 x 1 P3 gpx, x 1 q ď n ÿ k"1 ˇˇˇż R nd D x k D 2 x 1 pξ pt, T, x 1 `µpx ´x1 q, yq ¨px k ´x1 k q " g `y˘´g`m ξ T,t `x1 `µpx ´x1 q ˘@D x 1 g `mξ T,t px 1 `µpx ´x1 qq ˘, `y ´mξ T,t px 1 `µpx ´x1 qq ˘1D ´1 2 Tr ´D2 x 1 g `mξ T,t
`x1 `µpx ´x1 q ˘`y ´mξ T,t px 1 `µpx ´x1 qq ˘b2 1 ¯dy ˇˇˇ.

(5.11)

Because g P C 2`γ b,d pR nd , Rq, we readily deduce, reproducing the Taylor expansion on g employed for equations (5.4)-(5.6) above, that:

ˇˇˇg `y˘´g`m ξ T,t `x1 `µpx ´x1 q ˘@D x 1 gpm ξ T,t px 1 `µpx ´x1 qqq, py ´mξ T,t px 1 `µpx ´x1 qqq 1 D ´1 2 Tr ˆD2 x 1 g `mξ T,t `x1 `µpx ´x1 q ˘˘´y ´mξ T,t `x1 `µpx ´x1 q ˘¯b2 1 ˙ˇˇˇˇˇˇˇˇξ "x ď }g} C 2`γ b,d d 2`γ `y, m ξ T,t `x1 `µpx ´x1 q ˘˘ˇˇˇξ "x . (5.12)
Plugging this inequality into (5.11) yields:

∆ t,T,ξ,ξ 1 DD 2 x 1 P3 gpx, x 1 q ˇˇξ"x ď C}g} L 8 pC 2`γ b,d q n ÿ k"1 ż R nd pT ´tq ´1´pk´1 2 q pξ C ´1 pt, T, x 1 `µpx ´x1 q, yq|x k ´x1 k | ˆd2`γ `y, m ξ T,t px 1 `µpx ´x1 q ˘ˇˇξ "x ď C}g} L 8 pC 2`γ b,d q n ÿ k"1 pT ´tq ´pk´1 2 q`γ 2 |px ´x1 q k | ď C}g} C 2`γ , d γ px, x 1 q, (5.13)
using (3.8) for the second inequality and recalling that, since c 0 d 2 px, x 1 q ă pT ´tq, we indeed have pT ´tq ´pk´1 2 q`γ 2 |px ´x1 q k | ď pc 0 d 2 px, x 1 qq ´pk´1 2 q`γ 2 d 2k´1 px, x 1 q ď Cd γ px, x 1 q.

Hölder control for the degenerate variables

Of course, the analysis of the Hölder regularity of the frozen semi-group w.r.t. the degenerate variables is also based on the previous techniques. We still take advantage of cancellation tools. For the whole paragraph we consider two arbitrary given spatial points px, x 1 q P pR nd q 2 s.t. x 1 " x 1 1 , i.e. their first entry, corresponding to the non-degenerate variable, coincide.

' Off-diagonal regime. If T ´t ď c 0 d 2 px, x 1 q, we proceed to an expansion similar to (5.4) for D 2

x 1 P x T,t gpxq. In particular, with the notations introduced in (5.4), we write: P ξ T,t gpxq ´P ξ 1 T,t gpx 1 q ": ∆ t,T,ξ,ξ 1 P1 gpx, x 1 q `∆t,T,ξ,ξ P2 gpx, x 1 q.

(5.14)

We directly obtain from the Proposition 2, similarly to (5.5), that:

ˇˇ∆ t,T,ξ,ξ 1 P1 gpx, x 1 q ˇˇˇˇξ "x ď 2CpT ´tq 2`γ 2 }g} C 2`γ b,d ď 2C}g} C 2`γ b,d d 2`γ px, x 1 q.
(5.15)

We indeed recall that the difference w.r.t. (5.5) is that we do not have anymore the timesingularities coming therein from the spatial derivatives.

With the notations of (5.6), the second contribution of (5.14) writes:

∆ t,T,ξ,ξ 1 P2 gpx, x 1 q " ∆ t,T,ξ,ξ 1 P21 gpx, x 1 q `∆t,T,ξ,ξ 1 P22 gpx, x 1 q.
Proposition 2 again yields, similarly to (5.7), that:

ˇˇ∆ t,T,ξ,ξ 1 P21 gpx, x 1 q ˇˇp ξ,ξ 1 q"px,ξ 1 q ď 2CpT ´tq 2`γ 2 }g} C 2`γ b,d ď 2C}g} C 2`γ b,d d 2`γ px, x 1 q. (5.16)
On the other hand, we readily get from Proposition 3 that:

∆ t,T,ξ,ξ 1 P22 gpx, x 1 q " 1 2 Tr ´D2 x 1 gpm ξ T,t pxqqr Kξ T,t s 1,1 ´D2 x 1 gpm ξ 1 T,t px 1 qqr Kξ 1 T,t s 1,1 ¯.
Write now:

∆ t,T,ξ,ξ 1 P22 gpx, x 1 q " 1 2 Tr ´rD 2 x 1 gpm ξ T,t pxqq ´D2 x 1 gpm ξ 1 T,t px 1 qqsr Kξ T,t s 1,1 1 2 Tr ´D2 x 1 gpm ξ 1 T,t px 1 qq " r Kξ T,t s 1,1 ´r Kξ 1 T,t s 1,1 ‰ ¯. (5.17)
Since T ´t ď c 0 d 2 px, x 1 q, recalling as well that for ξ " x, ξ 1 " x 1 , m ξ T,t pxq " θ T,t pxq, m ξ 1 T,t px 1 q " θ T,t px 1 q, we readily deduce from Proposition 1 and Lemma 2 that:

1 2 ˇˇrD 2 x 1 gpm ξ T,t pxqq ´D2 x 1 gpm ξ 1 T,t px 1 qqsr Kξ T,t s 1,1 ˇď C}g} C 2`γ b,d d γ `θT,t pxq, θ T,t px 1 q ˘pT ´tq ď C}g} C 2`γ b,d d 2`γ px, x 1 q.
(5.18)

For the last contribution, we directly obtain from Lemma 1 (equation (3.18) for j " 1):

ˇˇ1 2 D 2 x 1 gpm ξ T,t px 1 qq `r Kξ T,t s 1,1 ´r Kξ 1 T,t s 1,1 ˘ˇˇˇˇp ξ,ξ 1 q"px,x 1 q ď C}g} C 2`γ b,d `pT ´tq 2`γ 2 `pT ´tqd γ px, x 1 q ď C}g} C 2`γ b,d d 2`γ px, x 1 q, (5.19)
using again that T ´t ď c 0 d 2 px, x 1 q for the last inequality. Plugging (5.18) and (5.19) into (5.17) gives the expected bound.

' Diagonal regime. If T ´t ě c 0 d 2 px, x 1 q, we write:

| P x T,t gpxq ´P x T,t gpx 1 q| ď ˇˇż R nd rp ξ pt, T, x, yq ´p ξ pt, T, x 1 , yqsgpyqdy ˇˇˇˇˇξ "x ď ˇˇˇż 1 0 dµ ż R nd
xDp ξ pt, T, x 1 `µpx ´x1 q, yq, px ´x1 qy

" g `y˘´g`m ξ T,t `x1 `µpx ´x1 q ˘@D x 1 gpm ξ T,t px 1 `µpx ´x1 qqq, py ´mξ T,t px 1 `µpx ´x1 qqq 1 D ´1 2 Tr ´D2 x 1 gpm ξ T,t px 1 `µpx ´x1 qqqpy ´mξ T,t px 1 `µpx ´x1 qqq b2 1 ¯dy ˇˇˇˇˇˇˇˇξ "x
, with the same cancellation argument as in (5.10)-(5.11). Observe anyhow that the cancellation involving the gradient in the above equation is possible precisely because x 1 " x 1 1 and therefore D pξ pt, T, x 1 `µpx´x 1 q, yq¨px´x 1 q " D x 2:n pξ pt, T, x 1 `µpx´x 1 q, yq¨px´x 1 q 2:n . We then obtain thanks to the previous identity and (5.12):

| P x T,t gpxq ´P x T,t gpx 1 q| ď }g} C 2`γ b,d n ÿ k"2 ż R nd ˇˇD x k pξ pt, T, x 1 `µpx ´x1 q, yq ˇˇ|x k ´x1 k |d 2`γ `y, m ξ T,t `x1 `µpx ´x1 q ˘˘dy ď }g} C 2`γ b,d n ÿ k"2 ż R nd C pT ´tq 1`pk´1 2 q´2 `γ 2 px C ´1 pt, T, m x T,t px 1 `µpx ´x1 qq, yq|x k ´x1 k |dy ď C}g} C 2`γ b,d d 2`γ px, x 1 q, (5.20)
reproducing the arguments used to establish (5.13) for the last inequality. Lemma 8 is proved.

Hölder norms associated with the Green kernel

Let us recall that in (2.36), for a source f P L 8 `r0, T s, C γ b,d pR nd , Rq ˘, we have to control the Hölder norms of the Green kernel which we split into two parts according to the position of the time integration variable w.r.t. the change of regime time t 0 (see (2.37)) a posteriori chosen to be t 0 :" `t `c0 d 2 px, x 1 q ˘^T. This is again the splitting according to the off-diagonal and diagonal regime. The point is that for the Green kernel, if t 0 ă T both regimes appear.

Lemma 9 Under (A), for fixed spatial points px, x 1 q P pR nd q 2 , we have that there exists a constant C :" CppAq, T q, s.t. for all f P L 8 `r0, T s, C γ b,d pR nd , Rq ˘:

sup tPr0,T s `|D 2 x 1 Gξ t 0 ,t f pt, xq ´D2 x 1 Gξ 1 t 0 ,t f pt, x 1 q| `|D 2 x 1 Gξ T,t 0 f pt, xq ´D2 x 1 Gξ 1 T,t 0 f pt, x 1 q| ď C}f } L 8 pC γ b,d q d γ px, x 1 q, and sup tPr0,T s `| Gξ t 0 ,t f pt, xq ´G ξ 1 t 0 ,t f pt, x 1 q| `| Gξ T,t 0 f pt, xq ´G ξ1 T,t 0 f pt, x 1 q| ď C}f } L 8 pC γ b,d q d 2`γ px, x 1 q, if x 1 " x 1 1 , where ξ " x, ξ 1 " x 1 , ξ1 " x.
We now turn to the proof of Lemma 9.

Proof: Lemma 9

Let us begin with the statement concerning the second order derivatives of the frozen Green kernel w.r.t. the non-degenerate variable x 1 .

For the off-diagonal regime, involving the term D 2

x 1 Gξ 1 t 0 ,t f pt, x 1 q, we readily get from Lemma 7 that

ˇˇD 2 x 1 Gξ t 0 ,t f pt, xq´D 2 x 1 Gξ 1 t 0 ,t f pt, x 1 q ˇˇˇˇˇp ξ,ξ 1 q"px,x 1 q ď ˇˇż t 0 t dsD 2 x 1 P ξ s,t f ps, xq ˇˇˇˇˇξ "x `ˇż t 0 t dsD 2 x 1 P ξ 1 s,t f ps, x 1 q ˇˇˇˇˇξ 1 "x 1 ď C}f } L 8 pC γ b,d q ż t 0 t dsps ´tq ´1`γ 2 ď C}f } L 8 pC γ b,d q d γ px, x 1 q. ( 5.21)
For the diagonal regime, involving the term D 2

x 1 Gξ 1 T,t 0 f pt, x 1 q, we have to be more subtle and perform again a Taylor expansion of D 2

x 1 P ξ s,t f ps, ¨q. Namely:

ˇˇD 2 x 1 Gξ T,t 0 f pt, xq´D 2 x 1 Gξ 1 T,t 0 f pt, x 1 q ˇˇˇˇˇp ξ, ξ1 q"px,xq ď ż T t 0 ds ˇˇż 1 0 dµDD 2 x 1 P ξ s,t f ps, x 1 `µpx ´x1 qq ¨px ´x1 q ˇˇˇˇˇξ "x ď n ÿ i"1 |px ´x1 q i | ż T t 0 ds ż 1 0 dµ ˇˇD x i D 2 x 1 P ξ s,t f `s, x 1 `λpx ´x1 q ˘ˇˇˇˇξ "x ď C}f } L 8 pC γ b,d q n ÿ i"1 |px ´x1 q i | ż T t 0
dsps ´tq ´1´pi´1 2 q`γ 2 , using again Lemma 7 and the arguments of (5.13) for the last inequality. This finally yields, recalling the definition of d in (1.9): Let us now turn to the Hölder controls on the degenerate variables. The idea is here again to perform a Taylor expansion at order one for x 1 " x 1 1 . Namely, from Lemma 7, the diagonal control is direct. We get

ˇˇD 2 x 1 Gξ T,t 0 f pt, xq´D 2 x 1 1 Gξ 1 T,t 0 f pt, x 1 q ˇˇˇˇˇp ξ,ξ 1 q"px,xq ď C}f } L 8 pC γ b,d q n ÿ i"1 |px ´x1 q i |pd 2 px, x 1 qq ´pi´1 2 q`γ 2 ď C}f } L 8 pC γ b,d q d γ px, x 1 q. ( 5
ˇˇG ξ T,t 0 f pt, xq´G ξ1 T,t 0 f pt, x 1 q ˇˇˇˇˇp ξ, ξ1 q"px,xq ď ż T t 0 ds ż 1 0 dµ ˇˇ@ D P ξ s,t f `s, x 1 `µpx ´x1 q ˘, px ´x1 q Dˇď C}f } L 8 pC γ b,d q ż T pt`c 0 d 2 px,x 1 qq^T ds n ÿ i"2 |px ´x1 q i |ps ´tq ´pi´1 2 q`γ 2 ď C}f } L 8 pC γ b,d q n ÿ i"2 |px ´x1 q i | loooomoooon ďdpx,x 1 q 2i´1 dpx, x 1 q 2´p2i´1q`γ ď C}f } L 8 pC γ b,d q d 2`γ px, x 1 q. (5.23)
Now, for the off-diagonal bound, associated with the term Gξ t 0 ,t f pt, ¨q, we precisely need to exploit the smoothness of f associated to the fact that the semi-group P ξ s,t has a density. Indeed, we cannot take advantage of the cancellation tools of Lemma 7, but we have for all x, x 1 P R nd s.t. x 1 " x 1 1 :

ˇˇż t 0 t ds ´P ξ s,t f ps, xq ´P ξ 1 s,t f ps, x 1 q ¯ˇˇˇˇξ "x ď ˇˇż t 0 t ds P ξ s,t f ps, xq ´f ps, m ξ t 0 ,t pxqq ˇˇˇˇˇξ "x `ˇż t 0 t ds P ξ 1 s,t f ps, x 1 q ´f ps, m ξ 1 t 0 ,t px 1 qq ˇˇˇˇˇξ 1 "x 1 `ˇż t 0 t
dsf ps, m ξ t 0 ,t pxqq ´f ps, m ξ 1 t 0 ,t px 1 qq ˇˇˇˇˇp ξ,ξ 1 q"px,x 1 q .

(5.24)

Note that the first two terms in the r.h.s. of inequality (5.24) are handled like in the previous section. Precisely, we write for the first contribution:

ˇˇż t 0 t ds P ξ s,t f ps, xq ´f ps, m ξ t 0 ,t pxqqdy ˇˇˇˇˇξ "x " ˇˇż t 0 t ds ż R nd
pξ pt, T, x, yqrf ps, yq ´f ps, θ t 0 ,t pxqs ˇˇˇˇˇξ

"x ď }f } L 8 pC γ b,d q ż t 0 t ds ż R nd
pξ pt, T, x, yqd γ py, θ t 0 ,t pxqqdy ˇˇξ

"x ď C}f } L 8 pC γ b,d q ż t 0 t dsps ´tq γ 2 dy ď C}f } L 8 pC γ b,d q d 2`γ px, x 1 q,
by definition of t 0 in (2.37). The second term of (5.24) is handled similarly. We thus obtain:

ˇˇż t 0 t ds P ξ s,t f ps, xq ´f ps, m ξ t 0 ,t pxqdy ˇˇˇˇˇξ "x `ˇż t 0 t ds P ξ 1 s,t f ps, x 1 q ´f ps, m ξ 1 t 0 ,t px 1 qdy ˇˇˇˇˇξ 1 "x 1 ď C}f } L 8 pC γ b,d q d 2`γ px, x 1 q.
(5.25)

For the last contribution in (5.24), we have directly that:

ˇˇż t 0 t dsf ps, m ξ t 0 ,t pxqq´f ps, m ξ 1 t 0 ,t px 1 qq ˇˇˇˇˇp ξ,ξ 1 q"px,x 1 q ď C}f } L 8 pC γ b,d q ż t 0 t dsd γ `θt 0 ,t pxq, θ t 0 ,t px 1 q ˘.
(5.26) Lemma 2 and (5.26) eventually yield:

ˇˇż t 0 t dsf ps, m ξ t 0 ,t pxqq ´f ps, m ξ t 0 ,t px 1 qq ˇˇˇˇˇp ξ,ξ 1 q"px,x 1 q ď C}f } L 8 pC γ b,d q `pt 0 ´tq 1`γ 2 `pt 0 ´tqd γ px, x 1 q " 2C}f } L 8 pC γ b,d q d 2`γ px, x 1 q. 5.3

Hölder norms of the perturbative contribution

To derive the spatial Hölder continuity of the perturbative term in (5.3) (see also the splitting of equation (2.28) of the user's guide) we proceed with the previous regime separation successfully used to establish in the previous sections the regularity of the semigroup and the Green kernel. Namely, we recall that the critical time giving the change of regime is (chosen after potential differentiation) t 0 " t `c0 d 2 px, x 1 q ^T . We can assume here w.l.o.g. that t 0 ă T (otherwise there is a globally off-diagonal regime and the analysis becomes easier).

The point of the current section is that we also precisely need to adapt the duality analysis on Besov spaces already exploited for the supremum norm of the spatial derivatives w.r.t. the non-degenerate variable (see Section 4.2).

' Control of rD 2

x 1 upt, ¨qs γ . From (2.28) and (2.36), let us recall the following notations: We recall that in the diagonal regime, i.e. c 0 d 2 px, x 1 q ď s ´t, we choose pξ, ξ1 q " px, xq. The idea is to expand the density pξ pt, s, ¨, yq between x and x 1 . The associated time singularity can then be bounded by a space singularity, recall that on the considered regime ps ´tq ´1 ď pc 0 d 2 px, x 1 qq ´1 which will in turn be exactly compensated by the difference of spatial points induced by the Taylor expansion and the Hölder regularity of the coefficients.

∆ ξ,ξ 1 pt, T, x, x 1 q " ∆ ξ,ξ 1 diag pt, T, x, x 1 q `∆ξ,ξ 1 off-diag pt, x, x 1 q, with ∆ ξ,ξ 1 off-diag pt, x, x 1 q " ż T t ds ż R nd dyp ξ pt, s, x, yqI sďt 0 pL s ´L ξ s qups, yq ´ż T t ds ż R nd dyp ξ 1 pt, s, x 1 , yqI sďt 0 pL s ´L ξ 1 s qups, yq, ∆ ξ, ξ1 diag pt, T, x, x 1 q " ż T t ds ż R nd dyp ξ pt, s, x,
We decompose from definitions (2.28) and (2.36):

| D 2 x 1 ∆ ξ, ξ1 diag pt, T, x, x 1 q| ˇˇp ξ, ξ1 q"px,xq ď ˇˇż T t ds ż R nd dy " D 2 x 1 pξ pt, s, x, yq ´D2 x 1 pξ pt, s, x 1 , yq ı" xF 1 ps, yq ´F1 ps, θ s,t pξqq, D y 1 y `1 2
Tr ´`aps, yq ´aps, θ s,t pξqq ˘D2 y 1 ¯ıups, yqI sąt 0 ˇˇˇˇˇˇξ

"x `ˇˇn ÿ i"2 ż T t ds ż R nd dy " D 2
x 1 pξ pt, s, x, yq ´D2

x 1 pξ pt, s, x 1 , yqI sąt 0 ı ˆAF i ps, yq ´rF i ps, θ s,t pξqq `Dx i´1 F i ps, θ s,t pξqqpy ´θs,t pξqq i´1 s, D y i ups, yq

E I sąt 0 ˇˇˇˇˇˇξ "x
, which readily yields with the notations of (4.3) that:

|D 2 x 1 ∆ ξ, ξ1 diag pt, T, x, x 1 q| ˇˇp ξ, ξ1 q"px,xq ď ˇˇż T t ds ż R nd dy ż 1 0 dµDD 2 x 1 pξ pt, s, x `µpx 1 ´xq, yq¨px´x 1 q∆ 1,F,σ pt, s, y, θ s,t pξq, uqI sąt 0 ˇˇˇˇˇˇξ "x `ˇˇż T t ds ż R nd dy ż 1 0 dµDD 2
x 1 pξ pt, s, x `µpx 1 ´xq, yq ¨px ´x1 q ˆ@∆ i,F pt, s, θ s,t pξq, yq, D y i ups, yq

D I sąt 0 ˇˇˇˇˇˇξ "x ": ˇˇD 2 x 1 ∆ 1 diag pt, T, x, x 1 q ˇˇ`ˇˇD 2 x 1 ∆ 2:n diag pt, T, x, x 1 q ˇˇ.
(5.28)

We will now control the first term of the above right hand side. In other words, we specify the control of (2.14). We obtain directly thanks to the smoothness assumption (S) on the coefficients and Proposition 2 (see also equation (3.8)) that for all k P rr1, nss:

|F 1 ps, yq ´F1 ps, θ s,t pξqq||D x k D 2 x 1 pξ pt, s, x `µpx 1 ´xq, yq| ˇˇξ "x ď C ´}F 1 ps, ¨q} C γ d d γ `y, θ s,t pξq ˘¯ˆps ´tq ´1´pk´1 2 q pC ´1 pt, s, x, yq ˇˇξ "x
ď C}F 1 } L 8 pC γ d q ps ´tq ´1´pk´1 2 q`γ{2 pC ´1 pt, s, x, yq.

(5.29) Similarly, |aps, yq ´aps, θ s,t pξqq|D x k D 2

x 1 pξ pt, s, x `µpx 1 ´xq, yq| ˇˇξ

"x ď C ´}aps, ¨q} C γ d d γ `y, θ s,t pξq ˘¯ˆps ´tq ´1´pk´1 2 q pC ´1 pt, s, x, yq ˇˇξ "x
ď C}a} L 8 pC γ d q ps ´tq ´1´pk´1 2 q`γ{2 pC ´1 pt, s, x, yq.

(5.30)

We carefully point-out that the indicated bound only depend on the supremum in time of the Hölder modulus of the coefficients (denoted }F 1 } L 8 pC γ d q , }a} L 8 pC γ d q respectively) and not on their supremum norm. In particular, we get from (5.29), (5.30):

|D 2 x 1 ∆ 1 diag pt, T, x, x 1 q| ď n ÿ k"1 ż T t 0 ds ż R nd dyps ´tq ´1´pk´1 2 q`γ{2 ˆr}D x 1 u} L 8 }F 1 } L 8 pC γ d q `}D 2 x 1 u} L 8 }a} L 8 pC γ d q sp C ´1 pt, s, x, yq|x k ´x1 k | ď Cp}D x 1 u} L 8 `}D 2 x 1 u} L 8 qd γ px, x 1 q, (5.31)
where again the constant C is small provided the coefficients do not vary much. Thanks to Lemma 5 and the previous analysis of Section 4.2, we directly deduce:

|D 2 x 1 ∆ 2:n diag pt, T, x, x 1 q| ď ˇˇn ÿ i"2 ż T t ds ż R nd dy ż 1 0 dµD y i ¨"´D D 2
x 1 pξ pt, s, x `µpx 1 ´xq, yq ¨px ´x1 q b´F i ps, yq ´rF i ps, θ s,t pξqq `Dx i´1 F i ps, θ s,t pξqqpy ´θs,t pξqq i´1 s ¯*ups, yqI sąt 0 ˇˇˇˇˇˇξ

"x ď C}u} L 8 pC 2`γ b,d q n ÿ k"1 ż T t 0 |x k ´x1 k |ds ps ´tq 1`pk´1 2 q´γ 2 ď C}u} L 8 pC 2`γ b,d q d γ´p2k´1q px, x 1 q|x k ´x1 k | ď C}u} L 8 pC 2`γ b,d q d γ px, x 1 q, (5.32)
and C is again small provided the coefficients do not vary much † .

Plugging (5.31) and (5.32) into (5.28) yields the stated control for the diagonal contribution.

Let us now turn to the control of |D 2

x 1 ∆ ξ,ξ 1 off-diag pt, x, x 1 q| in (5.27) (or (2.28) in the user's †. Let us point out that in (5.31), (5.31) , the constant C :" CppAq, c 0 q, but involves negative exponents of c 0 meant to be small. This is not a problem and still gives C ă 1 provided the Hölder moduli of the coefficients are sufficiently small. guide). In the off-diagonal case, we choose ξ " x and ξ 1 " x 1 and

|D 2 x 1 ∆ ξ,ξ 1 off-diag pt, x, x 1 q| ď ˇˇż t 0 t ds ż R nd dyD 2 x 1 pξ pt, s, x, yqpL s ´L ξ s qups, yq ˇˇˇˇˇˇξ "x `ˇˇż t 0 t ds ż R nd dyD 2 x 1 pξ 1 pt, s, x 1 , yqpL s ´L ξ 1 s qups, yq ˇˇˇˇˇˇξ 1 "x 1 . ( 5.33)
We readily get thanks to Lemmas 4 and 5:

|D 2 x 1 ∆ ξ,ξ 1 off-diag pt, x, x 1 q| ď C n ÿ i"2 ż t 0 t ds ps ´tq 1´γ 2 }u} L 8 pC 2`γ b,d q ď Cd γ px, x 1 q}u} L 8 pC 2`γ b,d q .
(5.34)

We point out that the constant C in (5.34) involves positive exponents of c 0 . We have opposite impacts for this thresholds depending on the diagonal and off-diagonal regimes at hand.

' Control of sup zPR d rupt, z, ¨qs 2`γ . We proceed as above from definitions (5.27) considering spatial points px, x 1 q P pR nd q 2 s.t. x 1 " x 1 1 . In the diagonal case, we also choose ξ " ξ1 " x and we write similarly to (5.28):

|∆ ξ,ξ 1 2:n,diag pt, T, x, x 1 q| ˇˇp ξ,ξ 1 q"px,xq ď ˇˇż T t ds ż R nd dy ż 1 0 dµ @ D x pξ pt, s, x `µpx 1 ´xq, yq, x ´x1 D ∆ 1,F,σ pt, s, y, θ s,t pξqqI sąt 0 ˇˇˇˇˇˇξ "x `ˇˇn ÿ i"2 ż T t ds ż R nd dy ż 1 0 dµ @ D x pξ pt, s, x `µpx 1 ´xq, yq, x ´x1 D ∆ i,F pt, s, θ s,t pξq, yqD y i ups, yqI sąt 0 ˇˇˇˇˇˇξ "x . (5.35)
We have an expression rather similar to the one that appeared for the control of rD 2 x 1 upt, ¨qs γ but with a weaker time singularity. In other words, thanks to identities (5.29), (5.30) and Lemma 5 we obtain:

|∆ ξ,ξ 1 2:n,diag pt, T, x, x 1 q| ď C}u} L 8 pC 2`γ b,d q n ÿ k"1 ż T t 0 dsps ´tq ´pk´1 2 q`γ 2 |x k ´x1 k | ď C}u} L 8 pC 2`γ b,d q n ÿ k"1 d 2p1´pk´1 2 q`γ 2 q px, x 1 q|x k ´x1 k | ď C}u} L 8 pC 2`γ b,d q d 2`γ px, x 1 q, (5.36)
where again the constant C is small if the Hölder moduli of the coefficients are small.

For the off-diagonal contribution, we get for x 1 " x 1 1 , ξ " x and ξ 1 " x 1 :

|∆ ξ,ξ 1 2:n,off-diag pt, T, x, x 1 q| ď ˇˇż t 0 t ds ż R nd dyp ξ pt, s, x, yqpL s ´L ξ s qups, yq ˇˇˇˇˇˇξ "x `ˇˇż t 0 t ds ż R nd dyp ξ 1 pt, s, x 1 , yqpL s ´L ξ 1 s qups, yq ˇˇˇˇˇˇξ 1 "x 1 ď C n ÿ k"2 ż t 0 t dsps ´tq γ 2 }u} L 8 pC 2`γ b,d q " Cc 0 d 2`γ px, x 1 q}u} L 8 pC 2`γ b,d q .
(5.37)

The last but one inequality is a consequence of Lemmas 4 and 5.

Controls of the discontinuity terms arising from the change of freezing point

It now remains to control the contribution arising from the change of freezing point in equation (5.3). The main result of this section is the following lemma.

Lemma 10 (Control of the discontinuity terms) There exists C :" CppAqq s.t. for all pt, x, x 1 q P r0, T s ˆRnd ˆRnd taking ξ 1 " x 1 , ξ1 " x, ˇˇD 2

x 1 P ξ 1 t 0 ,t upt 0 , x 1 q ´D2

x 1

P ξ1 t 0 ,t upt 0 , x 1 q ˇˇď Cc γ p2n´1q 0 }u} L 8 pC 2`γ b,d q d γ px, x 1 q, ˇˇP ξ 1 t 0 ,t upt 0 , x 1 q ´P ξ1 t 0 ,t upt 0 , x 1 q ˇˇď Cc 0 }u} L 8 pC 2`γ b,d q d 2`γ px, x 1 q, for x 1 " x 1 1 .
We prove the above statement in the next paragraphs respectively dedicated to the control of the derivatives w.r.t. the non-degenerate variables and the control of the Hölder moduli associated with the degenerate ones.

Control of the derivatives w.r.t. the non-degenerate variables

We can directly write like for (5.4) and (5.6):

D 2 x 1 P ξ 1 t 0 ,t upt 0 , x 1 q ´D2 x 1 P ξ1 t 0 ,t upt 0 , x 1 q ˇˇˇp ξ 1 , ξ1 q"px 1 ,xq " « ż R nd D 2 x 1 pξ 1 pt, t 0 , x 1 , yqrupt 0 , yq ´upt 0 , y 1 , m ξ 1 t 0 ,t px 1 q 2:n qsdy ´żR nd D 2 x 1 pξ 1 pt, t 0 , x 1 , yqrupt 0 , yq ´upt 0 , y 1 , m ξ1 t 0 ,t px 1 q 2:n qsdy ff `« ż R nd D 2 x 1 pξ 1 pt, t 0 , x 1 , yq ż 1 0 dµp1 ´µq Tr ´"D 2 x 1 u `t0 , m ξ 1 t 0 ,t px 1 q 1 `µpy ´mξ 1 t 0 ,t px 1 qq 1 , m ξ 1 t 0 ,t px 1 q 2:n D2 x 1 u `t0 , m ξ 1 t 0 ,t px 1 q ˘‰`y ´mξ 1 t 0 ,t px 1 q ˘b2 1 ¯dy ´żR nd D 2 x 1 pξ 1 pt, t 0 , x 1 , yq ż 1 0 dµp1 ´µq
Tr ´"D 2 x 1 u `t0 , m ξ1 t 0 ,t px 1 q 1 `µpy ´mξ 1 t 0 ,t px 1 qq 1 , m ξ1 t 0 ,t px 1 q 2:n

D2

x 1

1 u `t0 , m ξ 1 t 0 ,t px 1 q ˘‰`y ´mξ 1 t 0 ,t px 1 q ˘b2 1 ¯dy ff `1 2 « ż R nd D 2 x 1 pξ 1 pt, t 0 , x 1 , yqTr ´D2 x 1 upt 0 , m ξ 1 t 0 ,t px 1 qqpy ´mξ 1 t 0 ,t px 1 qq b2 1 ¯dy ´żR nd D 2 x 1 pξ 1 pt, t 0 , x 1 , yqTr ´D2 x 1 upt 0 , m ξ1 t 0 ,t px 1 qqpy ´mξ 1 t 0 ,t px 1 qq b2 1 ¯dy ff ": ∆ t,t 0 ,ξ 1 , ξ1 D 2 x 1 P1 upt 0 , x, x 1 q `∆t,t 0 ,ξ 1 , ξ1 D 2 x 1 P21 upt 0 , x, x 1 q `∆t,t 0 ,ξ 1 , ξ1 D 2 x 1 P22 upt 0 , x, x 1 q. (5.38)
We first directly write from (5.5) and (5.7):

ˇˇ∆ t,t 0 ,ξ 1 , ξ1 D 2 x 1 P1 upt 0 , x, x 1 q `∆t,t 0 ,ξ 1 , ξ1 D 2 x 1 P21 upt 0 , x, x 1 q ˇˇˇˇp ξ 1 , ξ1 q"px 1 ,xq ď 2Cpt 0 ´tq γ 2 }u} C 2`γ b,d ď Cc γ 2 0 }u} C 2`γ b,d d γ px, x 1 q.
(5.39)

Let us now deal with the last term in (5.38). We proceed similarly to equation (5.8) (control of the frozen semigroup). Write:

ˇˇ∆ t,t 0 ,ξ 1 , ξ1 D 2 x 1 P22 upt 0 , x, x 1 q ˇˇp ξ 1 , ξ1 q"px 1 ,xq ď 1 2 ż R nd dy ps ´tq pξ 1 C ´1 pt, t 0 , x 1 , yq|D 2 x 1 upt 0 , m ξ 1 t 0 ,t px 1 qq´D 2 x 1 upt 0 , m ξ1 t 0 ,t px 1 qq||py´m ξ 1 t 0 ,t px 1 qq 1 | 2 `1 2 ˇˇż R nd D 2 x 1 pξ 1 pt, t 0 , x 1 , yqTr ´D2 x 1 upt 0 , m ξ1 t 0 ,t px 1 qqpy ´mξ 1 t 0 ,t px 1 qq b2 1 D2 x 1 pξ 1 pt, t 0 , x 1 , yqTr ´D2 x 1 upt 0 , m ξ1 t 0 ,t px 1 qqpy ´mξ 1 t 0 ,t px 1 qq b2 1 ¯dy ˇď C|D 2 x 1 upt 0 , m ξ 1 t 0 ,t px 1 qq ´D2 x 1 upt 0 , m ξ1 t 0 ,t px 1 qq|, (5.40)
exploiting Proposition 2 for the first contribution and identity (3.16) in Proposition 3 for second contribution in the last inequality.

From Lemma 3, we derive that for t 0 ´t " c 0 d 2 px, x 1 q:

ˇˇ∆ t,t 0 ,ξ 1 , ξ1 D 2 x 1 P22 upt 0 , x, x 1 q ˇˇp ξ 1 , ξ1 q"px 1 ,xq ď C}upt 0 , ¨q} C 2`γ b,d c γ 2n´1 0 d γ px, x 1 q.
In a nutshell, from (5.39) and (5.41), we get the following control:

ˇˇD 2 x 1 P ξ 1
t 0 ,t upt 0 , x 1 q ´D2

x 1

P ξ1 t 0 ,t upt 0 , x 1 q ˇˇˇˇˇp ξ 1 , ξ1 q"px,x 1 q ď Cc γ 2n´1 0 }u} L 8 pC 2`γ b,d q d γ px, x 1 q,
which gives the statement for the second order derivatives w.r.t. the non-degenerate variables.

Hölder controls for the degenerate variables

Again, for the Hölder norm w.r.t. the degenerate variables, the difficulty is that we cannot take any advantage of cancellation tools. We adapt here the arguments employed in Section 5.1 for the frozen semigroup. Precisely, for all px, x 1 q P pR nd q 2 , x 1 " x 1 1 , ξ 1 " x 1 , ξ1 " x, we have similarly to (5.38) (but without the spatial derivatives D 2

x 1 ):

P ξ 1 t 0 ,t upt 0 , x 1 q ´P ξ1 t 0 ,t upt 0 , x 1 q " ∆ t,t 0 ,ξ 1 , ξ1 P1 upt 0 , x 1 , x 1 q `∆t,t 0 ,ξ 1 , ξ1 P21 upt 0 , x 1 , x 1 q `1 2 Tr ´rD 2 x 1 upt 0 , m ξ 1 t 0 ,t px 1 qq ´D2 x 1 upt 0 , m ξ1 t 0 ,t px 1 qqsr Kξ 1 t 0 ,t s 1,1 1 2 Tr ´D2 x 1 upt 0 , m ξ1 t 0 ,t px 1 qq `r Kξ 1 t 0 ,t s 1,1 ´r Kξ 1 t 0 ,t s 1,1 ˘¯, (5.41)
where accordingly with (5.38):

∆ t,t 0 ,ξ 1 , ξ1 P1 upt 0 , x 1 , x 1 q :" « ż R nd
pξ 1 pt, t 0 , x 1 , yqrupt 0 , yq ´upt 0 , y 1 , m ξ 1 t 0 ,t px 1 q 2:n qsdy ´żR nd pξ 1 pt, t 0 , x 1 , yqrupt 0 , yq ´upt 0 , y 1 , m ξ1 t 0 ,t px 1 q 2:n qsdy ff , and

∆ t,t 0 ,ξ 1 , ξ1 P21 upt 0 , x 1 , x 1 q :" « ż R nd pξ 1 pt, t 0 , x 1 , yq ż 1 0 dµp1 ´µq Tr ´"D 2 x 1 u `t0 , m ξ 1 t 0 ,t px 1 q 1 `µpy ´mξ 1 t 0 ,t px 1 qq 1 , m ξ 1 t 0 ,t px 1 q 2:n D2 x 1 u `t0 , m ξ 1 t 0 ,t px 1 q ˘‰`y ´mξ 1 t 0 ,t px 1 q ˘b2 1 ¯dy ´żR nd pξ 1 pt, t 0 , x 1 , yq ż 1 0 dµp1 ´µq Tr ´"D 2 x 1 u `t0 , m ξ1 t 0 ,t px 1 q 1 `µpy ´mξ 1 t 0 ,t px 1 qq 1 , m ξ1 t 0 ,t px 1 q 2:n D2 x 1 u `t0 , m ξ1 t 0 ,t px 1 q ˘‰`y ´mξ 1 t 0 ,t px 1 q ˘b2 1 ¯dy ff .
Reproducing the arguments that led to Equations (5.15) and (5.16), we derive:

ˇˇ∆ t,t 0 ,ξ 1 , ξ1 P1 upt 0 , x 1 , x 1 q `∆t,t 0 ,ξ 1 , ξ1 P21 upt 0 , x 1 , x 1 q ˇˇˇˇˇp ξ 1 , ξ1 q"px 1 ,xq ď Cpt 0 ´tq 2`γ 2 }u} C 2`γ b,d ď Cc 2`γ 2 0 }u} L 8 pC 2`γ b,d q d 2`γ px, x 1 q.
(5.42)

Let us now turn to the last two contributions in (5.41). Like for (5.18), from Proposition 1 and Lemma 3 we obtain:

1 2 ˇˇrD 2 x 1 upt 0 , m ξ 1 t 0 ,t px 1 qq ´D2 x 1 upt 0 , m ξ1 t 0 ,t px 1 qqsr Kξ 1 t 0 ,t s 1,1 ˇˇˇˇˇp ξ 1 , ξ1 q"px 1 ,xq ď C}u} L 8 pC 2`γ b,d q pt 0 ´tqd γ pm x t 0 ,t px 1 q, θ t 0 ,t px 1 qq ď Cc 1`γ 2n´1 0 }u} L 8 pC 2`γ b,d q d 2`γ px, x 1 q.
(5.43)

The last term of (5.41) is handled like in (5.19). Namely, by Lemma 1 (equation (3.18) for j " 1), we obtain:

ˇˇ1 2 D 2 x 1 upt 0 , m ξ1 t 0 ,t px 1 qq `r Kξ 1 t 0 ,t s 1,1 ´r Kξ 1 t 0 ,t s 1,1 ˘ˇˇˇˇp ξ 1 , ξ1 q"px 1 ,xq ď C}upt 0 , ¨q} C 2`γ b,d `pt 0 ´tq 2`γ 2
`pt 0 ´tqd γ px, x 1 q ď Cc 0 }u} L 8 pC 2`γ b,d q d 2`γ px, x 1 q.

(5.44) Plugging (5.42), (5.43) and (5.44) into (5.41), recalling as well that c 0 ă 1, we derive the second statement of Lemma 10.

Remark 5 (Concluding remark concerning the a priori estimates) Gathering all the controls of Sections 4 and 5 in equations (4.1) and (5.1)-( 5.3), we actually derive the following bound. There exist constants C :" CppTqq and C 0 :" C 0 ppAq, T q s.t.

}u} L 8 pC 2`γ b,d q ď Cp}g} C 2`γ b,d
`}f } L 8 pC γ b,d q q `C0 }u} L 8 pC 2`γ b,d q .

(5.45)

It can been observed from the previous technical lemmas that provided T, c 0 , and the Hölder moduli of the coefficients }a} L 8 pC γ d q ,

ř n i"2 }F i } L 8 pC 2i´3`γ d
q (with the definition of (3.20)) are small enough, then C 0 ă 1, which in turn directly yields the expected Schauder estimate for the mollified dynamics.

Of course, the remaining delicate part consists in getting rid of the small Hölder moduli constraint. This can be done through suitable scaling arguments that are exposed in Section 6.

Scaling issues and final proof of Theorem 1

The purpose of this section is to first introduce a suitable scaling procedure for the system with mollified coefficients for which we will be able to show equation (5.45). This intuitively means that the scaling has to make the Hölder moduli of the considered coefficients small. The expected control is then obtained going back to the initial variables through the inverse scaling procedure. Also, once the estimate is established for small final time horizon T , it can be deduced through iteration up to an arbitrary given time precisely because it provides a kind of stability for the solution in the space L 8 pr0, T s, C 2`γ b,d pR nd , Rqq. We then conclude the proof of our main result, Theorem 1 through compactness arguments.

Scaling settings and controls

We start here from the smooth solution u to equation (1.1) with mollified coefficients (that we again denote by a slight abuse of notation without the mollifying parameter m). For an additional parameter λ ą 0 to be specified later (but meant to be small ), introducing the scaled function u λ pt, xq " upt, λ ´1{2 T λ xq, it is then clear that this latter satisfies

$ ' & ' % B t u λ pt, xq `xFpt, λ ´1{2 T λ xq, λ 1{2 T ´1 λ Du λ pt, xqy `λ´1 2 Tr `D2 x 1 u λ pt, xqapt, λ ´1{2 T λ xq ˘" ´f pt, λ ´1{2 T λ xq, pt, xq P r0, T q ˆRnd , u λ pT, xq " gpλ ´1{2 T λ xq, x P R nd . (6.1)
This choice of rescaling is natural in view of the invariance by dilatation property (1.7), i.e. each single variable x i is scaled by the parameter λ according to its corresponding intrinsic scale.

We rewrite in short form the above equation as for all x P R nd # B t u λ pt, xq `xF λ pt, xq, Du λ pt, xqy `1 2 Tr `D2

x 1 u λ pt, xqa λ pt, xq ˘" ´fλ pt, xq, t P r0, T q, u λ pT, xq " g λ pxq, (

where

f λ pt, xq :" f pt, λ ´1{2 T λ xq, g λ pxq :" gpλ ´1{2 T λ xq, a λ pt, xq :" λ ´1apt, λ ´1{2 T λ xq, F λ pt, xq :" λ 1{2 T ´1 λ Fpt, λ ´1{2 T λ xq. (6.3)
Accordingly, we introduce the spatial operator pL λ s q sě0 appearing in (6.2) which writes explicitly for all ϕ P C 2 0 pR nd , Rq as:

L λ t ϕ " xF λ pt, ¨q, Dϕy `1 2 Tr ´aλ pt, ¨qD 2 x 1 ϕ ¯, λ ą 0.
The dynamics of the SDE associated with the second order differential operator pL λ t q tPr0,T s appearing in (6.1)-( 6.2) writes for a given starting point pt, xq P r0, T s ˆRnd :

X λ,t,x s " x `ż s t F λ pu, X λ,t,x u qdu `ż s t Bσ λ pu, X λ,t,x u qdW u , s ě t, (6.4)
where pW u q uě0 is a d-dimensional Brownian motion on some filtered probability space pΩ, F, pF t q tě0 , Pq and σ λ is a square root of the diffusion matrix a λ introduced in (6.3).

Equation (6.4) then naturally leads to consider, for fixed ps, yq P rt, T s ˆRnd , and with the notations of Section 3, the corresponding linearized model

d Xξ,λ v " rF λ pv, θ λ v,t pξ λ qq `DF λ pv, θ λ v,t pξ λ qqp Xξ,λ v ´θλ v,t pξ λ qqsdv `Bσ λ pv, θ λ v,t pξ λ qqdW v , Xξ,λ t " x, (6.5)
where θ λ v,t pξ λ q " λ 1 2 T ´1 λ θ v,t pξ λ q, ξ λ " λ ´1 2 T λ ξ.

The associated generator writes for ϕ P C 2 0 pR nd , Rq and ps, yq P rt, T s ˆRnd as: Lλ,ξ s ϕpyq " xF λ ps, θ λ s,t pξ λ qq `DF λ ps, θ λ s,t pξ λ qqpy ´θλ s,t pξ λ qq, Dϕpyqy `1 2 Tr `aλ pt, θ λ s,t pξ λ qqD 2 y 1 ϕpyq ˘.

Observe from (2.6) and (6.5) that the following very important correspondence holds:

@v P rt, T s, Xξ,λ v :" λ 1{2 T ´1 λ Xξ λ v . (6.6)
We thus derive from Proposition 2 the following important correspondence for the densities. Denoting by pξ λ pt, s, x, yq the density of Xξ,λ v starting from x at time t at point y in s, and x λ " λ ´1 2 T λ x, we have

pξ λ pt, s, x, yq " λ n 2 d 2 pξ λ pt, s, λ ´1 2 T λ x, λ ´1 2 T λ yq " λ n 2 d 2 p2πq nd 2 detp Kξ λ s,t q 1 2 exp ´´λ ´1 2 @ rT λ p Kξ λ s,t q ´1T λ spλ 1 2 T ´1 λ m ξ λ s,t px λ q´yq, λ 1 2 T ´1 λ m ξ λ s,t px λ q´y D ¯. (6.7)
In particular, for ξ " x one derives:

pξ λ pt, s, x, yq " λ n 2 d 2 pξ λ pt, s, λ ´1 2 T λ x, λ ´1 2 T λ yq " λ n 2 d 2 p2πq nd 2 detp Kξ λ s,t q 1 2 exp ˆ´λ ´1 2 A "
T λ p Kξ λ s,t q ´1T λ ‰ pθ λ s,t px λ q ´yq, θ λ s,t px λ q ´yE ˙. (6.8) Equation (6.7) in turn yields the following important control:

|D α x pξ λ pt, s, x, yq| ď C ˆλ ps ´tq ˙řn i"1 α i pi´1 2 q`n 2 d 2 exp ˆ´C ´1 ps ´tq λ ˇˇT ´1 s´t λ `λ1 2 T ´1 λ m ξ λ s,t px λ q ´y˘ˇˇ2
":

C ˆλ ps ´tq ˙řn i"1 α i pi´1 2 q pξ C ´1,λ pt, s, x, yq. (6.9)
In the following, we will also denote pC ´1,λ pt, s, x, yq :" pξ C ´1,λ pt, s, x, yq ˇˇξ "x in (6.9).

Remark 6

We emphasize that (6.9) gives that, each derivation of the Gaussian kernel pξ λ makes the small parameter λ appear. Hence, up to the additional time singularities, the iterated derivatives become smaller and smaller.

Scaling properties

The point is now to reproduce the previous perturbative approach for the solution of (6.2). We will here focus on the Hölder norm of the remainder term associated with the second order derivatives w.r.t. the non-degenerate variables. It can indeed be seen from the previous computations that the other contributions can be dealt similarly. We aim at proving that the L 8 pr0, T s, C γ b,d pR nd , Rqq norm of the indicated term is small. This will follow from our scaling procedure. Precisely, we introduce for λ ą 0, pt, x, x 1 , x 1 q P r0, T s ˆpR nd q 2 the quantity:

pD 2 x 1 R λ u λ qpt, T, x, x 1 q :" ˜ż pt`c 0 λd 2 px,x 1 qq^T t ds ż R nd ´D2 x 1 pξ λ pt, s, x, yqpL λ s ´L λ,ξ s qu λ ps, yq ´D2 x 1 pξ 1 λ pt, s, x 1 , yqpL λ s ´L λ,ξ 1 s qu λ ps, yq ¯dy ¸ˇˇˇˇp ξ,ξ 1 q"px,x 1 q `˜ż T pt`c 0 λd 2 px,x 1 qq^T ds ż R nd

´D2

x 1 pξ λ pt, s, x, yqpL λ s ´L λ,ξ s qu λ ps, yq

´D2

x 1 pξ 1 λ pt, s, x 1 , yqpL λ s ´L λ,ξ 1 s qu λ ps, yq ¯dy ¸ˇˇˇˇp ξ,ξ 1 q"px,xq ": ´pD 2 x 1 R λ u λ q off ´diag `pD 2 x 1 R λ u λ q diag ¯pt, T, x, x 1 q, (6.10) using again as in (5.3) different freezing point according to the spatial regime w.r.t. integration time s. Note however carefully that the cutting threshold here depends on the scaling parameter λ. This is very important in order to balance the various scales that will appear. Pay attention as well that the parameter c 0 also remains. Actually a subtle balance between those two parameters will be needed to derive the expected control. Proceeding as in Section 5.3, we aim at showing that there exists C :" CppAqq s.t. for all px, x 1 q P pR nd q 2 :

sup tPr0,T s |pD 2 x 1 R λ u λ qpt, T, x, x 1 q| d γ px, x 1 q ď Cpc ´pn´1 2 q`γ 2 0 λ γ 2 `c γ 2n´1 0 q}u λ } L 8 pC 2`γ b,d q .
(6.11)

Let us first consider the diagonal term in (6.10) assuming w.l.o.g. that t`c 0 λd 2 px, x 1 q ď T (otherwise we only have the off-diagonal contribution). Write:

|pD 2 x 1 R λ u λ q diag pt, T, x, x 1 q| ď ˇˇż T t`c 0 λd 2 px,x 1 q ds ż R nd dy " D 2 x 1 pξ λ pt, s, x, yq ´D2 x 1 pξ λ pt, s, x 1 , yq ı " xF λ,1 pt, yq ´Fλ,1 pt, θ λ s,t pξ λ qq, D y 1 y `1 2 Tr ´`a λ pt, yq ´aλ pt, θ λ s,t pξ λ qq ˘D2 y 1 ¯ıu λ ps, yq ˇň ÿ i"2 ż T t`c 0 λd 2 px,x 1 q ds ż R nd dy " D 2 x 1 pξ λ pt, s, x, yq ´D2 x 1 pξ λ pt, s, x 1 , yq ı A F λ,i pt, yq´rF λ,i pt, θ λ s,t pξ λ qq`DF λ,i pt, θ λ s,t pξ λ qqpy´θ λ s,t pξ λ qq i´1 s,D y i u λ ps, yq Eˇˇˇˇˇˇˇp ξ,ξ 1 q"px,xq " ˇˇż T t`c 0 λd 2 px,x 1 q ds ż R nd dy ż 1 0 dµDD 2 x 1 pξ λ pt, s, x `µpx 1 ´xq, yq ¨px ´x1 q ˆAF λ,1 pt, yq´F λ,1 pt, θ λ s,t pξ λ qq, D y 1 E `1 2 Tr ´`a λ pt, yq ´aλ pt, θ λ s,t pξ λ qq ˘D2 y 1 ¯˙u λ ps, yq ˇň ÿ i"2 ż T t`c 0 λd 2 px,x 1 q ds ż R nd dy ż 1 0 dµDD 2 x 1 pξ λ pt, s, x `µpx 1 ´xq, yq ¨px ´x1 q A F λ,i pt, yq´rF λ,i pt, θ λ s,t pξ λ qq`DF λ,i pt, θ λ s,t pξ λ qqpy´θ λ s,t pξ λ qq i´1 s, D y i u λ ps, yq Eˇˇˇˇˇˇˇp ξ,ξ 1 q"px,xq ": pD 2 x 1 R λ u λ q diag,1 pt, T, x, x 1 q `pD 2
x 1 R λ u λ q diag,2:n pt, T, x, x 1 q. (6.12)

We will now control the first term of the above right hand side. A key point for the analysis, is to observe that, on the considered diagonal regime, we actually have from equations (6.8)-(6.9), recalling that z Þ Ñ m ξ λ s,t pzq is affine and using the good scaling property in (3.5) and (3.11), that:

|D x k D 2 x 1 pξ λ pt, s, x `µpx 1 ´xq, yq| ď Cλ 1`pk´1 2 q
ps ´tq 1`pk´1 2 q pC ´1,λ pt, s, x, yq exp ´n ÿ j"1 |px ´x1 q j | 2 λ 2j´1 ps ´tq 2j´1 ď Cλ 1`pk´1 2 q ps ´tq 1`pk´1 2 q pC ´1,λ pt, s, x, yq.

(6.13)

To obtain the last inequality, we indeed observe from the homogeneity of the metric (see (1.9)) that λ 1{2 dpx, x 1 q " d `λ´1{2 T λ x, λ ´1{2 T λ x 1 ˘" ř n j"1 |px ´x1 q j λ 2j´1 2 | 1 p2j´1q . For the diagonal regime λd 2 px, x 1 q ď ps ´tq in turn implies that for all j P rr1, nss,

|px´x 1 q j | 2 λ 2j´1 ps´tq 2j´1 ď 1.
Another key point is to observe that the contributions d `λ´1{2 T λ y,λ ´1{2 T λ θ λ s,t px λ q ps´tq 1{2 is homogeneous to the argument of the exponential term in pC ´1,λ pt, s, x, yq. Namely, for any given β 0 ą 0 and β P p0, β 0 s, there exists C β 0 s.t.

˜d`λ ´1{2 T λ y, λ ´1{2 T λ θ λ s,t px λ q ps ´tq 1{2 ¸β pC ´1,λ pt, s, x, yq ď C β 0 pC ´1 β 0
,λ pt, s, x, yq. (6.14) Equation (6.14) is a direct consequence of the expression of pt,x C ´1,λ in Proposition 2 and the definition of d in (1.9).

Hence, from the definition of a λ , F λ in (6.3) equations (6.13) and (6.14), we derive that, under (A) for all k P rr1, nss:

|F λ,1 pt, yq ´Fλ,1 pt, θ λ s,t pξ λ qq||D x k D 2

x 1 pξ λ pt, s, x `µpx 1 ´xq, yq| ˇˇξ

"x ď C ´λ´1{2 }F 1 pt, ¨q} C γ d d γ `λ´1{2 T λ y, λ ´1{2
T λ θ λ s,t pξ λ q ˘λ 1`k´1 2 ps ´tq ´1´pk´1 2 q pC ´1,λ pt, s, x, yq ˇˇξ

"x
ď C}F 1 } L 8 pC γ d q λ k ps ´tq ´1´pk´1 2 q`γ{2 pC ´1,λ pt, s, x, yq. (6.15)

Similarly,

|a λ pt, yq ´aλ pt, θ λ s,t pξ λ qq|D x k D 2 x 1 pξ λ pt, s, x `µpx 1 ´xq, yq| ˇˇξ "x ď C ´λ´1 }apt, ¨q} C γ d d γ `λ´1{2 T λ y, λ ´1{2
T λ θ λ s,t pξ λ q ˘λ 1`pk´1 2 q ps ´tq ´1´pk´1 2 q pC ´1,λ pt, s, x, yq ˇˇξ "x ď C}a} L 8 pC γ d q λ k´1 2 ps ´tq ´1´pk´1 2 q`γ{2 pC ´1,λ pt, s, x, yq. (6.16)

Observe that both r.h.s of (6.15) and (6.16) exhibit a positive power of λ. Hence, those quantities are small provided λ is. The key point in the above computations is that the potentially explosive Hölder norms of F λ,1 , a λ (when λ goes to zero) are compensated by the derivatives of pξ λ ps, t, x, yq (see equation (6.9)). We again carefully point-out that the previous bounds only depend on the supremum in time of the Hölder moduli of the coefficients (denoted }F 1 } L 8 pC γ d q , }a} L 8 pC γ d q respectively) and not on their supremum norm. In particular, we get from (6.15), (6.16) with the notation of (6.12):

pD 2 x 1 R λ u λ q diag,1 pt, T, x, x 1 q ď n ÿ k"1 ż T t`c 0 λd 2 px,xq ds ż R nd dyλ k´1 2 ps ´tq ´1´pk´1 2 q`γ{2 ˆrλ 1 2 }D x 1 u λ } L 8 }F 1 } L 8 pC γ d q `}D 2 x 1 u λ } L 8 }a} L 8 pC γ d q sp C ´1,λ pt, s, x, yq|x k ´x1 k | ď c ´pk´1 2 q`γ{2 0 Cp}D x 1 u λ } L 8 `}D 2 x 1 u λ } L 8 qλ γ 2 d γ px, x 1 q. (6.17)
Thanks to the inequality (4.22) and the previous analysis (to be performed according to the current scaling procedure replacing ps´tq in Sections 4 and 5 by ps´tq{λ), we deduce:

pD 2 x 1 R λ u λ q diag,2:n pt, T, x, x 1 q ď ˇˇn ÿ i"2 ż T t`c 0 λd 2 px,x 1 q ds ż R nd dy ż 1 0 dµ A D y i " ´Fλ,i pt, yq ´rF λ,i pt, θ λ s,t pξ λ qq `DF λ,i pt, θ λ s,t pξ λ qqpy ´θλ s,t pξ λ qqs @x ´x1 , D x D 2 x 1 pξ λ pt, s, x `µpx 1 ´xq, yq D ¯*, u λ ps, yq Eˇˇˇˇˇˇˇξ "x ď C}u λ } L 8 pC 2`γ b,d q n ÿ k"1 ż T t`c 0 λd 2 px,x 1 q λ k´1 2 |x k ´x1 k |ds ps ´tq 1`pk´1 2 q´γ 2 ď Cc ´pk´1 2 q`γ 2 0 }u λ } L 8 pC 2`γ b,d q λ pk´1 2 q´pk´1 2 q`γ 2 d γ´p2k´1q px, x 1 q|x k ´x1 k | ď Cc ´pk´1 2 q`γ 2 0 }u λ } L 8 pC 2`γ b,d q λ γ 2 d γ px, x 1 q ď Cc ´pk´1 2 q`γ 2 0 }u λ } L 8 pC 2`γ b,d q λ γ 2 d γ px, x 1 q. (6.18)
Plugging (6.17) and (6.18) into (6.12) gives a diagonal control which precisely matches the r.h.s. of the expected final bound (6.11).

It therefore remains to handle the off-diagonal contributions. With the notations of (6.10), we deduce from the analysis of Section 4 and the previous arguments that:

|pD 2 x 1 R λ u λ q off ´diag | ď C}u λ } L 8 pC 2`γ b,d q ż t`c 0 λd 2 px,x 1 q t ds ps ´tq 1´γ 2 ď Cc γ 2 0 }u λ } L 8 pC 2`γ b,d q λ γ 2 d γ px, x 1 q.
(6.19) We emphasize that making the diagonal and off-diagonal sets depend on λ is here crucial. To illustrate this, let us briefly focus on the non-degenerate diffusive contribution. We need to proceed from (6.16) except that we have here no differentiation w.r.t. k and therefore no positive power in λ anymore. We retrieve the dependence in λ through the time integration. The bound of equation (6.19) completes the proof of (6.11).

In some sense the controls in (6.18) and (6.19) can be seen as a mere consequence of the intrinsic scaling of the system. This is indeed the case, but, in order to avoid any ambiguity, we provide in Appendix 9 a proof of the rescaled key Besov Lemma 5.

We would also derive from (5.41) and the same previous arguments by denoting t 0 " t `c0 λd 2 px, x 1 q and using the bound of (7.18) for the difference of the scaled flows, that :

ˇˇD 2 x 1 P ξ 1 t 0 ,t u λ pt 0 , x 1 q ´D2 x 1 P ξ1 t 0 ,t u λ pt 0 , x 1 q ˇˇˇˇˇp ξ 1 , ξ1 q"px,x 1 q ď Cc 1 2n´1 0 }u λ } C 2`γ b,d d γ px, x 1 q. (6.20)
Exploiting (6.20) and (6.11), as well as the corresponding control sup tPr0,T s,px,x 1 qPpR nd q 2 , x 1 "x 1

1 |pR λ u λ qpt, T, x, x 1 q| d 2`γ px, x 1 q ď Cc 1 2n´1 0 }u λ } L 8 pC 2`γ b,d q ,
that would be established through the same procedure (exploiting the control (8.13) for the difference of the scaled covariances), we eventually derive similarly to (5.45) that: d q , for some constant C 0 :" C 0 ppAqq. Hence, for c 0 , T and λ small enough with λ ! c 0 , i.e. q ă 1 we derive the expect final control

}u λ } L 8 pC 2`γ b,d q ď Cp}g λ } C 2`γ b,d `}f λ } L 8 pC γ b,d q q `C0 pc ´p2n´1q 0 λ γ 2 `T γ 2 `c 1 2n´1 0 q}u λ } L 8 pC 2`γ b,
}u λ } L 8 pC 2`γ b,d q ď C 1 ´c 0 p}g λ } C 2`γ b,d
`}f λ } L 8 pC γ b,d q q. (6.21)

Conclusion: final proof of Theorem 1.

Equation (6.21) provide the expected Schauder estimate for the rescaled system with mollified coefficients in small time. Recalling that u λ pt, xq " upt, λ ´1{2 T λ xq, we then derive from (6.21) and (6.3) that for T ą 0 and λ small enough there exists C0 :" C0 ppAq, T, λq ą 1 s.t.

}u} L 8 pC 2`γ b,d q ď C0 p}g} C 2`γ b,d

`}f } L 8 pC γ b,d q q, (6.22)

which precisely provides the required estimate in small time for the initial system with mollified coefficients. The point is now to extend the previous bound to an arbitrary fixed time T ą 0 not necessarily small. The stability resulting from estimate (6.22) allows to proceed by simple iterative application of the bound changing the final condition.

Schauder estimates for the mollified system for a general time Equation (6.22) is valid for all T ă T 0 with T 0 P p0, `8q sufficiently small. Now, for a given T ą 0 (not necessary small), we can solve iteratively N " r T T 0 s (where r¨s is the ceiling function) Cauchy problems in the following way. Consider first for all x P R nd :

# B t u 1 pt, xq `xFpt, xq, Du 1 pt, xqy `1 2 Tr `D2
x 1 u 1 pt, xqapt, xq ˘" ´f pt, xq, t P rT p1 ´1 N q, T q, u 1 pT, xq " gpxq.

In other words, our previous analysis, and the previous inequalities, are still available for T ´p1 ´1 N qT " 1 N T ď T 0 small enough. Precisely, from (6.22):

}u 1 `T p1 ´1 N q, ¨˘} C 2`γ b,d ď C0 p}g} C 2`γ b,d
`}f } L 8 prT p1´1 N q,T s,C 2`γ b,d q q. (6.23) Also, for mollified coefficients, it is plain from the Feynman-Kac formula to identify u 1 and u on rT p1 ´1 N q, T s where u solves (1.1) with mollified coefficients on r0, T s. Hence, (6.23) gives in particular that u 1 `T p1 ´1 N q, ¨˘" u `T p1 ´1 N q, ¨˘P C 2`γ b,d pR nd , Rq so that it is in particular natural to consider the following second Cauchy problem for all x P R nd :

# B t u 2 pt, xq `xFpt, xq, Du 2 pt, xqy `1 2 Tr `D2
x 1 u 2 pt, xqapt, xq ˘" ´f pt, xq, t P rp1 ´2 N qT, p1 ´1 N qT q, u 2 pp1 ´1 N qT, xq " upp1 ´1 N qT, xq.

Hence u 2 satisfies identity (6.22) for the corresponding time interval and the associated source and final condition. It also coincides with u on rT p1 ´2 N q, T p1 ´1 N qs. We get:

}u 2 } L 8 prT p1´2 N q,T p1´1 N qs,C 2`γ b,d q ď C0 p}upT p1 ´1 N q, ¨q} C 2`γ b,d `}f } L 8 prT p1´2 N q,T p1´1 N qs,C 2`γ b,d q q ď C0 ´C 0 p}g} C 2`γ b,d `}f } L 8 prT p1´1 N q,T s,C 2`γ b,d q q `}f } L 8 prT p1´2 N q,T p1´1 N qs,C 2`γ b,d q ď C2 0 p}g} C 2`γ b,d `}f } L 8 prT p1´2 N q,T s,C 2`γ b,d q q.
Repeating the analysis N -times, introducing for k P rr3, nss the auxiliary Cauchy problems for all x P R nd :

# B t u k pt,
}u} L 8 pr0,T s,C 2`γ b,d q ď CN 0 p}g} C 2`γ b,d
`}f } L 8 pr0,T s,C 2`γ b,d q q. (6.24)

This precisely gives our main estimate for the system (1.1) with mollified coefficients. Again, even though the coefficients are smooth, all the constants appearing in (6.24) only depend on the Hölder setting of assumption (A).

Remark 7 (About the constants in the final estimate) We could actually have slightly better bounds than those in (6.24). Namely a direct induction shows that the following control holds.

}u} L 8 pr0,T s,C 2`γ b,d q ď CN 0 }g} C 2`γ b,d `N ÿ i"1 C i 0 }f } L 8 prT p1´N ´i`1 N q,T p1´N ´i N qs,C 2`γ b,d q q.
(6.25)

We chose to write the bound in the form of equation (6.24) for simplicity. Note however that in any case, equation (6.24) or (6.25), we still have geometric constants coming from the iterative procedure. This is the main drawback of our approach, which anyhow seems, to the best of our knowledge, the only available one to consider degenerate Kolmogorov systems with non-linear drifts.

Compactness arguments

We now make the mollifying parameter appear again using the notations introduced in the user's guide (see equation (2.4)). Equation (6.24) rewrites in the following way. There exists a constant C 0 s.t. for all m P N:

}u m } L 8 pr0,T s,C 2`γ b,d q ď CN 0 p}g} C 2`γ b,d
`}f } L 8 pr0,T s,C 2`γ b,d q q. (6.26) From Ascoli's theorem we deduce that there exists u P L 8 pr0, T s, C 2`γ b,d q and a sequence of smooth functions pu m k q kě1 solution of (2.4), with m k Ñ k `8, satisfying (6.26) and s.t. u m k Ñ k u where u also satisfies (6.26). Since we also have from [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] that

u m k pt, xq Ñ k ErgpX t,x
T qs `şT t Erf ps, X t,x s qsds where pX t,x s q denotes the unique in law solution to (1.4), we deduce that upt, xq " ErgpX t,x T qs `şT t Erf ps, X t,x s qsds corresponds to the martingale, or mild, solution of (1.1) which satisfies the stated Schauder estimate (6.24).

From mild to weak solutions

Let ϕ be a smooth given function with compact support, i.e. ϕ P C 8 0 pR nd , Rq. It is clear that for the solution of (1.1) with mollified coefficients one indeed has:

ż T 0 dt ż R nd dxf m pt, xqϕpt, xq " ż T 0 dt ż R nd dxϕpt,
since }u´u m } L 8 pC 2`γ b,d q ÝÑ m 0.
For the terms of R 2 m pT, uq which involve the adjoint, the point is again to use the Besov duality to control the remainders. Namely, from the previous analysis we get that:

|R 2 m,2:n pT, uq| :" | n ÿ i"2 ż T 0 dt ż R nd dxD x i `ϕpt, xqF i pt, xq ˘`u m pt, xq ´upt, xq ˘| ď n ÿ i"2 ż T 0 dt ż R pn´1qd ź j‰i dx j }D x i `ϕF i pt, x zi , ¨q˘} B ´2`γ 2i´1 1,1 pR d ,Rq }pu m ´uqpt, x zi , ¨q} C 2`γ 2i´1 b pR d ,Rq
, denoting x zi " px 1 , ¨¨¨, x i´1 , x i`1 , ¨¨¨, x n q and ϕF i pt, x zi , ¨q : x i P R d Þ Ñ ϕF i pt, xq, pu úm qpt, x zi , ¨q : x i P R d Þ Ñ pu ´um qpt, xq. Again }u ´um } L 8 pC 2`γ b,d q ÝÑ m 0 and from the arguments of the proof of Lemma 5, there exists C s.t. for all i P rr2, nss,

}D x i `ϕF i pt, x zi , ¨q˘} B ´2`γ 2i´1 1,1 pR d ,Rq ď Cψ i pt, x zi q,
where ψ i has compact support on R pn´1qd . We thus readily derive |R 2 m,2:n pT, uq| ÝÑ m 0.

Eventually,

|R 2 m,1 pT, uq| :" | ż T 0 dt ż R nd dx " D x 1 `ϕpt, xqF 1 pt, xq ˘`D 2 x 1 `ϕpt, xqapt, xq ˘ı`u m pt, xq ´upt, xq ˘| " | ż T 0 dt ż R nd dx " `ϕpt, xqF 1 pt, xq ¯Dx 1 ``ϕpt, xqapt, xq ˘D2 x 1 ı `um pt, xq ´upt, xq ˘|,
which again tends to 0 with m since }u ´um } L 8 pC 2`γ b,d q ÝÑ m 0.

The contributions involving `pL m t q ˚´L t ˘ϕ in R 1 m pT, uq defined in (6.30) can be handled as in the proof of Lemma 5 exploiting that }a ´am } C γ b,d `}F 1 ´Fm,1 } C γ b,d

`řn i"2 }F i 266 F m,i } C 2i´3`γ b,d
Ñ m 0. We now deduce from (6.29), (6.30) and the previous controls that R m pT, uq Ñ m 0. The same computations also give that the term

ż T 0 dt ż R nd dx ´´B t `pL t q ˚¯ϕpt, xqupt, xq
is well defined under (A). From (6.28), (6.29), we thus finally derive:

ż T 0 dt ż R nd dxϕpt, xqf pt, xq " ż T 0 dt ż R nd
dx `´B t `pL t q ˚˘ϕpt, xqupt, xq, which gives the statement.

7 Appendix: Proof of technical results

Technical results associated with the flow

We begin this paragraph stating and proving a key result for the sensitivity of Hölder flows, i.e. when the coefficients satisfy (A). Those results are of course uniform w.r.t. a mollification procedure of the coefficients as the one previously considered from Section 3 to 5. Also, Lemma 2 is a direct consequence of Lemma 11 below and Young/convexity inequalities.

A first sensitivity result for the flow Lemma 11 Under (A), there exists C :" CppAq, T q s.t. for all px, x 1 q P pR nd q 2 , 0 ď t ă s ď T and i P rr1, nss:

|pθ s,t pxq ´θs,t px 1 qq i | ď C ´ps ´tq i´1 2 `d2i´1 px ´x1 q ¯.
The previous bound can be interpreted as follows. We somehow have the expected bound involving the spatial points, up to an additional contribution in time, which is precisely due to the pseudo-distance d. Indeed, this type of result already appeared (for Lipschitz drifts) in Proposition 4.1 of [START_REF] Menozzi | Martingale problems for some degenerate Kolmogorov equations[END_REF]. Through an appropriate mollifying procedure, this result remains unchanged Proof: The main idea to prove this control relies on Grönwall's Lemma. However, under (A), the function F is not Lipschitz (solely Hölder continuous). We have then to mollify suitably F. Let us denote by δ P R n , a vector with positive entries δ i ą 0 for i P rr2, nss. Define as well for all v P r0, T s, z P R nd , i P rr2, nss,

F δ i pv, z i´1:n q :" F i pv, ¨q ‹ ρ δ i pzq " ż R d F i pv, z i´1 , z i ´w, z i`1 ¨¨¨, z n qρ δ i pwqdw, (7.1)
with ρ δ i pwq :" 1 δ d i ρ ´w δ i ¯where ρ : R d Ñ R `is a usual mollifier, namely ρ has compact support and ş R d ρpzqdz " 1. Finally, we define F δ pv, zq :" pF 1 pv, zq, F δ 2 pv, zq, ¨¨¨, F δ n pv, zqq.

In this definition, we make a slight abuse of notation since the first component F 1 is not mollified. Due to the final control we want to prove and the intrinsic scale of the first component, the sublinearity of F 1 (implied by its Hölder property) is enough and it is not needed to mollify this component.

To be at the good current time scale for the contributions associated with the mollification, we pick δ i in order to have C :" CppAq, T q ą 0 s.t. for all z P R nd , u P rt, ss: 3) for the last inequality. We proceed similarly for the pn ´1q th component, but in this case we have to handle the non-Lipschitz continuity of F δ n´1 in its n th variable. For the rescaled flows see e.g. Lemma 2 in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] this difficulty could also be circumvented through mollification, the situation is here slightly different and it seems The last inequality is a consequence of our choice of δ n´1 in (7.4), identity (7.5) and convexity inequality.

ˇˇps ´tq 1 2 T ´1
We aim, now, to proceed with Grönwall's Lemma. To do so, first of all we need to use a Young inequality. Namely, we write for all δn´1,n ą 0 (where the two indexes in the subscript respectively denote the stage of the chain, i.e. n´1, and the considered variable, i.e. n): q for the last identity, recalling as well that dpx, x 1 q ď 1, and therefore |px ´x1 q n | 2n´3 2n´1 ď 1, for the last identity. The purpose of (7.8) is that each entry of the difference of the starting points appears at its intrinsic scale for the homogeneous distance d.

´ż v t ˇˇ`θ w,
Plugging the above inequality into (7.5) we derive:

|pθ s,
ż v 2 t dv 1 ˇˇ`θ v 1 ,t px 1 q ´θv 1 ,t pxq ˘1ˇˇ¸.
Similarly, for i P rr2, nss we derive:

|pθ s,t pxq ´θs,t px 1 qq i | (7.10)

ď C ˜ps ´tq i´1 2 `n ÿ j"2 |px ´x1 q j | 2i´1 2j´1 `ż v i "s t dv i´1 . . . ż v 2 t dv 1 ˇˇ`θ v 1 ,t px 1 q ´θv 1 ,t pxq ˘1ˇˇ¸.
Remark 8 We importantly point out that equations (7.9) and (7.10) actually hold not only for the fixed time s but also for any v P rt, T s.

The term for i " 1 is treated slightly differently. Namely, for all s P rt, ss, write:

|pθ s,t pxq ´θs,t px 1 qq 1 | ď |px ´x1 q 1 | `C n ÿ j"1 ż s t |pθ v,
| γ 2k´1 ď Cpps ´tq 1 1´γ `|px ´x1 q k | 1 2k´1 q.
We eventually derive from (7.11) that:

sup vPrt,ss |pθ v,t pxq ´θv,t px 1 qq 1 | ď C `ps ´tq 1 2 `dpx, x 1 q ˘,
which gives the stated bound for i " 1. It now remains to plug this control into (7.10).

We obtain for all i P rr2, nss:

|pθ s,t pxq ´θs,t px 1 qq i | ď C `ps ´tq i´1 2 `d2i´1 px, x 1 q `ps ´tq i´1 sup vPrt,ss |pθ v,t pxq ´θv,t px 1 qq 1 | ď C ´ps ´tq i´1 2 `d2i´1 px, x 1 q `ps ´tq i´1 `ps ´tq 1 2 `dpx, x 1 q ˘ď C `ps ´tq i´1 2 `d2i´1 px, x 1 q ˘, using again the Young inequality to derive that ps ´tq i´1 dpx, x 1 q ď C `ps ´tq i´1 2 d2i´1 px, x 1 q ˘. The proof is complete.

Again, Lemma 2 is a direct consequence of the previous Lemma 11 and Young/convexity inequalities.

We are now in position to prove the sensitivity results for the linearized system w.r.t. the freezing parameter.

Sensitivity results for the mean Proof: Technical Lemma 3

We assume w.l.o.g. that dpx, x 1 q ď 1. The idea of the proof is to separate the term to control into two contributions. Namely, we write: m x s,t px 1 q ´θs,t px 1 q " rm x s,t px 1 q ´θs,t pxqs `rθ s,t pxq ´θs,t px 1 qs. (7.12)

The definition of the proxy (3.1) yields that the mean value of Xm,ξ v , m ξ v,t is s.t. m x s,t px 1 q ´θs,t pxq " x 1 ´x `ż s t dvDFpv, θ v,t pxqqrm x v,t px 1 q ´θv,t pxqs. (7.13)

The sub-triangular structure of DF yields that for all i P rr2, nss:

`mx s,t px 1 q ´θs,t pxq ˘i " x 1 i ´xi `ż s t dvD i´1 F i pv, θ v,t pxqqrm x v,t px 1 q i´1 ´θv,t pxq i´1 s.

Also, since m x v,t px 1 q 1 " x 1 1 `şs t F 1 pv, θ v,t pxqqdv, so that rm x v,t px 1 q 1 ´θv,t pxq 1 s " x 1 1 ´x1 , we then obtain by iteration that: `mx s,t px 1 q ´θs,t pxq ˘i

" x 1 i ´xi `i ÿ k"2 " ż v i "s t dv i´1 . . . ż v k t dv k´1 i ź j"k D j´1 F j pv j , θ v j ,t pxqq ı rx 1 k´1 ´xk´1 s,
with the convention that for i " 1, ř i k"2 " 0. From the above control, equation (7.12) and the dynamics of the flow, recalling that the starting points are the same, so that the contributions involving differences of the spatial points or flows only appear in iterated time integrals, we derive:

| `mx s,t px 1 q ´θs,t px 1 q ˘i| ď ˇˇˇi ÿ k"2 " ż v i "s t dv i´1 . . . ż v k t dv k´1 i ź j"k D j´1 F j pv j , θ v j ,t pxqq ı rx 1 k´1 ´xk´1 s ˇˇż s t |F i pv, θ v,t pxqq ´Fi pv, θ v,t px 1 qq|dv (7.14) ď C ´i´1 ÿ k"2 ps ´tq i´k |x k ´x1 k | `ż s t ´n ÿ j"i
ˇˇ`θ v,t pxq ´θv,t px 1 q ˘jˇˇ2i´3`γ 2j´1 `ˇ`θ v,t pxq ´θv,t px 1 q ˘i´1 ˇˇ¯d v ¯.

From the previous Lemma 11, we thus obtain:

| `mx s,t px 1 q ´θs,t px 1 q ˘i| ď C ˆi´1 ÿ k"2 ps ´tq i´k |x k ´x1 k | `ps ´tq 2i´3`γ 2 `1
`d2i´3`γ px, x 1 qps ´tq ``ps ´tq pi´1q´1 2 `d2pi´1q´1 px, x 1 q ˘ps ´tq ˙. (7.15)

In particular, for s " t 0 " t `c0 d 2 px, x 1 q with c 0 ă 1, the previous equation yields:

| `mx t 0 ,t px 1 q ´θt 0 ,t px 1 q ˘i| ď C ´c0 d 2i´1 px, x 1 q `pc i´1 2 `γ 2 0 `c0 qd 2i´1`γ px, x 1 q `pc i´1 2 0 `c0 qd 2i´1 px, x 1 q ¯.
So, after summing and by convexity inequalities, for dpx, x 1 q ď 1:

d `mx t 0 ,t px 1 q, θ t 0 ,t px 1 q ˘ď Cc 1 2n´1 0 dpx, x 1 q.

Sensitivities for the scaled flows

For the scaling analysis of Section 6 we also need the scaled versions of the previous Lemmas. Recalling the notations introduced therein, i.e. for λ ą 0, 0 ď t ď v ď T , θ λ v,t px λ q " λ 1 2 T ´1 λ θ v,t px λ q, x λ :" λ ´1 2 T λ x, as corollaries of Lemmas 11 and 2, we readily get:

dpθ λ v,t px λ q, θ λ v,t px 1 λ qq ď C ´dpx, x 1 q ``v ´t λ ˘1 2 ¯. (7.16)
Setting now v " t `c0 λd 2 px, x 1 q, recalling that

d `λ1 2 T ´1 λ m x λ v,t px 1 λ q, θ λ v,t px 1 λ q ˘" λ ´1 2 d `mx λ v,t px 1 λ q, θ λ v,t px 1 λ q ˘, (7.17)
and reproducing the above arguments, we derive from (7.16) and (7.15) the bound:

d `λ1 2 T ´1 λ m x λ v,t px 1 λ q, θ λ v,t px 1 λ q ˘ď Cpc 1 2n´1 0 `λγ 2 qdpx, x 1 q. (7.18)
which follows exploiting the intrinsic scaling properties of system. Observe that the scaling procedure does not provide for the mean sensitivity a small constant (i.e. proportional to λ γ 2 ). B Indeed, in inequality (7.14), the terms involving |F i pv, θ v,t pxqq ´Fi pv, θ v,t px 1 qq| and the products `śi j"k D j´1 F j pv j , θ v j ,t pxqq ˘kPrr2,i´1ss do not allow us to take advantage from the 1 `γ 2i´3 -Hölder continuity of F i w.r.t the pi ´1q th spatial component nor from the γ 2j´1 -Hölder continuity of D j´1 F j pv j , θ v j ,t pxqq w.r.t. the pi ´1q th spatial component. We can only exploit respectively the Lipschitz continuity and the boundedness of such terms which are precisely those who do not scale. Plugging x λ " λ ´1 2 T λ x, x 1 λ " λ ´1 2 T λ x 1 in (7.15) indeed yields for v " t `c0 λd 2 px, x 1 q:

|pm x λ v,t px 1 λ q ´θλ v,t px 1 λ qq i | ď λ i´1 2 d 2i´1 px, x 1 qC ˆi´1 ÿ k"2 pc 0 q i´k `ci´1 2 0 pc 0 λd 2 px, x 1 qq γ 2 `λγ 2 d γ px, x 1 qc 0 ``c i´1 2 0 `c0 ˘˙,
which from (7.17) yields (7.18) recalling that c 0 and dpx, x 1 q are both less than 1.

8 Appendix: Sensitivity results for the resolvent and covariance

Sensitivity Lemma for the resolvent

Lemma 12 (Controls of the Sensitivities for the Resolvents) There exists C s.t. for all 0 ď t ď s ď T , px, x 1 q P pR nd q 2 , the following control holds. For all 1 ď j ă i ď n, with the notation of (3.20):

ˇˇ`R pτ,xq pt, sq ´R pτ,x 1 q pt, sq ˘i,j ˇˇď Cps ´tq i´j ´n ÿ Pay attention that we only know from our smoothness assumption (S) that for all i P rr2, nss, @z i:n " pz i , ¨¨¨, z n q P R pn´i`1qd , z i´1 Þ Ñ D x i´1 F i pz i´1 , z i:n q is C γ 2pi´1q´1 pR d , R d bR d q-Hölder continuous for η ą 0. Hence, we proceed carefully like in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF].

k"2 }F k } L 8 pC 2k´3`γ d q ¯`
We directly get that:

|∆ p Rs,t,x,x 1 1 | " | p Rs,t,x 1 ´p Rs,t,x 1 1 | " |T ´1 s´t p Rx ´R x 1 qps, tqT s´t | ď C ż s t ps ´tq ´1 n ÿ i"2
´|D i´1 F i pv, θ v,t pxqq ´Di´1 F i pv, θ v,t pxq i´1 , pθ v,t px 1 qq i:n q| `}pD x i´1 F i q i´1 } L 8 pC γ d q |pθ v,t pxq ´θv,t px 1 qq i´1 | η i ¯dv ": pR 1 `R2 qps, t, x, x 1 q, (8.3)

where η i :" γ 2pi´1q´1 and the notation pD x i´1 F i q i´1 indicates that D x i´1 F i is viewed as a function of its variable pi ´1q and the supremum is taken over the other ones. From Lemma 2 and the definition of d (see also Lemma 11), we readily get

|R 2 ps, t, x, yq| ď C}DF} L 8 pC γ d q n ÿ i"2 ż s t ps ´tq ´1´p v ´tq pi´1´1 2 q `d2pi´1q´1 px, x 1 q ¯ηi dv ď C}DF} L 8 pC γ d q `ps ´tq γ 2 `dγ px, x 1 q ˘, (8.4)
denoting with a slight abuse of notation }DF} L 8 pC γ d q :"

ř n i"2 }pD x i´1 F i q i´1 } L 8 pC γ d q .
To control the difference of the gradients terms in R 1 ps, t, x, x 1 q in (8.3), we need the following result whose proof is postponed to Appendix 8.3.

Lemma 13 (Reverse Taylor expansion)

There is a constant C ą 0 s.t, for all pz, z 1 q P R nd ˆRnd , v P r0, T s : ˇˇD x i´1 F i pv, zq ´Dx i´1 F i pv, z 1 q ˇˇď C}F i } L 8 pC 2i´3`γ d q dpz, z 1 q γ , with the notations of equation (3.20).

From the reverse Taylor expansion of Lemma 13 and the definition in (8.3), we obtain:

|R 1 ps, t, x, yq| ď Cps ´tq ´1 ż s t dv n ÿ i"2 }F i } L 8 pC 2i´3`γ d q d γ pθ v,t pxq, θ v,t px 1 qq ď C n ÿ i"2 }F i } L 8 pC 2i´3`γ d q `ps ´tq γ 2 `dγ px, x 1 q ˘, (8.5)
using again Lemma 2 for the last inequality. Gathering (8.4), (8.5) into (8.3) and recalling the definition in (3.20), we obtain:

|∆ p Rs,t,x,x 1 1 | ď C `n ÿ i"2 }F i } L 8 pC 2i´3`γ d q ˘`ps ´tq γ 2 `dγ px, x 1 q ˘. (8.6)
The result follows from the previous bound, the definition in (8.2) and the scalings of equation (8.1).

8.2

Sensitivity Lemma for the covariances |p Kξ s,t ´K ξ 1 s,t q i,j | ď Cps ´tq i`j´1 `ps ´tq γ 2 `dγ pξ, ξ 1 q ˘, (8.12) which precisely gives (3.18) for i " j " 1. Sensitivities for the scaled covariance.

In connection with Section 6, we recall the identity in law (6.6), i.e. Xξ,λ v :" λ 1{2 T ´1 λ Xξ λ v , v P rt, T s, which readily gives:

Kξ,λ v,t :" Covp Xξ,λ v q " λT ´1 λ Kξ λ v,t T ´1 λ .
In particular, we thus derive from the analysis of the previous paragraph:

r Kξ,λ v,t s 1,1 ď Cλ ´1pv ´tq, ˇˇr Kξ,λ v,t s 1,1 ´r Kξ 1 ,λ v,t s 1,1 ˇˇď Cλ ´1pv ´tq ´dγ pξ λ , ξ 1 λ q `pv ´tq

γ 2 ď Cλ ´1pv ´tq ´λγ 2 d γ pξ, ξ 1 q `pv ´tq γ 2 ¯,
recalling that ξ λ " λ ´1 2 T λ ξ and using the homogeneity properties of d for the last inequality. Also, for v " t `c0 λd 2 px, x 1 q and taking ξ " x, ξ 1 " x 1 , the above controls rewrite:

r Kξ,λ v,t s 1,1 ď Cc 0 d 2 px, x 1 q, ˇˇr Kξ,λ v,t s 1,1 ´r Kξ 1 ,λ v,t s 1,1 ˇˇď Cc 0 λ γ 2 d 2`γ px, x 1 q. (8.13)
Note that the sensitivity of the scaled covariance yields a contribution of the scaling coefficient in λ γ 2 . Unlike for the control of the scaled mean in (7.18), where, as previously noticed, we could not exploit the full regularity of F i w.r.t. the pi ´1q th variable, we can here precisely take advantage of such a regularity. Indeed, this follows from the expression of the covariance (8.7) which only involves D i´1 F i so that one can exploit the associated γ 2i´3 -Hölder regulariry w.r.t. x i´1 .

Reverse Taylor formula

Proof: Lemma 13

We assume here, for the sake of simplicity and without loss of generality, that d " 1 (scalar case). When d ą 1, the proof below can be reproduced componentwise. Let us decompose the expression around the variables which do/do not transmit the noise. Namely, we write for all δ i ą 0:

D x i´1 F i pv, zq ´Dx i´1 F i pv, z 1 q " ż 1 0 dµtD x i´1 F i pv, zq ´Dx i´1 F i pv, z i´1 `µdpz, z 1 q δ i
, z i:n qu `tD x i´1 F i pv, z i´1 `µdpz, z 1 q δ i , pz 1 q i:n q ´Dx i´1 F i pv, z 1 qu `tD x i´1 F i pv, z i´1 `µdpz, z 1 q δ i , z i:n q ´Dx i´1 F i pv, z i´1 `µdpz, z 1 q δ i , pz 1 q i:n qu ":

3 ÿ "1 ∆F i pv, z, z 1 q.
(8.14)

The first two terms can be dealt directly. From (A) we get:

|∆F 1 i pv, z, z 1 q| ď }pD x i´1 F i q i´1 } L 8 pC γ d q dpz, z 1 q δ i γ 2pi´1q´1 . (8.15)
Similarly,

|∆F 2 i pv, z, z 1 q| ď }pD x i´1 F i q i´1 } L 8 pC γ d q `dpz, z 1 q δ i γ 2pi´1q´1 `|pz ´z1 q i´1 | γ 2pi´1q´1 ď C}pD x i´1 F i q i´1 } L 8 pC γ d q dpz, z 1 q δ i γ 2pi´1q´1 . (8.16)
For ∆F 3 i pt, z, z 1 q, we use an explicit reverse Taylor expansion which yields together with the smoothness assumption of F i in (A):

|∆F 3
i pt, z, z 1 q| " dpz, z 1 q ´δi ˇˇ"F i pt, z i´1 `dpz, z 1 q δ i , z i:n q ´Fi pt, z i´1 `dpz, z 1 q δ i , pz 1 q i:n q `Fi pt, z i´1 , z i:n q ´Fi pt, z i´1 , pz 1 q i:n q ıˇˇď 2}F i } L 8 pC 2i´3`γ d q dpz, z 1 q 2i´3`γ´δ i . (8.17) Taking δ i s.t. δ i γ 2pi´1q´1 " 2i ´3 `γ ´δi , which implies that δ i " 2i ´3, gives in (8.15), (8.16) and (8.17) a global bound of order C}F i } L 8 pC 2i´3`γ d q d γ pz, z 1 q. The result then follows from (8.14). 9

Scaling Control of the degenerate part of the perturbative term

This section is dedicated to the proof of the scaled version of the key Besov Lemma 5. We recall that, with the definitions of Section 6, for all multi-index α " pα 1 , . . . , α n q P R nd , i P rr2, nss, we aim to control the terms

ż T t ds ż R nd D α D 2 x 1 pξ λ pt, s, x, yq @ ∆ λ i,F pt, s, θ s,t pξq, yq, D y i u λ ps, yq D dy,
5 can be adapted replacing ps ´tq therein by ps ´tq{λ in the computations involving the thermic characterization of Besov spaces. We also point out that, w.l.o.g., we assume that T {λ ď 1 so that in particular for 0 ď t ă s ď T , λ ´1ps ´tq ď 1. Indeed, the parameter λ is meant to be small (at least λ ď 1) but macro as well. From the previous analysis and the statement of Lemma 14 it can be seen that the optimal λ, i.e. the largest one, actually depends on the Hölder moduli of the coefficients. Hence, the condition T {λ ď 1 is, up to a possible modification of T , not restrictive.

Let us first introduce some notation: qc,λ pt, s, x, yq :"

n ź j"1 N cλ 2j´1
2 ps´tq 2j´1 `pθ s,t pxq ´yq j ˘" pc ´1 pt, s, x, yq, where for a ą 0, z P R d , like before N a pzq " 1 (9.5)

We recall from (4.14), that the parameter β i " p2i´3qp2i´1q 2i´3´γ . The first contribution of the scaled Besov control is:

ż 1 rλ ´1ps´tqs β i dv v v αi 2 }h v ‹ Ψ α,λ i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq ď ż 1 rλ ´1ps´tqs β i dv v v αi 2 ż R d dz ˇˇż R d D α pξ λ pt, s, x, yq @ ∆ λ i,F pt, s, θ λ s,t pξq, yq, D z h v pz ´yi q D dy i ˇˇˇˇˇˇξ "x ď C ż 1 rλ ´1ps´tqs β i dv v v αi 2 ż R d dz ż R d dy i h cv pz ´yi q v 1 2 λ ř n
j"1 α j pj´1 2 q qc,λ pt, s, x, yq ps ´tq

ř n j"1 α j pj´1 2 q ˆλ´i`1 2 d 2i´3`γ pλ ´1{2 T λ θ λ s,t pxq, λ ´1{2 T λ yq ď Cλ ř n j"1 α j pj´1 2 q´pi´1 2 q ż 1 rλ ´1ps´tqs β i dv v v αi 2 ż R d dz ż R d dy i ˆhcv pz ´yi q v 1 2
qc,λ pt, s, x, yq ps ´tq

ř n j"1 α j pj´1 2 q ps ´tq 2i´3`γ 2
, exploiting (6.13) for the last inequality. Then

ż 1 rλ ´1ps´tqs β i dv v v αi 2 }h v ‹ Ψ α,λ
i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq ďCλ ř n j"1 α j pj´1 2 q´pi´1 2 q qczi,λ pt, s, x, py 1:i´1 , y i`1:n qq

ż 1 rλ ´1ps´tqs β i dvv ´3 2 `α i 2 ps´tq ´řn j"1 α j pj´1 2 q`2 i´3`γ 2 ďCλ ř n
j"1 α j pj´1 2 q´pi´1 2 q qczi,λ pt, s, x, py 1:i´1 , y i`1:n qqλ r 1 2 ´α i 2 sβ i ps´tq r´1 2 `α i 2 sβ i ´řn j"1 α j pj´1 2 q`2 i´3`γ 2 ďCλ ř n j"1 α j pj´1 2 q´1 qczi,λ pt, s, x, py 1:i´1 , y i`1:n qqps ´tq ´řn j"1 α j pj´1 2 q`γ 2 , (9.6) the third inequality is a consequence of Proposition 2, and the last identity comes from the pick of β i which in particular gives ´pi ´1 2 q `r1 ´α i s β i 2 " ´1. Let us now consider the second contribution of the scaled Besov control, i.e. we now take v P " 0, rλ ´1ps ´tqs

β i ‰ . Write: ż R d h v pz ´yi qD y i ¨`Θ α,λ i,

˘,

D z h v pz ´yi q E dy i ": ´Tλ,1 `Tλ,2 ¯`v, t, s, x, py 1:i´1 , z, y i`1:n q ˘, (9.7)

thanks to the definition in (9.2) for the last identity.

|T λ,1 `v, t, s, x, py 1:i´1 , z, y i`1:n q ˘| (9.8)

ď C ż R d h cv pz ´yi q v 1 2 λ ř n j"1 α j pj´1 2 q ps ´tq ř n j"1 α j pj´1 2 q qc,λ pt, s, x, yqλ ´i`1 2 pλ 2i´1 2 |z ´yi |q 2i´3`γ 2i´1 dy i ď Cλ ř n j"1 α j pj´1 2 q´pi´1 2 q`2 i´3`γ 2 ż R d h cv pz ´yi q v 2´γ 4i´2
qc,λ pt, s, x, yq ps ´tq ř n j"1 α j pj´1 2 q dy i .

ď Cλ ř n j"1 α j pj´1 2 q´1`γ 2 v 2´γ 4i´2 ps ´tq ř n j"1 α j pj´1 2 q qczi,λ pt, s, x, py 1:i´1 , y i`1:n qqN cv`λ 2i´1 2 ps´tq 2i´1 `z ´θs,t pxq i ˘.

Write now from (9.7):

|T λ,2 `v, t, s, x, py 1:i´1 , z, y i`1:n q

˘| ď C ż R d dy i h cv pz ´yi q v 1 2 ż 1 0 dµ λ ř n j"1 α j pj´1 2 q`2 i´1
2 qc,λ pt, s, x, y 1:i´1 , z `µpy i ´zq, y i`1:n q ps ´tq ps ´tq ps ´tq ř n j"1 α j pj´1 2 q`2 i´1 2 `1 ps ´tq ř n j"1 α j pj´1 2 q`1´γ 2 ¯.

ř n j"1 α j pj´1 2 q`2 i´1 2 ˆ|y i ´z|λ ´i`1 2 ´ˇˇF i `s, λ ´1 2 T λ py 1:i´1 , z, y i`1:n q ˘´F i `s, λ ´1 2 T λ py 1:i´1 , θ s,t pxq i:n q ˘ˇF i `s, λ ´1 2 T λ py 1:i´1 , θ s,t pxq i:n q ˘´F i `s, λ ´1 2 T λ θ s,t pξq Dx i´1 F i `s, λ ´1 2 T λ θ s,
ř n j"1 α j pj´1 2 q`2 i´1 2 `1 ps ´tq ř n j"1 α j pj´1 2 q`1´γ 2 ď Cλ ř n j"1 α j pj´1 2 q qczi,λ pt, s, x, y 1:i´1 , y i`1:n q ˆż 1 0 dµ ż R d h cv pz
(9.9)

From (9.7), (9.9) and (9.9) we deduce, with the notation of (9.5):

}h v ‹ Ψ α,λ i,pt,xq,ps,y 1:i´1 ,y i`1 q } L 1 pR d ,Rq ď Cλ ř n j"1 α j pj´1 2 q qczi,λ pt, s, x, py 1:i´1 , y i`1:n qq ˆˆλ ´1`γ 2 v 2´γ 4i´2 ps ´tq

ř n j"1 α j pj´1 2 q `λ2i´3`γ 2 v 2i´3`γ 2p2i´1q
ps ´tq

ř n j"1 α j pj´1 2 q`2 i´1 2 `1 ps ´tq ř n j"1 α j pj´1 2 q`1´γ 2 ż 1 0 dµ ż R d dz ż R d dy i h cv pz ´yi qN cλ 2i´1
2 ps´tq 2i´1 pz `µpy i ´zq ´pθ s,t pxqq i q ď Cλ ř n j"1 α j pj´1 2 q qczi,λ pt, s, x, py 1:i´1 , y i`1:n qq ˆˆλ ´1`γ 2 v 2´γ 4i´2 ps ´tq

ř n j"1 α j pj´1 2 q `λ2i´3`γ 2 v 2i´3`γ 2p2i´1q
ps ´tq

ř n j"1 α j pj´1 2 q`2 i´1 2 `1 ps ´tq ř n j"1 α j pj´1 2 q`1´γ 2 ˙.
The last identity is a again consequence of the change of variable pw 1 , w 2 q " pz ´yi , z μpy i ´zq ´pθ s,t pxqq i q.

Partie III: Degenerate Kolmogorov chains

We now write:

ż rλ ´1ps´tqs β i 0 dvv αi 2 ´1}h v ‹ Ψ α,λ
i,pt,xq,ps,y 1:i´1 ,y i`1 q } L 1 pR d ,Rq ď Cλ ř n j"1 α j pj´1 2 q qczi pt, s, x, py 1:i´1 , y i`1:n qq

ż rλ ´1ps´tqs β i 0 dv v v αi 2 ˆˆλ ´1`γ 2 v 2´γ 4i´2 ps ´tq ř n j"1 α j pj´1 2 q `λ2i´3`γ 2 v 2i´3`γ 2p2i´1q
ps ´tq ř n j"1 α j pj´1 2 q`2 i´1 2 `1 ps ´tq ř n j"1 α j pj´1 2 q`1´γ 2 ":

Cλ ř n j"1 α j pj´1 2 q qczi,λ pt, s, x, py 1:i´1 , y i`1:n qqB λ α,β i pt, sq.

(9.10)

Let us now prove that

B λ α,β i pt, sq ď Cλ ´1 ps ´tq ř n j"1 α j pj´1 2 q´γ 2 . (9.11)
Integrating in v in (9.10) we derive:

B λ α,β i pt, sq ď Cps ´tq ´řn j"1 α j pj´1 2 q « λ ´1`γ 2 rλ ´1ps ´tqs β i p αi 2 ´2´γ 4i´2 q `λ2i´3`γ 2 rλ ´1ps ´tqs β i p αi 2 `2i´3`γ
2p2i´1q qq ps ´tq i´1 2 `rλ ´1ps ´tqs β i αi 2 ps ´tq ´1`γ 2 ff .

Recall now from the proof of Lemma 5 that:

β i ˆα i 2 ´2 ´γ 4i ´2 ˙´γ 2 ě 0, β i ˆα i 2 `2i ´3 `γ 2p2i ´1q ˙´2i ´1 2 ´γ 2 ě 0, β i αi 2 ´1 ě 0,
with β i " p2i´3qp2i´1q 2i´3´γ , αi " 2`γ 2i´1 . Therefore, since ps ´tq{λ ď 1:

B λ α,β i pt, sq ď Cps ´tq ´řn j"1 α j pj´1 2 q " λ ´1`γ 2 rλ ´1ps´tqs γ 2 `λ´1`γ 2 rλ ´1ps ´tqs γ 2 `rλ ´1ps ´tqsps ´tq ´1`γ 2 ı ,
which precisely gives (9.11). Plugging (9.11) into (9.10) and from (9.6) we eventually get:

ż 1 0 dv v v αi 2 }h v ‹ Ψ α,λ i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq ď Cλ ř n j"1 α j pj´1 2 q´1
ps ´tq ř n j"1 α j pj´1 2 q´γ 2 qczi,λ pt, s, x, py 1:i´1 , y i`1:n qq, which is precisely the stated control. The term }ϕpDqΨ α,λ i,pt,xq,ps,y 1:i´1 ,y i`1:n q } L 1 pR d ,Rq appearing in the Besov norm could be handled similarly. The result is complete.

Chapter 7

Strong Uniqueness for a chain of Degenerate Kolmogorov Equations Abstract: We establish strong uniqueness for a class of degenerate Kolmogorov SDEs under suitable Hölder regularity conditions for the associated drift term. In particular, we exhibit some sharp thresholds on the Hölder exponents for the strong uniqueness to hold. Our approach is based on the forward parametrix perturbation technique and also heavily relies on appropriate duality properties on Besov spaces.

Introduction

Statement of the problem and related results

In this work, we aim at establishing a strong well posedness result outside the classical Cauchy-Lipschitz framework for the following degenerate Stochastic Differential Equation (SDE) of Kolmogorov type:

dX 1 t " F 1 pt, X 1 t , .
. . , X n t qdt `σpt, X 1 t , . . . , X n t qdW t , dX 2 t " F 2 pt, X 1 t , . . . , X n t qdt, dX 3 t " F 3 pt, X 2 t , . . . , X n t qdt, . . .

dX n t " F n pt, X n´1 t , X n t qdt, t ě 0, (1.1)
where, pW t q tě0 stands for a d-dimensional Brownian motion on some filtered probability space pΩ, pF t q tě0 , Pq and for all i P rr1, nss * , t ě 0 the component X i t is R d -valued as well (i.e. X t P R nd ). We suppose that the pF i q iPrr2,nss satisfy a kind of weak Hörmander condition, i.e. the matrices `Dx i´1 F i pt, ¨q˘i Prr2,nss have full rank. However, the coefficients pF i q iPrr2,nss can be rather rough in their other entries, namely, Hölder continuous. We assume as well that the diffusion coefficient σ is bounded from above and below and spatially Lipschitz continuous.

For a system of Ordinary Differential Equation (ODE) it may be a real challenge to prove the well posedness outside the Lipschitz framework (see e.g. [START_REF] Di Perna | Ordinary Differential Equations, transport theory and Sobolev Spaces[END_REF]) and, as shown by Peano's example, uniqueness may fail as soon as the drift of the system of interest is only Hölder continuous. For an SDE, the story is rather different since the presence of the noise may allow to restore well posedness. Such a phenomenon, called regularization by noise (see the Saint Flour Lecture notes of Flandoli [START_REF] Flandoli | Random perturbation of PDEs and fluid dynamic models[END_REF] and the references therein for an overview of the topic), has been widely studied since the pioneering unidimensional work of Zvonkin [START_REF] Zvonkin | A transformation of the phase space of a diffusion process that will remove the drift[END_REF] and its generalization to the multi-dimensional setting by Veretennikov [START_REF] Veretennikov | Strong solutions and explicit formulas for solutions of stochastic integral equations[END_REF] where SDEs driven by a non-degenerate Brownian noise and a bounded drift are shown to be well-posed. We mean by non-degenerate that the noise has the same dimension as the underlying system on which it acts.

Let us mention, among others, and still within the non-degenerate setting, the works of Krylov and Röckner [START_REF] Krylov | Strong solutions of stochastic equations with singular time dependent drift[END_REF] (L q ´Lp drift), Zhang [START_REF] Zhang | Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients[END_REF] (L q ´Lp drift and weakly Lipschitz diffusion matrix) and also Fedrizzi and Flandoli [START_REF] Fedrizzi | Pathwise Uniqueness and Continuous Dependence for SDEs with Nonregular Drift[END_REF] (L q ´Lp and Hölder drift).

The crucial point in the aforementioned results lies into the non-degeneracy assumption assumed on the noise added in the considered system: we can benefit from the regularization by noise phenomenon only in the directions submitted to the noise and when the noise degenerates it is useless to expect a generalization of the previous results without any additional assumptions.

In our current setting, the non-degeneracy assumption on the Jacobian `Dx i´1 F i pt, ¨q˘, i P rr2, nss precisely allows the noise to propagate through the chain passing from the i th to the pi `1q th level thanks to the drift, hence leading to a propagation of the noise in the whole considered space. The main idea is then to take advantage of this particular propagation, known as weak Hörmander setting (in reference to the work of Hörmander on hypoelliptic differential operator [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF]), to restore strong well posedness under our current Hölder framework. This feature has already been considered in the literature for the system (1.1) in the particular case n " 2, see the works of Chaudru de Raynal [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF], Wang and Zhang [START_REF] Wang | Degenerate SDE with Holder-Dini drift and non-Lipschitz noise coefficient[END_REF], Fedrizzi, Flandoli, Priola and Vovelle [START_REF] Fedrizzi | Regularity of stochastic kinetic equations[END_REF], Zhang [START_REF] Zhang | Stochastic hamiltonian flows with singular coefficients[END_REF]. In any cases, in addition to the weak Hörmander structure, the regularity of the drift with respect to (w.r.t.) the second space argument is required to be of regularity index superior or equal (depending on the work) to 2{3 (critical Hölder index or critical weak differentiation index). As a generalization of these results, we prove in this paper that strong well posedness holds as soon as each drift component F i is β j -Hölder continuous in the j th variable for some β j P ´p2j ´2q{p2j ´1q, 1 ı so that we recover the critical index mentioned above when j " 2.

In comparison with the results obtained in the non-degenerate framework, the increasing value of the critical Hölder index can be understood as the price to pay to balance the degeneracy of the noise. Indeed, there is a strong competition between the irregularity of the drift and the fluctuations of the noise, which has been highlighted in the work of Delarue and Flandoli [START_REF] Delarue | The transition point in the zero noise limit for a 1d Peano example[END_REF]: as the regularity of the drift decreases the fluctuations of the noise have to increase more and more in order to regularize the system.

In our setting, the way the noise is allowed to propagate implies a loss of the fluctua-tions at each level of the chain and therefore, a loss of its regularization property. A good manner to understand this phenomenon is the following case. Given a level i in rr2, nss of the chain (1.1), assume that F 1 " 0, σ " Id, F pt, x ´1, ¨¨¨, x n q " x ´1, P rr2, i ´1ss, F i pt, x i´1 , ¨¨¨, x n q ": x i´1 `F i pt, x i q, and F j " 0, j P rri `1, nss. In such a case, the noise added at the i th level of the chain is only the pi ´1q th iterated time integral of a Brownian motion whose fluctuations are of order t i´1{2 , meaning that the Hölder regularity of Fi has to increase with i. This simple feature illustrates the loss of fluctuations of the noise from level to level through its propagation to the chain and consequently in the corresponding spatial direction. This is the reason why the index of Hölder regularity increases when considering direction further and further away from the source of noise.

Thanks to Delarue and Flandoli's result [START_REF] Delarue | The transition point in the zero noise limit for a 1d Peano example[END_REF], Chaudru de Raynal [START_REF] Chaudru | Weak regularization by stochastic drift: result and counter example[END_REF] and then Chaudru de Raynal and Menozzi [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] have proposed a class of counter-examples to weak well posedness for the chain (1.1) when n " 2 and in full generality, respectively. This class of counter-examples says that weak well posedness does not hold as soon as for any i in rr2, nss the drift F i has an Hölder regularity index with respect to the j th variable, denoted by βj i , below p2i ´3q{p2j ´1q e.g. regarding the regularity of the drift w.r.t. the second space variable we hence obtain β2

2 " 1{3. Also, in the latest work [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF], weak well posedness is shown to hold as soon as the regularity indexes p βj i q 2ďi,jďn satisfy p2i ´3q{p2j ´1q ă βj i ď 1 so that the result is almost sharp † , in the sense that the critical case corresponding to an equality for the thresholds is still open.

In a nutshell, when n " 2 and d " 1, to exhibit the critical Hölder threshold β2 2 (Hölder continuity index of F 2 in its second variable) needed for the weak uniqueness to hold, one can go back to the key example of Peano. The point is to compare the maximal solutions of the ODE with dynamics 9 x t " sgnpx t q|x t | β2 2 starting from 0 (which is precisely the singular point of the ODE), which writes as ˘ct 1{p1´β 2 2 q , and the typical magnitude of the integral of the Brownian motion which is t 3{2 . For the noise to dominate in small time, in order to leave the point where uniqueness fails for the deterministic equation, one must take 3{2 ă 1{p1 ´β 2 2 q ô β2 2 ą 1{3 (see [START_REF] Chaudru | Weak regularization by stochastic drift: result and counter example[END_REF]). Through a perturbative approach, the threshold can also be deduced from the behavior of the ratio |x 2 | β2 2 {t 3{2 where x 2 is a spatial variable which has the same typical order as the integral of the Brownian motion. Namely |x 2 | -t 3{2 . Hence, |x 2 | β2 2 {t 3{2 -t ´3{2p1´β 2 2 q . For weak uniqueness to hold, the point is then to impose that the previous ratio gives an integrable singularity in time. This precisely leads to the condition ´3{2p1 ´β 2 2 q ą ´1 ô β2 2 ą 1{3. The weak uniqueness thresholds for the full chain were derived similarly in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] studying for i P rr2, nss, |x j | βj i {t i´1{2 , j P rri, nss, where x j is a spatial variable which has the same typical order as the pj ´1q th iterated integral of the Brownian motion. Namely |x j | -t j´1{2 . The corresponding ratio thus satisfies |x j | βj i {t i´1{2 -t ´pi´1{2q`β j i pj´1{2q . In order to obtain an integrable singularity one must take ´pi ´1{2q `β j i pj ´1{2q ą ´1 ô βj i ą p2i ´3q{p2j ´1q which is precisely the indicated threshold. Of course, the previous ratios appear through rather lengthy and technical perturbative procedures: the forward parametrix expansion in [START_REF] Chaudru | Weak regularization by stochastic drift: result and counter example[END_REF] and the backward one in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] (which allows to †. the drift F 1 may be assumed to be only in L q ´Lp space for any n 2 d{p `2{q ă 1, p ě 2, q ą 2 or Hölder continuous for any strictly positive Hölder index.

improve the Hölder thresholds on F 1 w.r.t. [START_REF] Chaudru | Weak regularization by stochastic drift: result and counter example[END_REF]).

When comparing the weak well posedness result in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] with the strong well posedness proved in this work, one may wonder why in the strong case the critical Hölder exponent are greater and why they do not take into account the level in the chain (recall that we investigate strong well posedness for drifts F i which are β j -Hölder continuous in the j th variable for some β j independent of i satisfying p2j ´2q{p2j ´1q ă β j ď 1). This in particular gives β 2 ą 2{3 while we obtained β2 2 ą 1{3 for weak well posedness. One reason for these notable differences relies on Partial Differential Equation (PDE) results. Indeed, it is well known that the family of generators pL t q tě0 of (1.1) is a family of linear partial differential operators of second order so that any SDE are connected with a PDE (the solution of the SDE being, in fact, the characteristic of the solution of the Cauchy problem associated with L). For instance, this connexion has been widely used in the last past five decades allowing to pass from regularization properties of PDE to well posedness result for SDE and conversely, see e.g. the book of Bass [START_REF] Bass | Diffusions and Elliptic Operators[END_REF].

Keeping in mind this connexion, one may view the difference of critical Hölder indexes between strong and weak well posedness as a consequence of the different regularization properties required on the underlying PDE to prove the probabilistic result: roughly speaking the weak well posedness requires gradient bounds for the solution of the associated Cauchy problem while the strong well posedness requires in addition to control some of the cross derivatives of the solution. Since the regularization properties of the operator are the same in both cases, this additional control explains the critical Hölder indexes increase for the strong uniqueness to hold.

The reason why the Hölder regularity index is blind to the level of the chain is a bit more involved at this stage of the presentation and we refer to subsections 1.3 and 1.4 below for a more detailed discussion on the subject. Let us only mention that in the weak case, the underlying Cauchy problem is investigated with a global source term f of independent regularity while in the strong case the source term should be any component of the drift of (1.1) itself. This is a consequence of the initial Zvonkin approach. Each of the pF i q iPrr1,nss has to be considered as a source term for the corresponding PDE. W.r.t. to the previous heuristic derivation of the thresholds for the weak uniqueness, let us indicate that the Zvonkin approach to strong uniqueness, coupled to a forward perturbative approach actually leads to consider the ratios |x j | β j {t pj´1{2q`1{2 , j P rr1, nss, i.e. the time singularity is higher of degree 1{2. We recall that, formally, for the weak uniqueness we are led to control the gradients of the Green kernel associated with a degenerate Ornstein-Uhlenbeck like process and an associated perturbation term involving the difference of the initial generator of (1.1) and the one of the Ornstein-Uhlenbeck proxy. For the strong uniqueness the quantities of interest are the derivatives of the gradient w.r.t. the non-degenerate variables for the Green kernel and the perturbative term. The additional singularity in the previous ratio precisely corresponds to the typical scale induced by the derivative of a Gaussian kernel w.r.t. its non-degenerate entry. Considering again the associated typical scale for the variable x j , which is t j´1{2 , we write |x j | β j {t pj´1{2q`1{2 -t ´j`β j pj´1{2q . In order to get an integrable singularity in time, we must have ´j `βj pj ´1{2q ą ´1 ðñ β j ą p2j ´2q{p2j ´1q, which precisely gives the stated thresholds. These features, Zvonkin tranform and perturbative analysis are detailed re-spectively in Section 1.3 and 1.4 below. One could wonder whether the thresholds we find are linked or not to the Zvonkin approach. A partial answer is that other techniques, based on pathwise arguments, developed by Catellier and Gubinelli [START_REF] Catellier | Averaging along irregular curves and regularisation of ODEs[END_REF] to address well-posedness of non-degenerate fractional driven SDEs with singular drift, lead to the same thresholds. In the non-degenerate scalar case, let us also mention mention the works of Gradinaru and Offret [START_REF] Gradinaru | Existence and asymptotic behaviour of some time-inhomogeneous diffusions[END_REF] or Bass and Chen [START_REF] Bass | Brownian motion with singular drift[END_REF] who obtained better thresholds than we do. Anyhow, the one dimensional case must be considered apart.

Let us eventually conclude this part by emphasizing that we were unfortunately not able to produce a counter example for strong uniqueness when the Hölder index of regularity of the drift is below the critical value given above. Recall indeed that weak well posedness holds in our considered case so that, if the critical exponent exhibited is sharp (or almost sharp), this means from the Yamada Watanabe Theorem that strong existence fails as well. It hence seems rather difficult to produce a counter example based on the irregularity of the drift in such a case and we are not aware of any result of this type in the literature. Usual counter examples, based on measurability arguments, typically rely on the regularity of the diffusion matrix, as it is the case for the so called Tanaka's example [START_REF] Tanaka | Stochastic differential equations with reflecting boundary condition in convex regions[END_REF] (see also the counter examples in the book of Cherny [START_REF] Cherny | Singular Stochastic Differential Equations[END_REF]). Nevertheless, as we will discussed in subsection 1.4, these thresholds appear to be almost sharp for the method implemented in this work to prove the strong well posedness result.

Notations, assumptions and main result

Some notations. We will denote by a bold letter x, y any element of R nd , writing as well x " px 1 , ¨¨¨, x n q where for i P rr2, nss, x i P R d . For practical purpose we will be led in our analysis to consider subcomponents of a vector x P R nd . Namely, for any 0 ď i ď j ď n and x P R nd , we introduce the notation x i:j :" px i , ¨¨¨, x j q. Accordingly, we write the drift as the mapping ps, xq P R `ˆR nd Þ Ñ Fps, xq " pF 1 ps, xq, ¨¨¨, F n ps, xqq " pF 1 ps, xq, F 2 ps, xq, F 3 ps, x 2:n q ¨¨¨, F n ps, x n´1:n qq, from the specific structure of the drift appearing in (1.1).

For f P C 1 pR nd , R k q, k P t1, du, we denote for all i P rr1, nss, by D x i f pxq the Jacobian matrix of the derivative of f w.r.t. to its R d -valued variable x i . As shortened form, and when no ambiguity is possible, we also write for all x, y P R nd , D x i f pxq " D i f pxq and D y i f pyq " D i f pyq. Also, if k " 1 we denote by Df pxq " pD 1 f pxq ¨¨¨D n f pxqq ˚the full gradient of the function f at point x.

Let f : R nd Ñ R k and β :" pβ 1 , ¨¨¨, β n q P p0, 1s n be a multi-index. We say that f is uniformly β-Hölder continuous if for all j P rr1, nss rf j s β j :" sup pz 1:j´1 ,z j`1:n qPR pn´1qd , z‰z 1 ,pz,z 1 qPpR d q 2

|f ps, z 1:j´1 , z, z j`1:n q´f ps, z 1:j´1 , z 1 , z j`1:n q| |z ´z1 | β j ă `8.

(1.2) For a smooth function Ψ : r0, T s ˆRnd Ñ R nd , where T ą 0 is a fixed given time, writing for pt, xq P r0, T s ˆRnd , Ψpt, xq " `Ψ1

Ψ i qpt, xq P R nd b R d b R d .
Assumptions We will assume throughout the paper that the following conditions hold.

(ML) The coefficients F and σ are measurable in time. Also, the diffusion coefficient σ is uniformly Lipschitz continuous in space, uniformly in time, i.e. there exists κ ą 0 s.t. for all t ě 0, px, x 1 q P pR nd q 2 : |σpt, xq ´σpt, x 1 q| ď κ|x ´x1 |.

(UE) The diffusion matrix a :" σσ ˚is uniformly elliptic and bounded, uniformly in time, i.e. there exists Λ ě 1 s.t. for all t ě 0, px, ζq P R nd ˆRd :

Λ ´1|ζ | 2 ď xapt, xqζ, ζy ď Λ|ζ| 2 .
(T β ) For all j P rr1, nss, the functions pF i q iPrr1,pj`1q^nss are uniformly β j -Hölder continuous in the j th spatial variable with β j P p 2j´2 2j´1 , 1s, uniformly w.r.t. the other spatial variables of F i and in time. In particular, there exists a finite constant C β s.t. max pi,jqPrr1,nss 2 sup sPr0,T s rpF i q j ps, ¨qs β j ď C β .

(H η ) For all i P rr2, nss, D x i´1 F i is non-degenerate (weak type of Hörmander condition) and η -Hölder continuous w.r.t. x i´1 uniformly in x i:n and time. We also assume without loss of generality that η P ´0, inf jPrr2,nss β j ´2j´2 2j´1 ( ¯, i.e. η is meant to be small.

From now on, we will say that assumption pAq is in force provided that (ML), (UE), (T β ), (H η ) hold.

Main result. The main result of this work is the following theorem.

Theorem 1 (Strong uniqueness for the degenerate system (1.1)) Under (A) there exists a unique strong solution to system (1.1).

Remark 1 Still in comparison with the results obtained in the non-degenerate cases, and especially the one of Krylov and Röckner [KR05], we do not tackle the case of drift in L q ´Lp w.r.t. the first (and then non-degenerate) variable. This is only to keep our result as clear as possible and to concentrate on the novelty of the approach we use here. We are anyhow confident that these specific drifts could be handled. Indeed, all the intermediate results needed to perform the analysis in that setting seem to be already available. We refer to subsection 1.4 for further details.

1.3 Proof of the main result: Zvonkin Transform and smoothing properties of the PDE associated with (1.1)

We emphasize that under our assumptions, it follows from [CdRM17] that (1.1) is well posed in the weak sense. Hence, from Yamada-Watanabe theorem it is sufficient to prove that strong (or pathwise) uniqueness holds to prove strong well posedness. To do so, our main strategy rests upon the Zvonkin transform initiated by Zvonkin in [START_REF] Zvonkin | A transformation of the phase space of a diffusion process that will remove the drift[END_REF] which has been widely used during the last decade to prove strong well posedness, see e.g., and heavily relies on the connexion between SDE and PDE as we already emphasized.

Eventually, let us mention [START_REF] Veretennikov | Stochastic equations with diffusion that degenerates with respect to part of the variables[END_REF], where the author extended the result in [START_REF] Veretennikov | Strong solutions and explicit formulas for solutions of stochastic integral equations[END_REF] to the considered chain for n " 2. Note however that this extension only concerns the nondegenerate part, the coefficients being supposed to be twice continuously differentiable functions with bounded derivatives w.r. where a " σσ ˚denotes the generator associated with (1.1). We then formally associate the SDE (1.1) with the following systems of PDEs:

# pB t u i `Lt u i qpt, xq " F i pt, xq, pt, xq P r0, T q ˆRnd , u i pT, xq " 0 d , i P rr1, nss, (1.5)
and we denote by U " pu 1 , ¨¨¨, u n q its global solution. Let now pF m q mě0 , pa m q mě0 denote two sequences of mollified coefficients satisfying assumption (A) uniformly in m that are infinitely differentiable functions with bounded derivatives of all, order greater than 1 for F m , and converging in supremum norm to pF, aq (such sequences are easily obtained from [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF]). Then, for each m, the regularized systems of PDEs associated with (1.5) write:

# pB t u m i `Lm t u m i qpt, xq " F m i pt, xq, pt, xq P r0, T q ˆRnd , u m i pT, xq " 0 d , i P rr1, nss, (1.6)
where L m t is obtained from (1.4) replacing F by F m and a by a m . The above system (1.6) is well posed and admits a unique smooth solution U m " pu m 1 , ¨¨¨, u n m q. Hence, applying Itô's Formula, one easily deduces that Then, the main idea consists in taking advantage of the regularization properties of the operator L m (uniformly in m) and expect that the solutions U m , m ě 0 will be smoother than the source term F so that the right hand side of (1.7) is smoother than the integrand of the left hand side of the considered equation. In other words, we are looking for a good regularization theory for the PDE (1.6) uniformly on the mollification argument. This good regularization theory is summarized in the following crucial result whose proof is, in fact, the main subject of this work and is postponed to Section 2.

Theorem 2 For T ą 0 small enough ‡ , there exists a constant C T :" C T ppAqq ą 0 satisfying C T Ñ 0 when T Ñ 0 such that for every m ě 0, the solution U m satisfies with the notation of (1.3):

}DU m } 8 `}DpD 1 U m q} 8 ď C T . (1.8)
We also point out that, from the uniformity in m in the previous theorem, we could also derive some regularizing properties for the system (1.5) through appropriate compactness arguments.

Let now X and X 1 be two solutions of (1.1). Using the representation (1.7) to express the difference of the bad drift in terms of the function U m and its derivative, we write:

X t ´X1
t " U m pt, X t q ´Um pt, X 1 t q ´ż t 0 rDU m ps, X s qBσps, X s q ´DU m ps, X 1 s qBσps, X 1 s qs dW s `"R m t pXq ´Rm t pX 1 q ı `ż t 0 B rσps, X s q ´σps, X 1 s qs dW s .

Take then the supremum in time of the square of the difference. Passing to the expectation, a convexity inequality then leads to the following estimate:

E " sup tďT |X t ´X1 t | 2  ď 5 ´E " sup tďT |U m pt, X t q ´Um pt, X 1 t q| 2  `E "ż T 0 |rDU m Bs ps, X s q ´rDU m Bs ps, X 1 s q| 2 }σ} 2 8 ds  `E "ż T 0 p}DU m B} 8 `1q |rσps, X s q ´σps, X 1 s qs| 2 ds  `2||R m ¨p¨q|| 2 8
¯. ‡. By "small enough" we mean that there exists a time T ą 0 depending on known parameters in (A) s.t. for all T ď T the statement of the theorem holds.

it consists in a perturbation argument of the operator L m which is expanded around a good proxy, usually denoted by Lm (we keep here the super-scripts in m to emphasize that the perturbative technique we perform will concern the system (1.6) with mollified coefficients). The terminology of good in our setting relies to the fact that the operator Lm is the generator of the "closest" Gaussian approximation Xm of X m which has generator L m . In our case, such a process is well known and is the linearized (with respect to the source of noise) version of (1.1) whose coefficients are frozen along the curve pθ m s,t q sPrt,T s that solves the deterministic counterpart of (1.1) with mollified coefficients (i.e. with σ m " 0 d,d ) namely, 9 θ m s,t " F m ps, θ m s,t q. This process may be seen as a (non-linear) generalization of the so-called Kolmogorov example [START_REF] Kolmogorov | Zufällige Bewegungen (zur Theorie der Brownschen Bewegung)[END_REF] and we refer the reader to the work of Delarue and Menozzi [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF] and Menozzi [START_REF] Menozzi | Parametrix techniques and martingale problems for some degenerate Kolmogorov equations[END_REF] for more explanations. Having this proxy at hand, the parametrix procedure consists in deriving the desired estimates for the proxy and control the expansion error.

In [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF], Chaudru de Raynal and Menozzi successfully used this approach in its backward form to prove weak well posedness of (1.1) under less restrictive assumptions (the critical thresholds for the Hölder exponents being smaller as indicated above). In that case, the curve along which the system is frozen for the proxy is the solution of the backward deterministic counterpart of (1.1)). This backward approach is very suitable when investigating the martingale problem associated with our main system since it allows to control subtly the expansion error associating precisely the coefficients F i with their corresponding differentiation operator D i and does not require any mollification of the coefficients. Unfortunately, when trying to obtain estimates on the derivatives of the solutions of the PDE, the backward approach is not appropriate since the corresponding proxy does not provide an exact density and this fact does not allow to benefit from cancellation techniques which are very helpful in this context (see paragraph below).

Hence, our parametrix approach will be of forward ¶ form as done in the work of Chaudru de Raynal [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF]. This is, in fact, a non-trivial generalization of the approach developed in the aforementioned paper where the strong well posedness of (1.1) is obtained when n " 2. Indeed, the strategy used in [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF] is not adapted to this general case because of some subtle phenomena appearing only when n ě 3. In particular, the singularities appearing when considering the remainder term of the parametrix were in [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF] equilibrated at hand through elementary cancellation arguments, whereas the current approach takes advantage of the full-force duality results between Besov spaces (see Sections 1.4 and 2.2 below). This forward perturbative approach has also been successfully used in Chapter 6 to establish some weaker regularization properties of the PDE (1.5) through appropriate Schauder estimates.

Regularizing properties of the degenerate Ornstein-Uhlenbeck proxy

When exploiting such a forward parametrix approach, a good primer to understand what could be, at best, expected, consists in investigating the regularization property of the proxy operator L. To be as succinct as possible, let us consider the case where p Lt q tě0 ¶. Meaning that the freezing curve θ solves the corresponding ODE associated with (1.1) in a forward form.

is the generator of a degenerate Ornstein-Uhlenbeck process p Xt q tě0 with dynamics: where, up to a modification of the constants involved, ş R nd dyp C ´1 pt, s, x, yq " 1. Similarly, the derivatives of p will be bounded by a density of the form pC ´1 up to an additional multiplicative contribution reflecting the time-singularities associated to the derivation index. Precisely, for any multi-index α " pα 1 , ¨¨¨, α n q P N n , there exists C α s.t.

|D α

x ppt, s, x, yq| ď

C α ps ´tq n 2 d 2 `řn i"1 α i pi´1 2 q
exp ´´C ´1ps ´tq|T ´1 s´t p Rs,t x ´yq| 2 ď Cα ps ´tq ř n i"1 α i pi´1 2 q pC ´1 pt, s, x, yq.

(1.18)

We refer to the proof of Proposition 2 for a complete version of this statement.

To prove estimate (1.8) of Theorem 2 for the current system (1.12), it follows from the specific structure of the matrix B that we have to estimate for any l P rr1, nss the quantities D x l D r

x 1 ũi pt, xq, r P t0, 1u. From (1.18), we thus have

|D x l D r x 1 ũi pt, xq| ď C ż T t ds ż R nd dy|F i ps, yq|ps ´tq ´pl´1{2q´r{2 pC ´1 pt, s, x, yq. (1.19)
We now face two problems: first the F i are unbounded, second the above time singularity is, as is, not integrable. Let us consider the worst case i.e. when r " 1. To smoothen the time singularity, the main idea consists in using the regularity of the source term F i by exploiting precisely the fact that, once integrated through the variables y l to y n , the transition density p does not depend on the variable x l anymore. This is due to the structure of A in (1.11), which in particular yields that the resolvent p Rs,t q 0ďtďsďT is lower triangular. It is hence equal to 0 when we differentiate it w.r. dy l:n ppt, s, x, yq " 0.

When using this property, we obtain that

|D x 1 D x l ũi pt, xq| " ˇˇˇż T t ds ż R nd
dy `Fi ps, yq ´Fi pt, y 1 , ¨¨¨, y l´1 , θ l s,t pxq, ¨¨¨, θ n s,t pxqq ˘Dx 1 D x l ppt, s, x, yq ˇˇˇ.

We thus obtain from (1.18):

|D x 1 D x l ũi pt, xq| ď C ż T t ds ż R nd
dy ˇˇF i ps, yq ´Fi pt, y 1 , ¨¨¨, y l´1 , θ l s,t pxq, ¨¨¨, θ n s,t pxqq ˇps ´tq ´pl´1{2q´1{2 pC ´1 pt, s, x, yq.

Then, using the regularity assumed of F i , which satisfies (T β ), we get that for some constant C (which possibly change from line to line) ps ´tq ´pl´1{2q´1{2`β j pj´1{2q pC ´1 pt, s, x, yq, which is integrable only if for all j P rrl, nss, ´pl ´1{2q ´1{2 `βj pj ´1{2q ą ´1 ðñ β j ą `p2l´2q{p2j´1q ˘. This condition actually holds if for any i P rr1, nss, β i ą `p2i´2q{p2i´1q which is exactly the infimum assumed in (T β ). As we can see, there is no hope to obtain better thresholds with such a strategy. This is the reason why we said that these thresholds are almost sharp for the approach used here.

|D x 1 D x l ũi pt, xq| ď C ż T t ds ż R nd

Back to the perturbative analysis

Let us now briefly explain what happens when one wants to control the approximation error in the forward parametrix expansion. Coming back to our general setting and denoting by pm the transition density of our proxy, we obtain from the first order parametrix expansion the following representation for each regularized component u m i , i P rr1, nss of our solution U m : for all pt, xq P r0, T s ˆRnd

u m i pt, xq " ż T t ds ż R nd dy ! ´Fm i ps, yq `pL m s ´L m s qu m i ps, yq
) pm pt, s, x, yq.

(1.20)

Above there is an additional term in the right hand side, in comparison with (1.13), which is precisely the approximation error due to the parametrix expansion. It thus appears that the solution has an implicit representation which makes its derivatives themselves appear. Hence, when differentiating the above representation to derive the estimate (1.8) in Theorem 2, we obtain bounds that depend themselves on the derivatives of the solution.

We then have to estimate each derivative appearing in the right hand side and use a circular argument. Namely, when differentiating u m i pt, xq, we will obtain the required estimate provided the multiplicative constants associated with the terms }Du i m } 8 and }D 1 Du i m } 8 , that will appear in the corresponding upper-bound for the above right hand side, are small enough (see also Section 2 of [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF] and Section 2.2 below for details).

Moreover, as we have already seen, in order to smoothen the time singularity appearing when we apply a cross differentiation operator in the l th and 1 st direction to the term ş T t ds ş R nd dypL m s ´L m s qu m i ps, yqp m pt, s, x, yq corresponding to the approximation error, we will have to center this term around the derivatives of the solution itself (in the sense given in the above discussion). This procedure allows us, thanks to Taylor expansions, to weaken the singularities and provides integrable (in time) terms. The dramatic point is that, when doing so, our bound involves the cross derivatives D Du m i , P rr1, nss whose control in supremum norm is, as suggested by the discussion done in the explicit case of a simple degenerate Ornstein-Uhlenbeck process, definitely out of reach. In fact, as told by the results in [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF], the only thing we could hope is that the gradient in the degenerate directions viewed as a function of the degenerate variables, i.e. D 2:n u m i pt, x 1 , ¨q :" `D2 u m i pt, x 1 , ¨q, ¨¨¨, D n u m i pt, x 1 , ¨q˘f or any pt, x 1 q P r0, T sˆR d , belongs to an appropriate anisotropic Hölder space with regularity indexes strictly less than 1. Such spaces can as well be viewed as particular cases of anisotropic Besov spaces with corresponding positive regularity indexes. Thus, the generalized derivative of D 2:n u m i pt, x 1 , ¨q should belong to some anisotropic Besov space of negative regularity indexes, strictly bigger than ´1.

Here is the main novelty of our approach: to tackle this problem, our main idea, in order to balance the lack of differentiation property of the full gradient, consists in putting precisely in duality the anisotropic Besov norm with negative exponent of `D D 2:n u m i pt, x 1 , ¨q˘ Prr2,nss , with the corresponding anisotropic Besov norm with positive exponent of the remaining terms coming from the differentiation of (1.20), which in particular involve the coefficients of the operator L m ´L m and contain the time singularities coming from the derivatives of the frozen Gaussian kernel pm . To the best of our knowledge, this approach is quite new in this parametrix setting and it appears to be very robust. We refer to the proof of the main Theorem 2 in Section 2 for details and to Proposition 3.6 in the book of Lemarié-Rieusset [START_REF] Lemarie-Rieusset | Recent developments in the Navier-Stokes problem[END_REF] for duality results on Besov spaces.

We are thus led to control on the one hand the Besov norm with negative exponent of the cross derivatives of the solution, see Lemma 4, and on the other hand the Besov norm . In other words, the regularity index depends on the considered variable with positive exponent of the remaining terms in (1.20) (involving the coefficients of the operator L m ´L m ), see Lemma 3. The first control (Besov norm with negative exponent) is crucial and appears to be quite delicate. Indeed, due to the implicit representation (1.20), this estimate also involves supremum norms of the full gradient Du m i and of the cross derivatives themselves. This again reflects the circular nature of the arguments needed to derive the result.

To conclude this discussion on Besov duality, let us mention that a similar strategy has been implemented in Chapter 6 in order to derive sharp Schauder estimates for the PDE (1.5) (with possibly non-trivial final condition). Therein, since we were interested in controlling the Hölder norm of the solution, the duality was anyhow used the other way round: positive regularity indexes for the solution and negative ones for the remaining terms of the perturbative expansion.

Let us close this discussion coming back to Remark 1. As we emphasized, in comparison with the non-degenerate result, Theorem 1 should hold assuming that the drift F 1 belongs to a suitable L q ´Lp space w.r.t. time and the non-degenerate variable x 1 . We are convinced that this is the case but we deliberately decide not to tackle this setting in order to keep this work shorter and more coherent. Indeed, in this case, the difficulty comes from the estimate on the second order derivative in the non-degenerate direction of the first component of the solution U m , namely D x 1 D x 1 u m 1 (which is a part of the main estimate (1.8) in Theorem 2). The point is to establish for this quantity an L q ´Lp control. This cannot be derived from the previously described approach and requires harmonic analysis techniques (see also [START_REF] Krylov | Strong solutions of stochastic equations with singular time dependent drift[END_REF]). The main problem to establish the estimate is mainly due to the source term, which is actually F 1 . To prove it, the main idea consists in exploiting the results of Menozzi [START_REF] Menozzi | Martingale problems for some degenerate Kolmogorov equations[END_REF] (where such an estimate is proved under the assumption that the drift is Lipschitz) through the tools developed in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF] (backward parametrix approach for drift F whose first component may be in L q ´Lp and the other ones in Hölder spaces). Then, the Zvonkin Transform should also be tuned a little bit following the strategy developed by Veretennikov (see e.g. [START_REF] Veretennikov | Strong solutions and explicit formulas for solutions of stochastic integral equations[END_REF] and [START_REF] Fedrizzi | Regularity of stochastic kinetic equations[END_REF]). Such a program would surely toughen our paper without adding any surprising result and we prefer to focus on the novelty of the approach based on duality results for Besov spaces and the generalization of the strong uniqueness result to the whole chain (i.e. to any arbitrary n ě 1) rather than drowning the reader into additional technical considerations.

2 Perturbation techniques for the PDE : proof of Theorem 2

In order to keep the notations as clear as possible, we forget the superscript m standing for the mollifying procedure and we suppose that the following assumptions hold:

Assumption (AM). We say that assumptions (AM) hold if the assumptions gathered in (A) hold true and the coefficients F, a are infinitely differentiable functions with bounded derivatives of all order for a and greater than 1 for the coefficient F.

In the whole section, we consider a fixed final time T ą 0 which is meant to be small, i.e. T ! 1. Let us consider for this section a generic PDE with generator corresponding to (1.4) and scalar source f having the same Hölder regularity than the drift terms in (1.1) (i.e. the scalar function f below can be any of the entries of the R d -valued pF i q iPrr1,nss in the dynamics (1.1)). Namely, we concentrate on # pB t u `Lt uqpt, xq " ´f pt, xq, pt, xq P r0, T q ˆRnd , upT, xq " 0, (2.1) where pL t q tě0 is defined in (1.4) and stands for the generator associated with (1.1) when the coefficients are smooth.

The key result to prove strong uniqueness for the SDE (1.1) is actually the following theorem from which one easily derive Theorem 2 for each component of the solution of the systems (1.5) and then the result in full generality.

Theorem 3 (Pointwise bounds for the derivatives of the PDE (2.1)) There exists γ :" γppAqq ą 0 and C :" CppAqq ą 0 s.t.

}Du} 8 `}DpD 1 uq} 8 ď CT γ , (2.2)
with obvious extension of the definition in (1.3) to the current scalar case.

The proof of Theorem 3 is performed in Section 2.2 through the forward parametrix approach consisting in considering a suitable proxy semi-group around which the initial solution of (2.1) can be expanded. To this end we first investigate in Section 2.1 below the linearized Gaussian process deriving from the dynamics in (1.1) which will provide the suitable model for the parametrix.

Gaussian proxy and associated controls

Linearization of the dynamics

Fix some freezing points pτ, ξq P r0, T s ˆRnd . For fixed initial conditions pt, xq P r0, T s ˆRnd , a natural linearization associated with the mollified version of (1.1) writes

d Xpτ,ξq v " rFpv, θ v,τ pξqq `DFpv, θ v,τ pξqqp Xpτ,ξq v ´θv,τ pξqqsdv `Bσpv, θ v,τ pξqqdW v , @v P rt, ss, Xpτ,ξq t " x, (2.3)
where 9 θ v,τ pξq " Fpv, θ v,τ pξqq, v P r0, T s, θ τ,τ pξq " ξ, (2.4)

and DFpv, ¨q denotes the subdiagonal of the Jacobian matrix DFpv, ¨q. Namely, for z P R nd : as in Proposition 1. Also, there exists C :" CppAq, T q ą 0 s.t. for all multiindex α " pα 1 , ¨¨¨, α n q P N n , |α| ď 3 and denoting by D α

DFpv, zq " ¨0d,d ¨¨¨¨¨¨¨¨¨0 d,d D z 1 F 2 pv, zq 0 d,d ¨¨¨¨¨¨0 d,d 0 d,d D z 2 F 3 pv, z 2:n q 0 d,d 0
x :" D α 1 x 1 ¨¨¨D αn xn , we have: From now on, we will write with a slight abuse of notation pξ pt, s, x, yq :" ppt,ξq pt, s, x, yq, i.e. we omit the freezing parameter in time when it corresponds to the considered starting time. One can derive from Proposition 2 the following important regularization result.

|D α x ppτ,ξq pt, s, x, yq| ď C ps ´tq ř n i"1 α i pi´1 2 q`n 2
Lemma 1 (Regularization effects for the inhomogeneous semi-group) Let 0 ă γ ď 1. There exists C :" CppAqq such that for all P t0, 1u, l P rr1, nss, k P rr1, nss, x P R nd :

|D x 1 D x l P ξ s,t ´ˇ`¨´m pt,ξq s,t pxq ˘kˇˇγ ¯pxq|| ξ"x ď Cps ´tq ´ 2 ´pl´1 2 q`γpk´1 2 q .

(2.16)

Let f be a β-Hölder continuous functions where β :" pβ 1 , ¨¨¨, β n q P p0, 1s n is a multiindex and for i P rr1, nss, β i stands for the Hölder regularity of f in the variable x i . The following result then holds.

-There exists C :" CppAqq s.t. for all P t0, 1u, l P rr1, nss, x P R nd :

|D x 1 D x l P ξ s,t ´f p¨q ´f pm pt,ξq s,t pxqq ¯pxq|| ξ"x ď C n ÿ j"1
rf j ps, ¨qs β j ps ´tq ´ 2 ´pl´1 2 q`β j pj´1 2 q , (2.17)

where rf j ps, ¨qs β j :" sup z‰z 1 PR d ,pz 1:j´1 ,z j`1:n qPR pn´1qd

|f pz 1:j´1 ,z 1 ,z 1:j`1 q´f pz 1:j´1 ,z,z 1:j`1 q| |z´z 1 | β j stands for the Hölder continuity modulus of order β j of f in its j th variable.

-Centering arguments. For all l P rr1, nss, k ď l, it holds that: D x l P ξ s,t ´f p¨q ¯pxq " D x l P ξ s,t ´f p¨q ´f p¨1 :k´1 , θ k:n s,t pξqq ¯pxq.

(2.18)

-As particular cases of the previous items, we have that there exists C :" CppAqq s.t. for all l P rr1, nss 2 , x P R nd :

|D x 1 D x l P ξ s,t f pxq|| ξ"x ď C n ÿ j"l
rf j ps, ¨qs β j ps ´tq ´l`β j pj´1 2 q , (2.19)

|D x l P ξ s,t f pxq|| ξ"x ď C n ÿ j"l
rf j ps, ¨qs β j ps ´tq ´pl´1 2 q`β j pj´1 2 q .

Proof: Let us first mention that identities (2.16) and (2.17) are direct consequences of Proposition 2. Centering arguments like (2.18) will be a crucial tool in the analysis below. To justify such an identity, write: rf j ps, ¨qs β j ps ´tq ´l`β j pj´1 2 q .

D x l P ξ s,
The control for |D x l P ξ s,t f pxq| ˇˇξ"x is derived similarly. We state in the lemma below a useful control to obtain through Lemma 1 some smoothing effects for the degenerate part of the operator. The statement readily follow from (T β ).

Lemma 2 From the smoothness assumption on the drift coefficient in (T β ), there exists C :" CppAqq s.t. for all P rr2, nss, k ě , and for all ps, x, ξq P r0, T s ˆpR nd q 2 : ˇˇˇ´F ps, y 1:k´1 , θ k:n s,t pξqq ´F ps, θ s,t pξqq ´Dx ´1 F ps, θ s,t pξqq `y ´θs,t pξq

˘ ´1¯ď C ! k´1 ÿ j"
rpF q j ps, ¨qs β j |py ´θs,t pxqq j | β j ( `rpD x ´1 F q ´1ps, ¨qqs η |py ´θs,t pxqq ´1| 1`η

) .

Control of the sensitivities: proof of Theorem 3

To prove Theorem 3, the idea is to expand the solution of the PDE with regularized coefficients around a suitable proxy, as explained in Section 1.4. The proxy used here is the Gaussian process introduced in Section 2.1 for a suitable freezing parameter ξ to be specified later on. Then, the Duhamel formula (or first order parametrix expansion) yields: for any ξ in R nd .

To establish (2.2) we need to differentiate the above expression w.r.t. px l q lPrr1,nss and then w.r.t. x 1 and px l q lPrr1,nss in order to obtain estimates depending only on known parameters in (A). Differentiating first this expression w.r.t. x l , l P rr1, nss we obtain: The term H ξ l ps, xq gathers all the derivatives of the solution w.r.t. the non-degenerate variables whereas I ξ l ps, xq precisely gathers the derivatives w.r.t. the degenerate ones. We will now start from the representation (2.21) which we will again differentiate w.r.t. the non-degenerate variable x 1 in order to prove the estimates of Theorem 3 concerning the second order derivatives which are the trickiest ones. Indeed, as it has been succinctly explained in Section 1.4, when differentiating the kernel associated with the frozen semigroup defined by (2.15) we generate an a priori not integrable time singularity which then needs to be smoothen by using, among others, tools developed in Lemma 1 (centering or cancellation arguments). The worst case then corresponds to the higher order of differentiation, namely D x l `Dx 1 upt, xq ˘which, as suggested by Proposition 2, generates a time singularity of order 1{2 `pl ´1{2q in the time integrand of the r.h.s. of (2.21). We then only concentrate on this term and omit the proof of the statement concerning the boundedness of the gradient D x l upt, xq which could be shown more directly.

D x l upt, xq " ż T t D x l " P ξ s,t f
The proof will be divided into two parts: we first handle the non-degenerate part of the operator (i.e. the estimate for D x 1 H l ps, xq) and then the degenerate part (i.e. the estimate for D x 1 I l ps, xq) which is a bit more involved.

Finally, before entering into the proof, we introduce some notations to ease the reading. We point out that the contributions H ξ l , I ξ l do depend on the freezing variable ξ. We omit this dependence for notational convenience when, as default ξ coincides with the spatial argument x of the term. We write in this case H ξ l ps, xq| ξ"x ": H l ps, xq (resp. I ξ l ps, xq| ξ"x ": I l ps, xq). The freezing parameter explicitly appears when a more careful choice is needed (see Section 3.2). Accordingly with the definition in (1.2), we will also use, with a slight abuse of notation recalling from (1.1) that the j th variable appears with Hölder regularity β j up to the level j of the chain, the notation rpF 1:j q j ps, ¨qs β j :" max iPrr1,jss rpF i q j ps, ¨qs β j .

( For all l P rr1, nss, one readily derives from Lemma 1 (centering argument from the variables l to n) that for the source term:

|D x 1 H l,1 ps, xq| :" ˇˇD x 1 D x l " P ξ s,t f ps, ¨qı pxq ˇˇˇˇˇˇξ "x ď C n ÿ j"l
rf j ps, ¨qs β j ps ´tq ´l`β j pj´1{2q .

(2.23) Those terms are integrable in time as soon as

β j pj ´1 2 q ´l ą ´1, j P rrl, nss ð β j P ˆ2j ´2 2j ´1 ; 1  . (2.24)
These thresholds are precisely those appearing in assumption (T β ) and which lead to strong uniqueness for the associated SDE. Through our perturbative approach we will actually show that they are also precisely those leading to Theorem .

Thanks to Lemma 1, this again gives

|D x 1 H l,2 ps, xq| ď C ˆn ÿ j"l
}D 1 u} 8 rpF 1 q j ps, ¨qs β j ps ´tq ´l`β j pj´1 2 q (2.25)

`n ÿ j"l l´1 ÿ k"1
}D 1 D j u} 8 rpF 1 q k ps, ¨qs β k ps ´tq ´l`β k pk´1 2 q`pj´1 2 q ˙, leading precisely to the same integrability thresholds of equation (2.24) and assumption (T β ) (as for the source term). The idea behind this control is crucial. We first handle, with the sole Hölder properties of drift and the supremum norm of D 1 u, the variables which are at a good smoothing scale w.r.t. the induced singularity. For the remaining term, which exhibits for the drift non-sufficient smoothing effects, we then additionally exploit a cancellation argument involving the gradient of the solution itself, which consequently makes the cross derivatives appear. Eventually, we get for the diffusive part: , where y 1 " py 1 1 , ¨¨¨, y d 1 q, B j y 1 denotes the derivative w.r.t. to the j th scalar entry of the non-degenerate variable y 1 and a j¨d enotes the j th row of the diffusion matrix a. We therefore derive from Lemma 1 and the smoothness of a (by the Rademacher theorem, a is differentiable almost everywhere): .

|D x 1 H l,
|D x 1 H l,
Hence, Lemmas 1 and 2 yield:

|D x 1 I 1,l ps, xq| (2.31) ď C||D l`1:n u|| 8 n ÿ j"l`1
rpD x j´1 F j q j´1 s η ps´tq ´l`p1`ηqpj´3{2q `rpF l`1:n q j s β j ps´tq ´l`β j pj´1{2q ( , where D l`1:n u :" `Dl`1 u, ¨¨¨, D n u ˘using as well the notation of (2.22) for the last inequality. Observe that the most singular terms in the previous bounds are those associated with the exponents ´l `βj pj ´1{2q. It turns out that they are indeed integrable under assumption (T β ) giving again precisely the appropriate thresholds. On the other hand, Lemma 1 also yields

|D x 1 I 2,l ps, xq| " ˇˇˇˇD x l D x 1 « P ξ s,t ˜l ÿ i"2
@ F i ps, ¨q ´Fi ps, ¨1:l´1 , θ l:n s,t pξqq, D i ups, ¨qD ¸ff pxq ˇˇˇˇˇˇˇˇξ "x ď C||D i:l´1 u|| 8 n ÿ j"l rpF 1:l q j ps, ¨qs β j ps ´tq ´l`β j pj´1{2q , (2.32) and those terms are again integrable as soon as the thresholds of (T β ) hold.

It hence remains to control the terms in D x 1 I 3,l . These terms are the tricky ones since they are, a priori, not designed to smoothen the time singularities generated by the cross differentiation. Observe indeed that, if one tries to reproduce the above calculations, we obtain from Lemma 1 that

|D x 1 I 3,l ps, xq| " ˇˇˇˇD x l D x 1 « P ξ s,t ˆl ÿ i"2
@ pF i ps, ¨1:l´1 , θ l:n s,t pξqq ´Fi pθ s,t pξqq ´Di´1 F i ps, θ s,t pξqqp¨´θ s,t pξqq i´1 q, D i ups, ¨qD ˙ffpxq ˇˇˇˇˇˇˇˇξ "x ď C||D 2:l u|| 8 ps ´tq ´l ˆ´l ÿ j"2 rpF 2:j q j ps, ¨qs β j ps ´tq pj´1 2 qβ j `rpD j´1 F j q j´1 ps, ¨qs η ps ´tq p1`ηqpj´3 2 q ¯, ď C||D 2:l u|| 8 ps ´tq

´l´p s ´tq 3 2 β 2 `ps ´tq 1 2 p1`ηq ¯,
up to a modification of C and for T small enough. This leads, as soon as l ě 2, to a time singularity which is not integrable. Indeed, 1 2 p1 `ηq ă 1 (recall that η is meant to be small). To overcome this problem, the idea consists in writing, thanks to Lemma 1,

|D x 1 I 3,l ps, xq| " ˇˇˇD x l D x 1 " P ξ s,t ´l ÿ i"2
@ pF i ps, ¨1:l´1 , θ l:n s,t pξqq´F i ps, θ s,t pξqq´D i´1 F i ps, θ s,t pξqq `¨´θ s,t pξq ˘i´1 q, `Di ups, ¨q ´Di ups, ¨1:l´1 , θ l:n s,t pξqq ˘D¯ı pxq ˇˇˇˇˇˇˇˇξ "x

(2.33) and to take advantage of the additional smoothing effect from the solution of the regularized PDE itself through the above contribution D i ups, ¨q ´Di ups, ¨1:l´1 , θ l:n s,t pξqq, i P rr2, lss.

To do so we write, by expanding the gradients in (2.33) with the Taylor formula, for any l P rr2, nss. The above expansion hence allows to obtain the additional contribution p¨´θ s,t pξqq k which, thanks to Lemma 1, is precisely designed to smoothen the time singularity coming from the cross differentiation over the variables x l and x 1 of the semigroup (notice indeed that k ě l). Namely, introducing now for all i P rr2, lss, k P rrl, nss, py 1:i´1 , y i`1:n q P R pn´1qd , pt, xq P r0, T s ˆRnd : Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k

|D x 1 I 3,
:

y i P R d Þ Ñ Ψ
ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k py i q " " D x l D x 1 pξ pt, s, x, yq b ´pF i py 1:l´1 , θ l:n s,t pξqq ´Fi pθ s,t pξqq ´Di´1 F i pθ s,t pξqq `y ´θs,t pξq ˘i´1 q ¯pp¨´θ s,t pξqq k q ˚, (2.35)

where the subscript p1, lq in Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k is here to indicate the differentiation w.r.t. D x l D x 1 acting on the frozen density (we will also use for the analysis the notation Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,l,k which indicates that the sole derivation w.r.t. x l has to be taken into consideration, see Section 3.2 below). Pay attention that the above function is pR d q b4valued.

We can expect a time smoothing effect for Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k of order ps´tq ´1 2 `pi´1 2 qβ i ps ´tq ´1 2 `ppi´1q´1 2 qp1`ηq , where the exponent ´1 2 derives from the derivation D x 1 . This fact can be easily deduced from Lemma 1 and the previous computations when ξ " x. With these notations at hand, it thus remains to control

|D x 1 I 3,l ps, xq| ď l ÿ i"2 n ÿ k"l ˇˇˇˇż 1 0 dλ ż R pn´1qd
dpy 1:i´1 , y i`1:n q (2.36) ˆżR d dy i # `Ψps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k py i q ˘: D y i D y k ups, y 1:l´1 , θ l:n s,t pξq `λpy l:n ´θl:n s,t pξqqq

+ˇˇˇˇˇ,

where " : " stands for the double tensor contraction, in terms of known parameters in (A).

Unfortunately, as suggested by the computations in (2.23), the control in supremum norm of pD i D k uq 1ďiďkďn is definitely out of reach. Recall indeed that in this equation (which corresponds to the case i " 1) the thresholds are sharp in order to retrieve an integrable singularity (see (2.24)).

Roughly speaking, what we can expect is that for fixed py 1:i´1 , y i`1:n q P R pn´1qd , the partial application y i Þ Ñ D k u i ps, y i q :" D y k ups, yq, y :" py 1:i´1 , y i , y i`1:n q is as smooth as the Green kernel involving the source f in the indicated equation, i.e. Gξ f ps, yq :" ş T s dv P ξ v,s f pv, yq. The specific Hölder modulus α k i of y i Þ Ñ D y k Gi,ξ f ps, y i q :" D y k Gξ f ps, yq can be derived from Lemma 1 and we obtain that any α k i satisfying α k i ă

1´p1´β k qpk´1 2 q i´1 2 is attainable.
Our guess is hence that the same property should hold for y i Þ Ñ D y k u i ps, y i q.

Keeping in mind this objective, the next step consists in exploiting a Besov duality result on the variable y i recalling that C

α k i b pR d , Rq " B α k i 8,8 pR d , Rq
, where from now on the notation B s p,q stands for a Besov space with associated indexes p, q, s (see Triebel [START_REF] Triebel | Theory of function spaces[END_REF] and Appendix 5 below). The indexes p, q denote the integrability parameters and s the smoothness one. However, instead of focusing on the Hölder modulus, we will concentrate on }D i D k u i ps, ¨q}

B α k i ´1 8,8
(space of the derivatives of Hölder functions). This specific choice allows to naturally deal with the cancellation techniques which require to change the freezing points depending on the current regime of the underlying frozen heat-kernel (see the proof of Lemma 4 in Section 3.2 below). We also carefully mention that the quantity ). Exploiting this fact, we then derive from (2.36) and the multi-linearity of the tensors involved that:

}D i D k u i ps, ¨q} B α k i ´1 8,8 is equivalent to }D k u i ps, ¨q}
|D x 1 I 3,l ps, xq| ď C l ÿ i"2 n ÿ k"l ż 1 0 dλ ż R pn´1qd dpy 1:i´1 , y i`1:n q ˆ"› › › " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k p¨q ı› › › B 1´α k i 1,1 › › ›y i Þ Ñ D i D k ups, y 1:l´1 , θ l:n s,t pξq`λpy l:n ´θl:n s,t pξqqq › › › B α k i ´1 8,8 * ď C l ÿ i"2 n ÿ k"l ż R pn´1qd dpy 1:i´1 , y i`1:n q # › › › › " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k p¨q › › › › B 1´α k i 1,1 (2.37) ˆ" sup z j ,jPrr1,nss,j‰i › › › › D i D k ups, z 1:i´1 , ¨, z i`1:n q › › › › B α k i ´1 8,8 I iďl´1 `sup z j ,jPrr1,nss,j‰i,λPr0,1s › › › › D i D k ups, z 1:i´1 , θ i s,t pξq `λp¨´θ i s,t pξqq, z i`1:n q › › › › B α k i ´1 8,8 I i"l ı + .
Observe now that, for any λ P r0, 1s and the norm equivalence of Proposition 4, we have:

› › › › D i D k ups, z 1:i´1 , θ i s,t pξq `λp¨´θ i s,t pξqq, z i`1:n q › › › › B α k i ´1 8,8 ď C › › › › D k ups, z 1:i´1 , θ i s,t pξq `λp¨´θ i s,t pξqq, z i`1:n q › › › › B α k i 8,8 ď C › › › › D k ups, z 1:i´1 , ¨, z i`1:n q › › › › B α k i 8,8
, exploiting the scaling and shift invariance properties of the Hölder modulus. Using again the norm equivalence of Proposition 4, we eventually derive:

› › › › D i D k ups, z 1:i´1 , θ i s,t pξq `λp¨´θ i s,t pξqq, z i`1:n q › › › › B α k i ´1 8,8 ď C 2 › › › › D i D k ups, z 1:i´1 , ¨, z i`1:n q › › › › B α k i ´1 8,8
.

(2.38)

In other words, the dilations-translations of the variable y i , that appear when i " l in (2.37), do not affect the regularity estimate. We therefore obtain:

|D x 1 I 3,l ps, xq| ď C l ÿ i"2 n ÿ k"l ż R pn´1qd dpy 1:i´1 , y i`1:n q # › › › › " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k p¨q › › › › B 1´α k i 1,1 (2.39) ˆsup z j ,jPrr1,nss,j‰i › › › › D i D k ups, z 1:i´1 , ¨, z i`1:n q › › › › B α k i ´1 8,8
+ .

To conclude this proof we now need the following results whose proofs are postponed to the next subsection: Lemma 3 Let l P rr2, nss, i P rr2, lss and k P rrl, nss and let Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k : R d Ñ R d be the function defined by (2.35). There exist C :" CppAqq ą 0, α k i :" Lemma 4 Let u be the solution of (2.1).

1`η 4 2i´1 ă 1´p1´β k qpk´1 2 q i´1 2 , γ k i " γ i :" 1 2 `ηpi ´3 2 q ą 1 2 such that › › › › " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k p¨q › › › › B 1´α k i 1,1 ď Cps ´tq ´3 2 `γk i qczi pt, s, x,
There exists C :" CppAqq ą 0 such that for all i ď k P rr2, nss 2 and α k i :"

1`η 4 2i´1 ă 1´p1´β k qpk´1 2 q i´1 2 , sup y j ,jPrr1,nss,j‰i › › › › D i D k u `s, y 1:i´1 , ¨, y i`1:n ˘› › › › B α k i ´1 8,8
ď Cp}Du} 8 `}DD 1 u} 8 q.

(2.41)

Remark 2 (About Lemmas 3 and 4) There are some specific points to be emphasized about the indicated lemmas: We point out that the parameters γ k i , α k i actually do not depend on the index k. We keep the notations to remember that they are associated with the Hölder regularity of D x k u w.r.t. to its i th variable which we choose to investigate in the corresponding negative Besov space for technical reasons. We also emphasize that the norm

› › › › D y i D y k ups, y 1:i´1 , ¨, y i`1:n q › › › › B α k i ´1 8,8 is actually equivalent to › › › › D y k ups, y 1:i´1 , ¨, y i`1:n q › › › › B α k i 8,8
which corresponds to the usual Hölder norm on C α k i b (see Proposition 4 below). This can be seen again as a specific feature appearing when dealing with strong uniqueness through the Zvonkin approach. The regularity is associated with a variable independently of the level in the chain (1.1).

We can then deduce, using Lemma 3 and Lemma 4 that

|D x 1 I 3,l ps, xq| ď C l ÿ i"2 n ÿ k"l ps ´tq ´3 2 `γk i ´}Du} 8 `}DD 1 u} 8 ¯, (2.42)
which are integrable terms since γ k i ą 1 2 . With the notations of (2.21), (2.30), we eventually derive from (2.42), (2.32), (2.31) that there exists γ :" γppAqq ą 0 such that:

ˇˇˇż T t dsD x 1 I l ps, xq ˇˇˇď CT γ ´}Du} 8 `}DD 1 u} 8 ¯.
(2.43)

Conclusion. Bringing together (2.43) and (2.29) yields for all l P rr1, nss and all pt, xq P r0, T s:

|D x l D x 1 upt, xq| ď CpT γ `T δ q ´}Du} 8 `}DD 1 u} 8 ¯. (2.44)
It is clear that the previous analysis can be reproduced without differentiating w.r.t. x 1 , leading to improved singularity exponents (see also the proof of Lemma 4 which somehow exactly explicit these computations). We therefore get:

|D x l upt, xq| ď CpT γ 1 `T δ 1 q ´}Du} 8 `}DD 1 u} 8 ¯, (2.45)
for some positive exponents γ 1 , δ 1 (with γ 1 ą γ, δ 1 ą δ).

Taking the time-space supremum in the l.h.s of (2.44) and (2.45), recalling as well that T is meant to be small, i.e. s.t. 4CT δ^γ ď 1 2 , we derive:

}Du} 8 `}DD 1 u} 8 ď 2CpT γ `T δ q.
This concludes the proof.

This section is dedicated to the proofs of the main technical results needed to obtain Theorem 3. Namely, we prove the Besov estimates of Lemmas 3 and 4.

Proof of Lemma 3

We will here exploit the thermic characterization of Besov spaces (see Chapter 2.6.4 in [START_REF] Triebel | Theory of function spaces[END_REF]) which is also recalled in Appendix 5.

From (5.1), we are thus led to estimate, for any l P rr2, nss, i P rr2, l ´1ss and k P rrl, nss:

ż 1 0 dv v v 1`α k i ´1 2 }B v h v ‹ Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k p¨q} L 1 pR d ,Rq .
We split the time integral in the above equation into two parts writing:

ż ps´tq ρ i,k 0 dvv α k i ´1 2 }B v h v ‹ Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k p¨q} L 1 pR d ,Rq `ż 1 ps´tq ρ i,k dvv α k i ´1 2 }B v h v ‹ Ψ
ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k

p¨q} L 1 pR d ,Rq (3.1)
": Lower " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k ı `Upper " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k ı , for a parameter ρ i,k ą 0 to be specified. The term Upper corresponding to the upper-part of the integral w.r.t. v does not involve singularities. We will use this fact to calibrate the associated parameter ρ i,k in order to match the integrability constraint Upper " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k ı ď C ps ´tq 1`1 2 ´γk i qczi pt, s, x, py 1:i´1 , y i`1:n qq,

where qczi has been defined in (2.40) and γ k i ą 1{2 in order to obtain a time integrable singularity. For this term, we will only use crude upper-bounds on the derivatives of the heat-kernel and the coefficients satisfying (T β ). On the other hand, the contribution Lower in (3.1) precisely contains the singularities w.r.t. v. It is therefore crucial to use there suitable cancellation tools. The point will then be to prove that the associated estimates are compatible with the upper-bound in equation (3.2).

We now write: Recall from the definition in (2.35) that Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k py i q is pR d q b4 -valued. To proceed with the computations we assume w.l.o.g. for the rest of the proof that d " 1 to avoid tensor notations for simplicity. Writing explicitly the function Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k py i q leads to:

Upper " Ψ ps,
Upper " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k ı ď ż 1

ps´tq ρ i,k dvv α k i ´1 2 ż R d dz ˇˇˇˇż R d
B v h v pz ´yi q ´Dx 1 D x l pξ pt, s, x, yq " ppθ s,t pξq ´yq k q ´Fi ps, y 1:l´1 , θ l:n s,t pξqq ´Fi ps, θ s,t pξqq ´Dx i´1 F i ps, θ s,t pξqq `y ´θs,t pξq ˘i´1 ı¯ˇˇˇˇd y i ˇˇˇˇξ "x

.

From Lemma 2 and Proposition 2, we derive there exists a C :" CppAq, T q ą 0 such that introducing qc pt, s, x, yq " px c pt, s, x, yq, c " C ´1:

Upper " Ψ ˘, recalling that the lower bound of β j pj ´1{2q is increasing for the last inequality (recall indeed that we assumed that β j P ´2j´2 2j´1 , 1 ı ). We now want to choose the threshold ρ i,k in order to match the integrability condition in (3.2). This amounts to write: (3.

´1
3)

It therefore remains to check that such a choice is compatible with the time integral part for v P r0, ps ´tq ρ i,k s in the thermic characterisation of the Besov norm. We point out that for this term it is absolutely essential to get rid of the exponent v ´1 coming from the upper-bound of the thermic heat-kernel, i.e. B v h v pz ´yi q. In order to get an integrable singularity in v, we need to decrease the crude upper-bound on B v h v pz ´yi q. This is done through cancellation techniques exploiting the smoothness properties of Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k .

To investigate Lower " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k ı let us first recall from the definition in (2.35) that for all k P rrl, nss: Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k py i q " D x l D x 1 pξ pt, s, x, yq " py ´θs,t pξqq k ´Fi py 1:l´1 , θ l:n s,t pξqq ´Fi pθ s,t pξqq ´Di´1 F i pθ s,t pξqq `y ´θs,t pξq ˘i´1 ¯.

(3.4)

Let us now specify the dependence w.r.t. y i of the previous expression in function of the considered indexes i P rr1, lss, l P rr2, nss, k P rrl, nss. This will be useful to develop corresponding adapted cancellation arguments.

Observe first that the dependence in y i appears in (3.4) for any i P rr1, lss, l P rr2, nss, k P rrl, nss through the term D x 1 x 2 pξ pt, s, x, yq. For the term into brackets we distinguish two cases. If i ď l ´1, and then k ą i, the only bracket term containing y i is the one associated with F i . If now i " l ď k, for the contribution i " l " k, the only term into brackets in (3.4) that will also depend on y i is py ´θs,t pξqq i and the cancellation arguments need to be slightly modified. Eventually, when i " l ă k, there will be no dependence on y i for the terms into brackets.

' First case i ă l ď k (i P rr1, l ´1ss, l P rr2, nss, k P rrl, nss). With the notations of (3.1), we write: Lower " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k ı "

ż ps´tq ρ i,k 0 dvv α k i ´1 2 ż R d dz| ż R d
dy i B v h cv pz ´yi q pΨ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k py i q ´Ψps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k pzqq| ":

ż ps´tq ρ i,k 0 dvv α k i ´1 2 ż R d
dz|pT ps,y 1:i´1 ,y i`1:n q,pt,xq 1,i,pl,1q,k `T ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pl,1q,k q `v, z ˘|, (3.5) where: T ps,y 1:i´1 ,y i`1:n q,pt,xq 1,i,pl,1q,k `v, z ˘(3.6)

:" ż R d
B v h v pz ´yi q ´Dx 1 D x l pξ pt, s, x, yq " py ´θs,t pξqq k ´Fi ps, y 1:l´1 , θ l:n s,t pξqq ´Fi ps, y 1:i´1 , z, y i`1:l´1 , θ l:n s,t pξqq ¯ıdy i , with a slight abuse of notation when i " l ´1 and T ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pl,1q,k `v, z ˘(3.7)

:" ż R d B v h v pz ´yi q " D x 1 D x l
pξ pt, s, x, yq ´Dx 1 D x l pξ pt, s, x, y 1:i´1 , z, y i`1:n q ı" py ´θs,t pξqq k ´Fi ps, y 1:i´1 , z, y i`1:l´1 , θ l:n s,t pξqq´F i ps, θ s,t pξqq´D i´1 F i ps, θ s,t pξqqpy´θ s,t pξqq i´1 ¯ıdy i .

Write now from (3.7), Proposition 2 and Lemma 2 (recalling as well that we took ξ " x for the current analysis):

|T ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pl,1q,k

`v, z ˘| ď C ż R d
dy i h cv pz ´yi q v ż 1 0 dλ qc pt, s, x, y 1:i´1 , z `λpy i ´zq, y i`1:n q ps ´tq pi´1 2 q`pl´1 2 q`1 2 |py ´θs,t pxqq k ||y i ´z| ˆ´ˇˇˇF i ps, y 1:i´1 , z, y i`1:l´1 , θ l:n s,t pxqq ´Fi ps, θ s,t pxqq ´Di´1 F i ps, θ s,t pxqqpy ´θs,t pxqq i´1 ˇˇď

C ż R d dy i h cv pz ´yi q v 1 2
ż 1 0 dλ qc pt, s, x, y 1:i´1 , z `λpy i ´zq, y i`1:n q ps ´tq pi´1 2 q`1 2 ˆ#|z ´pθ s,t pxqq i | β i `l´1 ÿ j"i`1 ps ´tq β j pj´1{2q `ps ´tq p1`ηqpi´3{2q

`v, z ˘| ď C ż R d dy i h cv pz ´yi q v 1 2
ż 1 0 dλq c pt, s, x, y 1:i´1 , z `λpy i ´zq, y i`1:n q ˆ´v β i 2 ps ´tq ´pi´1 2 q´1 2 `ps ´tq ´pi´1 2 q´1 2 `βi pi´1{2q `l´1 ÿ j"i`1 ps ´tq ´pi´1 2 q´1 2 `βj pj´1{2q `ps ´tq ´pi´1 2 q´1 2 `p1`ηqpi´3{2q ¯.

ď C qczi pt, s, x, y 1:i´1 , y i`1:n q ż 1 0 dλ ż R d dy i h cv pz ´yi qN cps´tq 2i´1 pz `λpy i ´zq ´pθ s,t pxqq i q ˆv´1 2 ´v β i 2 ps ´tq ´pi´1 2 q´1 2 `ps ´tq ´pi´1 2 q´1 2 `p1`ηqpi´3{2q ¯, (3.8) recalling for the last inequality that for all j in rri, l ´1ss, β j pj ´1 2 q ą j ´1 ą i ´3 2 and η is supposed to be a small parameter. ¸, using the change of variable pw 1 , w 2 q " pz ´yi , z `λpy i ´zq ´pθ s,t pxqq i q for the last inequality.

For ρ i,k chosen as in (3.3) one gets from the definition in (3.1):

Lower " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k ı ď C qczi pt, s, x, py 1:i´1 , y i`1:n qq

ż ps´tq ρ i,k 0 dv v v α k i 2 ˆ˜1 v 1´β i 2 ps ´tq 1 2
`v β i 2 ps ´tq i `1 ps ´tq i´p1`ηqpi´3{2q ":

C qczi pt, s, x, py 1:i´1 , y i`1:n qqB 1,ρ i,k pt, sq.

It therefore remains to prove that, if B 1,ρ i,k pt, sq ď ps ´tq ´1´1 2 `γk i for γ k i ą 0 then: Eventually, for the exponent of the third contribution in (3.11), for the previous choice of ρ i,k " 2i ´1, we get

´1 ´1 2 `γk i ą ´1 ðñ γ k i ą 1 2 . ( 3
ρ i,k α k i 2 ´3 2 `ηpi ´3 2 q " 1 2 p2i ´1q 1 `η 4 2i ´1 ´3 2 `ηpi ´3 2
q " ´1 `η 8 `ηpi ´3 2 q ą ´1, (3.14) which means that criterion (3.10) is indeed satisfied, even though if this last contribution is rather critical in order to obtain the required smoothing effect with γ k i " 1{2`ηpi´3{2q. This concludes the proof of Lemma 3 for the indexes i P rr1, l ´1ss. ' Second case i " l ď k. Let us begin with the case i " l ă k for which the only contribution in y i in (3.4) appears through D x 1 D x i ppt, s, x, yq. Hence, for such indexes, Lower " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pi,1q,k ı "

ż ps´tq ρ i,k 0 dvv α k i ´1 2 ż R d
dz|T ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pi,1q,k q `v, z ˘|, where: T ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pi,1q,k `v, z " ż R d dy i B v h v pz ´yi q " D x 1 D x i pξ pt, s, x, yq´D x 1 D x i pξ pt, s, x, y 1:i´1 , z, y i`1:n q ı" py´θ s,t pξqq k ˆ´F i ps, y 1:i´1 , θ i:n s,t pξqq´F i ps, θ s,t pξqq´D i´1 F i ps, θ s,t pξqqpy ´θs,t pξqq i´1 ¯ıˇˇˇˇ.

(3.15)

The previous analysis for this term can be reproduced adapting the computations leading to (3.8). Precisely,

|T

ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pi,1q,k

`v, z ˘| ď C ż R d
dy i h cv pz ´yi q v ż 1 0 dλ qc pt, s, x, y 1:i´1 , z `λpy i ´zq, y i`1:n q ps ´tq pi´1 2 q`pl´1 2 q`1 2 |py ´θs,t pxqq k ||y i ´z| ˆˇˇp y ´θs,t pxqq i´1 ˇˇ1

`η ď C ż R d dy i h cv pz ´yi q v 1 2
ż 1 0 dλ qc pt, s, x, y 1:i´1 , z `λpy i ´zq, y i`1:n q ps ´tq pi´1 2 q`1 2 ps ´tq p1`ηqpi´3{2q .

This contribution has already been analyzed and yields the expected integrable singularity in time.

Let us now focus on the remaining case i " l " k, for which we write:

L " Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pi,1q,i ı "

ż ps´tq ρ i,i 0 dvv α i i ´1 2 ż R d dz| ż R d
dy i B v h v pz ´yi q pΨ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pi,1q,i py i q ´Ψps,y 1:i´1 ,y i`1:n q,pt,xq i,pi,1q,i pzqq| "

ż ps´tq ρ i,i 0 dvv α i i ´1 2 ż R d dz ˇˇˇż R d
dy i B v h v pz ´yi q " D x 1 D x i pξ pt, s, x, yqpy ´θs,t pξqq i ´Dx 1 D x i pξ pt, s, x, y 1:i´1 , z, y i`1:n qpz ´θs,t pξqq i ı ´Fi ps, y 1:i´1 , θ i:n s,t pξqq´F i ps, θ s,t pξqq´D i´1 F i ps, θ s,t pξqqpy ´θs,t pξqq i´1

¯ˇˇ"

:

ż ps´tq ρ i,i 0 dvv α i i ´1 2 ż R d
dz|pT ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pi,1q `T ps,y 1:i´1 ,y i`1:n q,pt,xq 3,i,pi,1q q `v, z ˘|, (3.16) where T ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pi,1q,i `v, z ˘is obtained from (3.15) taking k " i and T ps,y 1:i´1 ,y i`1:n q,pt,xq 3,i,pi,1q `v, z :" ż R d dy i B v h v pz ´yi q ´Dx 1 D x i pξ pt, s, x, y 1:i´1 , z, y i`1:n q (3.17) " py i ´zq ´Fi ps, y 1:i´1 , θ i:n s,t pξqq´F i ps, θ s,t pξqq´D i´1 F i ps, θ s,t pξqqpy ´θs,t pξqq i´1 ¯ıˇˇˇˇ.

Write first:

|T ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pi,1q,i

`v, z ˘| ď C ż R d
dy i h cv pz ´yi q v ż 1 0 dλ qc pt, s, x, y 1:i´1 , y i `λpz ´yi q, y i`1:n q ps ´tq pi´1 2 q`pi´1 2 q`1 2 |py ´θs,t pxqq i ||y i ´z| ˆˇˇp y ´θs,t pxqq i´1 ˇˇ1

`η ď C ż R d dy i h cv pz ´yi q v 1 2
ż 1 0 dλq c pt, s, x, y 1:i´1 , y i `λpy i ´zq, y i`1:n q ˆ´|y i `λpz ´yi q ´pθ s,t pxqq i | `λ|z ´yi | ps ´tq 2i´1 2 ¯ps ´tq pi´3 2 qp1`ηq .

We thus derive:

|T ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pi,1q,i

`v, z ˘| ď C ż R d dy i h cv pz ´yi q v 1 2
ż 1 0 dλq c pt, s, x, y 1:i´1 , y i `λpy i ´zq, y i`1:n q ´1 ps ´tq i `v 1 2 ps ´tq 2i´1 2 ¯ps ´tq pi´3 2 qp1`ηq

ď C ż R d dy i h cv pz ´yi q v 1 2
ż 1 0 dλq c pt, s, x, y 1:i´1 , y i `λpy i ´zq, y i`1:n q ´1 ps ´tq `v 1 2 ps ´tq i´1 2 `3 2 ´ηpi´3 2 q ¯. Finally,

ż ps´tq ρ i,i 0 dvv α i i ´1 2 ż R d
dz|pT ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pi,1q | ď Cps ´tq ´3 2 `ηpi´3 2 q qczi pt, s, x, y 1:i´1 , y i`1:n q ż ps´tq ρ i,i 0 dv ´v´1`α i i 2 `v´1 2 `αi i 2 ps ´tq i´1 2 ď Cps ´tq ´1`ηpi´3 2 q`η 8 qczi pt, s, x, y 1:i´1 , y i`1:n q ď Cps ´tq ´3 2 `γi i qczi pt, s, x, y 1:i´1 , y i`1:n q, recalling for the last inequality that, since from (3.12) and (3.13), α i i " p1 `η{4q{p2i 1q, ρ i,i " 2i ´1, α i i ρ i,i {2 " 1{2 `η{8, ρ i,i {2 " i ´1{2. We also refer to (3.14) for similar computations.

On the other hand, we readily get, similarly to the previous contributions from Lemmas 1 and 2: |T ps,y 1:i´1 ,y i`1:n q,pt,xq 3,i,pi,1q

`v, z ˘| ď C ż R d dy i h cv pz ´yi q v 1 2
|py ´θs,t pξqq i´1 | 1`η ps ´tq i qc pt, s, x, y 1:i´1 , z, y i`1,n q ď Cps ´tq ´3 2 `ηpi´3 2 q v ´1 2 ż R d dy i h cv pz ´yi qq c pt, s, x, y 1:i´1 , z, y i`1:n q.

Similarly to the contributions appearing in (3.5) this bound needs to be integrated w.r.t. v P r0, ps ´tq ρ i,i s. One gets:

ż ps´tq ρ i,i 0 dvv α i i ´1 2 ż R d
dz|pT ps,y 1:i´1 ,y i`1:n q,pt,xq 3,i,pi,1q | ď Cps ´tq ´3 2 `ηpi´3 2 q qczi pt, s, x, y 1:i´1 , y i`1:n q ż ps´tq ρ i,i 0 dvv ´1`α i i 2 ď Cps ´tq ´3 2 `ηpi´3 2 q`α i i 2 ρ i,i qczi pt, s, x, y 1:i´1 , y i`1:n q ď Cps ´tq ´3 2 `γi i qczi pt, s, x, y 1:i´1 , y i`1:n q, reproducing the previous computations done on T ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pi,1q `v, z ˘for the last inequality.

Proof of Lemma 4

We now tackle the Besov estimate of the cross derivative of the solution of (1.5). Fix t P r0, T s and px 1:i´1 , x i`1:n q P R pn´1qd . From the thermic characterization of Besov spaces recalled in equation ( 5.1), we actually have to control: For the proof we focus on the second contribution in the above definition. The first one could be handled similarly and more directly † † .

}D i D k upt, x 1,i´1 , ¨, x i`1,n q}
Using representation (2.20) we first write with the notations of (2.21): dx i D x i h v pz ´xi q b ´Dx k upt, x 1:i´1 , x i , x i`1:n q ´Dx k upt, x 1:i´1 , z, x i`1:n q ¯ˇˇˇˇ, integrating by parts and using a usual cancellation argument in the above equality. To handle the above expression we are led to consider separately the time integration interval corresponding to the diagonal and the off-diagonal regime for the considered running time. We hence introduce the time set S i " ts P rt, T s : ps ´tq ď c 0 |x i ´z| 2{p2i´1q u for a constant c 0 to be specified later on and meant to be small. The time set S i corresponds to the off-diagonal regime (i.e. the distance between the i th spatial is larger than the i th characteristic time scale).

|h v ‹ D i D k upt,

The complementary set S c

i " ts P rt, T s : ps ´tq ą c 0 |x i ´z| 2{p2i´1q u corresponds to the diagonal one (i.e. the distance between the i th spatial variable is smaller than the i th characteristic time.

Then, we need to expand suitably the terms in (3.19) with our perturbative approach. To this end, we proceed as in Chapter 6. Namely, we will expand the term D x k upt, x 1:i´1 , x i , x i`1:n q with (2.21) taking ξ " x, whereas we will expand differently the contribution D x k upt, x 1:i´1 , z, x i`1:n q, depending on the considered regime (off-diagonal or diagonal) for the current running time. Setting x 1 " px 1:i´1 , z, x i`1:n q, this term will indeed be expand as in (2.21) around the freezing point ξ 1 " x 1 in the off-diagonal regime and around ξ1 " x in the diagonal one. Denoting by t 0 " t`c 0 |x i ´z| 2{p2i´1q the transition time between the two regimes, we hence write (see also Section 2 in Chapter 6): upt, x 1 q " Gξ 1 t 0 ,t f pt, x 1 q `G ξ1 T,t 0 f pt, x 1 q `P ξ 1 t 0 ,t upt 0 , x 1 q ´P ξ1 t 0 ,t upt 0 , x 1 q `ż T t ds ż R nd dy ´Isďt 0 pξ 1 pt, s, x 1 , yqpL s ´L ξ 1 s q `Isąt 0 pξ 1 pt, s, x 1 , yqpL s ´L ξ1 s q ¯ups, yq,

where we denoted for all ξ 1 P R nd : ! " H ξ k ps, xq ´Hξ 1 k ps, x 1 q ı `"I ξ k ps, xq ´Iξ 1 k ps, x 1 q ı ) , ! " H ξ k ps, xq ´Hξ 1 k ps, x 1 q ı `"I ξ k ps, xq ´Iξ 1 k ps, x 1 q ı ) ,

@0 ď v ă r ď T, Gξ 1 r,v f pt, x 1 q " ż r v ds ż R nd dyp ξ 1 pt, s,
is the contribution of the diagonal regime and `hv ‹ D x i D x k upt, x 1:i´1 , ¨, x i`1:n qpzq ˘|BS i :" ż R d dx i D z h v pz ´xi q ! D x k P ξ 1 t 0 ,t upt 0 , x 1 q ´Dx k P ξ1 t 0 ,t upt 0 , x 1 q

) , (3.25)
is the resulting boundary term. This last term, arising from the change of freezing point, is particularly delicate to analyze.

Off-diagonal estimates: control of (3.23). On the time set S i , we cannot expect some regularization from the difference of the transition densities so that we bluntly estimate the terms appearing in (3.23), writing: and recalling for the last inequality that γ k i " 1{2 `ηpi ´3{2q so that 2γ k i {p2i ´1q " r1 `2ηpi ´3{2qs{p2i ´1q ą p1 `η{4q{p2i ´1q " α k i (see also the statements of Lemmas 3 and 4) and similarly for the contributions involving δ k i ą 1{2. We eventually get from (3.28) and (3.26): .

|h v ‹ D x i D x k upt,
|h v ‹ D x i D x k upt,
(3.30)

Diagonal estimates: control of the term (3.24). We are now going to handle the term (3.24) which correspond to the r.h.s. of (3.19) on S c i . In that case the points x i and x 1 i are close w.r.t. the characteristic time scale of the i th variable and the main idea consists in controlling the difference between the frozen densities at ξ " ξ1 " x with starting points x and x 1 respectively. Precisely, recalling that x and x 1 only differ in the i th component, we can write: exppcps ´tq|T ´1 s´t m ξ s,t px ´x1 q| 2 q expp´c 2 ps ´tq|T ´1 s´t pm ξ s,t pxq ´yq| 2 q.

D x k pξ pt,
Using the rescaling arguments of the proof of Proposition 2 on the resolvent (see equation (2.13)), we then get ps ´tq and let us discuss how the terms H ξ k ps, xq ´Hξ k ps, x 1 q, I ξ k ps, xq ´Iξ k ps, x 1 q in the above equation can be handled.

We first focus on the term I ξ k ps, xq ´Iξ k ps, x 1 q in (3.34). This contribution, associated with the degenerate components of perturbed operator, is again the most delicate to handle. From the definitions in (2.30) we are led to control the sum ř 3 "1 rI ξ ,k ps, xq Íξ ,k ps, x 1 qs. For the terms I ξ 1,k ps, xq ´Iξ 1,k ps, x 1 q, I ξ 2,k ps, xq ´Iξ 2,k ps, x 1 q we are going to reproduce the analysis leading to (2.31), (2.32). Observe first that the above terms do not involve D x 1 , therefore we gain a singularity of order 1{2 w.r.t. to the indicated equations (2.31), (2.32). On the other hand, the difference of the derivatives of the frozen densities w.r.t. x k can be handled with (3.33). This leads to: dsps ´tq ´pk´1 2 q´α k i pi´1 2 q ´ps ´tq β j pj´1 2 q `ps ´tq p1`ηqpj´1 2 q ¯, changing the summation variables from (2.31) for notational simplicity.

From the very definition of α k i " p1 `η{4q{p2i ´1q in Lemma 3 and the specific choice of η in assumption (A) (see (H η )) we derive

ˇˇˇˇ2 ÿ "1 ż T t`c 0 |px 1 ´xq i | 2 2i´1
ds ´Iξ ,k ps, xq ´Iξ ,k ps,

x 1 q ¯ˇˇˇˇď C|px 1 ´xq i | α k i }Du} 8 T δ , (3.35)
for some δ ą 0.

From the previous analysis it is therefore sufficient to focus on the tricky term, namely I 3,k ps, xq introduced in (2.30). We begin the proof considering first I 3,k ps, xq. Exploiting as well Lemma 1 for a centering argument w.r.t. the k th variable, we write: Let us reproduce now the arguments used in Section 2.2 to handle I ξ 3,k (see e.g. the computations from equation (2.33) to (2.34)). Expanding with the Taylor formula the difference ´Dy ups, yq ´Dy ups, y 1:k´1 , θ k:n s,t pξqq ¯, using the Schwarz theorem to exchange the order of differentiations ‡ ‡ , we obtain with the notations of (2.35) (see also the comments following this equation): py q ´Ψps,y 1: ´1,y `1:n q,pt,x 1 q ,k,m py q  D y D ym u `s, y 1:k´1 , θ k:n s,t pξq `λpy ´θs,t pξqq k:n ˘.

I ξ 3,
I ξ 3,
Thus, we derive similarly to (2.39): + , where ş R pn´1qd dpy 1: ´1, y `1:n q means that we integrate over y 1: ´1 and y `1:n . To conclude, we need the following appropriate version of Lemma 3 to handle the Besov norm with negative exponent in the above r.h.s. Its proof is postponed to the next section. Again, for the specific choice of α k i " p1 `η{4q{p2i ´1q performed in the proof of Lemma 3, we eventually derive from Lemma 5 and (3.36), with the notation of (3.29), that: for some δ ą 0 recalling that δ k i ą 1{2 for the last inequality. The arguments needed to control this term are actually those already exploited in [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF] when n " 2.

Gathering equations (3.38) and (3.39), we finally derive with the notations of (3.34): dx i D z h v pz ´xi q ! D x k P ξ 1 t 0 ,t upt 0 , x 1 q ´Dx k P ξ1 t 0 ,t upt 0 , x 1 q

|h v ‹ D x i D x k upt,
) , which we will actually handle like the off-diagonal components. Recall here that the transition time t 0 " t `c0 |px ´x1 q i | 2{p2i´1q and that pξ 1 , ξ1 q " px 1 , xq, where these values are again plugged in the expansions after the derivations. From Lemma 1 (cancellation argument), we write:

D x k
P ξ 1 t 0 ,t upt 0 , x 1 q ´Dx k P ξ1 t 0 ,t upt 0 , x 1 q (3.41)

" ż R nd D x k pξ 1
pt, t 0 , x 1 , yqrupt 0 , yq ´upt 0 , y 1:k´1 , pθ t 0 ,t px 1 qq k:n qsdy ´żR nd D x k pξ pt, t 0 , x 1 , yqrupt 0 , yq ´upt 0 , y 1:k´1 , pm ξ t 0 ,t px 1 qq k , θ t 0 ,t px 1 q k`1:n qsdy " # ż R nd D x k pξ 1 pt, t 0 , x 1 , yqrupt 0 , yq ´upt 0 , y 1:k , pθ t 0 ,t px 1 qq k`1:n qsdy ´żR nd D x k pξ pt, t 0 , x 1 , yq " upt 0 , yq ´upt 0 , y 1:k , pθ t 0 ,t px 1 qq k`1:n q ı dy + (3.42)

`#" ż R nd D x k pξ 1 pt, t 0 , x 1 , yq " upt 0 , y 1:k , pθ t 0 ,t px 1 qq k`1:n q ´upt 0 , y 1:k´1 , pθ t 0 ,t px 1 qq k:n q ´xD k upt 0 , y 1:k´1 , pθ t 0 ,t px 1 qq k:n q, py ´θt 0 ,t px 1 qq k y ı dy ı

´"ż R nd D x k pξ pt,t 0 ,x 1 ,yqrupt 0 ,y 1:k ,pθ t 0 ,t px 1 qq k`1:n q´upt 0 ,y 1:k´1 ,pm ξ t 0 ,t px 1 qq k ,θ t 0 ,t px 1 q k`1:n q ´xD k upt 0 , y 1:k´1 , pm ξ t 0 ,t px 1 qq k , θ t 0 ,t px 1 q k`1:n q, py ´mξ t 0 ,t px 1 qq k ysdy ı + (3.43)

`# ż R nd D x k pξ 1
pt, t 0 , x 1 , yqxD k upt 0 , y 1:k´1 , pθ t 0 ,t px 1 qq k:n q, py ´θt 0 ,t px 1 qq k ydy (3.44)

´żR nd D x k pξ pt,t 0 , x 1 , yqxD k upt 0 , y 1:k´1 ,pm ξ t 0 ,t px 1 qq k , θ t 0 ,t px 1 q k`1:n q, py´m ξ t 0 ,t px 1 qq k ydy + ": pB ξ 1 , ξ1 1 `Bξ 1 , ξ1 2 `Bξ 1 , ξ1 3 qpt 0 , x 1 q.

We now exploit the Hölder regularity of D k u w.r.t. the k th variable to control the terms in B ξ 1 , ξ1 2 pt 0 , x 1 q defined in (3.43). Let us first write from the previous decomposition:

|B ξ 1 , ξ1 2,1 pt 0 , x 1 q| :" ˇˇż R nd D x k pξ 1 pt, t 0 , x 1 , yq " upt 0 , y 1:k , pθ t 0 ,t px 1 qq k`1:n q ´upt 0 , y 1:k´1 , pθ t 0 ,t px 1 qq k:n q ´xD k upt 0 , y 1:k´1 , pθ t 0 ,t px 1 qq k:n q, py ´θt 0 ,t px 1 qq k y ı dy ˇ" ż R nd ż 1 0 dλ ˇˇD x k pξ 1 pt, t 0 , x 1 , yq ˇˇrD k upt 0 , y 1:k´1 , ¨, pθ t 0 ,t px 1 qq k`1:n qs α k k |py´θ t 0 ,t px 1 q ˘k| 1`α k k dy,

where r¨s α k k denotes the Hölder modulus of order α k k . From Proposition 2, we thus derive:

|B ξ 1 , ξ1 2,1 pt 0 , x 1 q| ď Cpt 0 ´tq α k k pk´1 2 q sup z j ,jPrr1,nss, j‰k rD k upt 0 , z 1:k , ¨, z k`1:n qs α k k ˆżR pn´1qd dy 1:k´1 dy k`1:n qC ´1zk pt, s, x, py 1:k´1 , y k`1:n qq " Cpt 0 ´tq 1 2 `η 8 rpD k uq k pt 0 , ¨qs α k k , (3.45) recalling from (3.12) that α k k " p1 `η{4q{p2k ´1q and using the notation of (1.2) for the last inequality. The same arguments readily give:

|B ξ 1 , ξ1
2,2 pt 0 , x 1 q| :" ˇˇż R nd D x k pξ pt, t 0 , x 1 , yqrupt 0 , y 1:k , pθ t 0 ,t px 1 qq k`1:n q ´upt 0 , y 1:k´1 , pm ξ t 0 ,t px 1 qq k , θ t 0 ,t px 1 q k`1:n q ´xD k upt 0 , y 1:k´1 , pm ξ t 0 ,t px 1 qq k , θ t 0 ,t px 1 q k`1:n q, py ´mξ t 0 ,t px 1 qq k ysdy ¯ˇď Cpt 0 ´tq 1 2 `η 8 rpD k uq k pt 0 , ¨qs α k k .

(3.46)

Let us now deal with the contribution B ξ 1 , ξ1 1 pt 0 , x 1 q in (3.42). Observe from this definition that this term is non zero if and only if k ă n. Write then

|B ξ 1 , ξ1 1 pt 0 , x 1 q| ď ż R nd |D x k pξ 1
pt, t 0 , x 1 , yq||upt 0 , yq ´upt 0 , y 1:k , pθ t 0 ,t px 1 qq k`1:n q|dy `żR nd |D x k pξ pt, t 0 , x 1 , yq||upt 0 , yq ´upt 0 , y 1:k , pθ t 0 ,t px 1 qq k`1:n q|dy ď C}Du} 8 ˆżR nd dy pt 0 ´tq k´1 2 pξ 1 C ´1 pt, t 0 , x 1 , yq|pθ t 0 ,t px 1 q ´yq k`1:n | (3.47)

`żR nd dy pt 0 ´tq k´1 2 pξ C ´1 pt, t 0 , x 1 , yq `|pm x t 0 ,t px 1 q ´yq k`1:n |`|pm x t 0 ,t px 1 q ´θt 0 ,t px 1 qq k`1:n | ˘˙.

To deal with the last contribution in the r.h.s., we will need some auxilliary lemmas already used in Chapter 6 for Schauder estimates. Namely, analogously to Lemmas 1 and 3 therein, we have the following result.

Lemma 6 There exists ϑ " ϑppAqq P p0, 1q s.t. for all j P rrk, nss: |pm x t 0 ,t px 1 q ´θt 0 ,t px 1 qq j | ď c ϑ 0 |px ´x1 q i | 2j´1 2i´1 .

(3.48)

In particular, recalling that t 0 " t `c0 |px ´x1 q i | 2 2i´1 with |px ´x1 q i | ď 1, we obtain }Du} 8 pt 0 ´tq k´1 2 |pm x t 0 ,t px 1 q ´θt 0 ,t px 1 qq k`1:n | ď ď }Du} 8 c ϑ´pk´1 2 q 0 |px ´x1 q i | α k i .

Plugging the above control in (3.47) we obtain

|B ξ 1 , ξ1 1 pt 0 , x 1 q| ď C}Du} 8 ! 2pt 0 ´tq `cϑ´pk´1 2 q 0 |px ´x1 q i | α k i ) . ( 3

.49)

Let us eventually control the term B ξ 1 , ξ1 3 pt 0 , x 1 q defined in (3.44) which we rewrite in the following way:

B ξ 1 , ξ1
3 pt 0 , x 1 q " # ż R nd dyD x k pξ 1 pt, t 0 , x 1 , yq A " D k upt 0 , y 1:k´1 , pθ t 0 ,t px 1 qq k:n q ´Dk upt 0 , θ t 0 ,t px 1 qq ‰ , py ´θt 0 ,t px 1 qq k E ´żR nd dyD x k pξ pt, t 0 , x 1 , yq A " D k upt 0 , y 1:k´1 , pm ξ t 0 ,t px 1 qq k , θ t 0 ,t px 1 q k`1:n q ´Dk upt 0 , pm ξ t 0 ,t px 1 qq 1:k , θ t 0 ,t px 1 q k`1:n q ‰ py ´mξ t 0 ,t px 1 qq k E + `# ż R nd dyD x k pξ 1 pt, t 0 , x 1 , yq ˆArD k upt 0 , θ t 0 ,t px 1 qq ´Dk upt 0 , pm ξ t 0 ,t px 1 qq 1:k , θ t 0 ,t px 1 q k`1:n q, py ´θt 0 ,t px 1 qq k

E +

´# ż R nd dyD x k pξ pt, t 0 , x 1 , yq A D k upt 0 , pm ξ t 0 ,t px 1 qq 1:k , θ t 0 ,t px 1 q k`1:n q, py ´mξ t 0 ,t px 1 qq k E ´żR nd dyD x k pξ 1 pt, t 0 , x 1 , yq A D k upt 0 , pm ξ t 0 ,t px 1 qq 1:k , θ t 0 ,t px 1 q k`1:n q, py ´θt 0 ,t px 1 qq k E + ,

where, thanks to Proposition 3 the last contribution is actually 0. For the first and second contributions in the above r.h.s. we have, thanks to the Hölder regularity of x 1:k Þ Ñ D k up¨, x 1:k , ¨q, Proposition 2 and Lemma 6: , with the notations of (3.29). This point is actually crucial to complete our circular argument.

B ξ 1 , ξ1 3 pt 0 , x 1 q ď C ż R nd dy ! pξ 1 C ´1 pt,
From the definition of t 0 " t `c0 |px ´x1 q i | 2{p2i´1q and α k i " p1 `η{4q{p2i ´1q, recalling that t 0 ´t is small as well (i.e. t 0 ´t ď Cpt 0 ´tq 1{2`η{8 ), we obtain from (3.51):

|D x k

P ξ 1 t 0 ,t upt 0 , x 1 q ´Dx k P ξ1 t 0 ,t upt 0 , ¯.

The main point to close our circular argument consists then in taking the supremums over w.r.t. x 1:i´1 , x i`1:n , i, k and t P r0, T s on the l.h.s. and to tune the constant c 0 and the terminal time T in order to obtain C `cδ 0 `T δ ˘ď 1{2. We then derive that for all 2 ď i ď k ď n:

Proof of the technical lemmas

Proof: [Proof of Lemma 5] We follow the proof of Lemma 3, concentrating on the case ď k ´1, the specific case " k could be treated similarly considering the slightly different cancellation terms already discussed in Lemma 3. The quantity to estimate is now: " Ψ ps,y 1: ´1,y `1:n q,pt,xq ,k,m p¨q ´Ψps,y 1: ´1,y `1:n q,pt,x 1 q ,k,m p¨q  .

(3.55)

Splitting the thermic part of the Besov norm as in (3.1), we obtain the same kind estimate for the non-singular in time part. Indeed, we point out that the difference (3.55) does not involve D x 1 , therefore we gain a singularity of order 1{2 w.r.t. to equation (3.2). On the other hand, the difference of the derivatives of the frozen densities w.r.t. x k can be handled with (3.33). Choosing ρ ,m " 2 ´1 as in the proof of Lemma 3, and recalling that α k i pi ´12{q " p1 `η{4q{2 (see (3.12)), it is plain to check that:

U " Ψ ps,y 1: ´1,y `1:n q,pt,xq ,k,m p¨q ´Ψps,y 1: ´1,y `1:n q,pt,x 1 q ,k,m p¨q  ď C|px ´x1 q i | α k i ps ´tq 1`α k i pi´1 2 q´γ m qcz pt, s, x, py 1: ´1, y `1:n qq, dz|pT ps,y 1: ´1,y `1:n q,pt,x,x 1 q 1, ,k,m `T ps,y 1: ´1,y `1:n q,pt,x,x 1 q 2, ,k,m q `w, z˘| , where ă k ď m and:

T ps,y 1: ´1,y `1:n q,pt,x,x 1 q 1, ,k,m `w, z˘( 3.57)

" ż R d
B w h w pz ´y q `Dx k pξ pt, s, x, yq ´Dx k pξ pt, s, x 1 , yq @F ps, y 1:k´1 , θ k:n s,t pξqq ´F ps, y 1: ´1, z, y `1:k´1 , θ Note now that when proceeding first as in (3.31), (3.32) and then control the resulting difference as in (3.7) we get, thanks to (3.33), that: ˇˇˇˇ´D

x k pξ pt, s, x, yq ´Dx k pξ pt, s, x, y 1: ´1, z, y `1:n q Dx k pξ pt, s, x 1 , yq ´Dx k pξ pt, s, x 1 , y 1: ´1, z, y `1:n q ¯ˇˇˇď C ps ´tq ´1 2 `i´1 2 `k´1 2 ż 1 0 dλp ξ C ´1 pt, s, x, y 1: ´1, z `λpy ´zq, y `1:n q|px 1 ´xq i ||z ´y | ď C ps´tq p ´1 2 q`α k i pi´1 2 q`pk´1 2 q ż 1 0 dλp ξ C ´1 pt, s, x, y 1: ´1, z `λpy ´zq, y `1:n q|px 1 ´xq i | α k i |z ´y |, using the fact we are in the diagonal regime in the last inequality. With this control at hand, together with estimate (3.33), to handle the contributions pT ps,y 1: ´1,y `1:n q,pt,xq 1, ,k,m T ps,y 1: ´1,y `1:n q,pt,xq 2, ,k,m q `w, z˘, we can mimic the proof of the estimation of the contributions pT ps,y 1:i´1 ,y i`1:n q,pt,xq 1,i,pl,1q,k `T ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pl,1q,k q `v, z ˘done in Lemma 3 to obtain L " Ψ ps,y 1: ´1,y `1:n q,pt,xq ,k,m p¨q ´Ψps,y 1: ´1,y `1:n q,pt,x 1 q ,k,m This complication for the rescaled flows could also be avoided through mollification, see e.g. Lemma 2 in [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF]. But in our framework, it seems that Young type controls are more suitable. Indeed, we write: The last inequality is a consequence of the assumption dpx, x 1 q ď 1 and by our pick of δ n´1 in (4.4), by inequality (4.5) and by convexity inequality.

|pθ s,
For all δn´1,n ą 0 (where the two indexes in the subscript respectively denote the level of the chain, i.e. n ´1, and the considered variable, i.e. n), we get by Young inequality: . We eventually derive from (4.10) that: sup vPrt,ss |pθ v,t pxq ´θv,t px 1 qq 1 | ď C `ps ´tq 1 2 `dpx, x 1 q ˘, which gives the statement for i " 1. Now, by plugging this inequality into (4.9), we get for all i P rr2, nss: |pθ s,t pxq ´θs,t px 1 qq i | ď C `ps ´tq i´1 2 `d2i´1 px, x 1 q `ps ´tq i´1 sup vPrt,ss |pθ v,t pxq ´θv,t px 1 qq 1 | ď C ´ps ´tq i´1 2 `d2i´1 px, x 1 q `ps ´tq i´1 `ps ´tq 1 2 `dpx, x 1 q ˘ď C `ps ´tq i´1 2 `d2i´1 px, x 1 q ˘, for the last identity, we use again the Young inequality to derive that ps ´tq i´1 dpx, x 1 q ď C `ps ´tq i´1 2 `d2i´1 px, x 1 q ˘. The proof is complete. Now, we have the tools to control the sensitivity of the mean of the linearized system w.r.t. the freezing parameter.

´ż v t ˇˇ
Sensitivity results for the mean Proof: Technical Lemma 6 Again through the analysis, we assume w.l.o.g. that dpx, x 1 q ď 1. The control is done with a distinction of two contributions to handle. m x s,t px 1 q ´θs,t px 1 q " rm x s,t px 1 q ´θs,t pxqs `rθ s,t pxq ´θs,t px 1 qs. (4.11)

By the proxy definitionin (2.3), we deduce that the mean value of Xm,ξ v , m ξ v,t is s.t. m x s,t px 1 q ´θs,t pxq " x 1 ´x `ż s t dvDFpv, θ v,t pxqqrm x v,t px 1 q ´θv,t pxqs. (4.12)

The sub-triangular structure of DF yields that for all i P rr2, nss:

`mx s,t px 1 q ´θs,t pxq ˘i " x 1 i ´xi `ż s t dvD i´1 F i pv, θ v,t pxqqrm x v,t px 1 q i´1 ´θv,t pxq i´1 s.

Also, since m x v,t px 1 q 1 " x 1 1 `şs t F 1 pv, θ v,t pxqqdv, so we obtain that rm x v,t px 1 q 1 ´θv,t pxq 1 s " x 1 1 ´x1 , we then obtain by iteration that: `mx s,t px 1 q ´θs,t pxq ˘i

" x 1 i ´xi `i ÿ k"2
" ż v i "s t dv i´1 . . .

ż v k t dv k´1 i ź j"k D j´1 F j pv j , θ v j ,t pxqq ı rx 1 k´1 ´xk´1 s,
with the convention that for i " 1, ř i k"2 " 0. From the above control, equation (4.11) and the dynamics of the flow, and because the starting points are the same, the contributions involving differences of the spatial points (x 1 ´xq) or flows only appear in iterated time integrals, we obtain: | `mx s,t px 1 q ´θs,t px 1 q ˘i| ď ˇˇˇi ÿ k"2 " ż v i "s t dv i´1 . . . ˇˇ`θ v,t pxq ´θv,t px 1 q ˘jˇˇ2j´2 2j´1 `ˇ`θ v,t pxq ´θv,t px 1 q ˘i´1 ˇˇ¯d v ¯.

ż
We derive from the previous Lemma 7:

| `mx s,t px 1 q ´θs,t px 1 q ˘i| ď C `d2i´2 px, x 1 qps ´tq ``ps ´tq pi´1q´1 2 `d2pi´1q´1 px, x 1 q ˘ps ´tq ˙.

In particular, for s " t 0 " t `c0 d 2 px, x 1 q with c 0 ă 1, the previous equation yields:

| `mx t 0 ,t px 1 q ´θt 0 ,t px 1 q ˘i| ď C ´c0 d 2i´1 px, x 1 q `pc 2i´1`γ 0 `c0 qd 2i´1`γ px, x 1 q `pc i´1 2 0 `c0 qd 2i´1 px, x 1 q ¯.

Still for dpx, x 1 q ď 1, after summing and by convexity inequalities, we eventually deduce: d `mx s,t px 1 q, θ s,t px 1 q ˘ď Cc 1 2n´1 0 dpx, x 1 q. 5 Appendix: Some reminders about Besov spaces

Thermic characteristic of the Besov space

Let us now recall some definitions/characterizations from Section 2.6.4 of Triebel [START_REF] Triebel | Theory of function spaces[END_REF]. For α P R, q P p0, `8s, p P p0, 8s, B α p,q pR d q :" tf P S 1 pR d q : }f } H α p,q ă `8u where SpR d q stands for the Schwartz class and }f } H α p,q :" }ϕpDqf } L p pR d q `´ż 1 0

v pm´α 2 qq }B m v h v ‹ f } q L p pR d q dv v ¯1 q , (5.1)
with ϕ P C 8 0 pR d q (smooth function with compact support) is s.t. ϕp0q ‰ 0, ϕpDqf :" pϕ f q _ where f and pϕ f q _ respectively denote the Fourier transform of f and the inverse Fourier transform of ϕ f . The parameter m is an integer s.t. m ą α 2 and for v ą 0, z P R d , h v pzq :" 1 p2πvq d 2 exp ´´|z| 2 2v ¯is the usual heat kernel of R d . We point out that the quantities in (5.1) are well defined for q ă 8. The modifications for q " `8 are obvious and can be written passing to the limit.

Observe that the quantity }f } H s p,q , where the subscript H stands to indicate the dependence on the heat-kernel, depends on the considered function ϕ and the chosen m P N. It also defines a quasi-norm on B s p,q pR d q. The previous definition of B α p,q pR d q is known as the thermic characterization of Besov spaces and is particularly well adapted to our current framework. By abuse of notation we will write as soon as this quantity is finite }f } H α p,q ": }f } B α p,q .

Equivalence of Besov norms

Proposition 4 There is a constant C ą 1 such that for all f P B α 8,8 pR d q, α P p0, 1q we have: Abstract : In the first part of this thesis, we aim to estimate the invariant distribution of an ergodic process driven by a Stochastic Differential Equation. The ergodic theorem suggests us to consider the empirical measure associated with a discretization scheme of the process which can be regarded as a discretization of the occupation measure of the process. In [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF], Lamberton and Pagès introduced an algorithm of discretization with decreasing time steps which allows the convergence of the empirical measure toward the invariant distribution of the process, they also provide a central limit theorem (CLT) which asymptotically quantifies the deviations between these both measures. We establish nonasymptotic concentration inequality for the empirical measure deviations (in accordance with the previously mentioned CLT), and also we give some controls of the solution of the associated Poisson equation which is useful for this concentration inequalities.

C ´1}f } B α
In a second part, we establish some Schauder controls associated with parabolic equations related with a degenerate stochastic system, where the drift is a vector field satisfying a weak Hörmander condition like. But we aim to suppose only the minimal Hölder regularity. This work is an extension of the estimates given by Delarue and Menozzi [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF]. Finally, our approach allows us to proof the strong uniqueness of the considered stochastic equation in a Hölder regularity framework. Our results extend the controls of Chaudru de Raynal [START_REF] Chaudru | Strong existence and uniqueness for degenerate SDE with Hölder drift[END_REF] for the dimension equal to 2.

  γ k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A the infinitesimal generator of the diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . C β the usual (homogeneous) Hölder space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C k,β the usual (homogeneous) Hölder space with k `β as parameter . . . . C k,β b the usual (inhomogeneous) Hölder space with k `β as parameter . .

  Assumption : Chapitre 1 (C1), (GC), (C2), (L V ), (U), (S) . . . . . . 77 (A)a Assumption : Chapitre 2 (C1), (GC), (C2), (L V ), (U), (S), (T β )128 (A)b Assumption : Chapitre 3 (C0), (C1), (GC), (C2), (L V ), (U), (S), (T β ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .171 (C0) Assumption : diffusion coefficients are Lipschitz continuous . . . . . . . . 169 (C1) Assumption : X 0 is exponentially integrable . . . . . . . . . . . . . . . . . . . . . . . . 62 (C2) Assumption : Boundess of σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 (C2) Assumption : boundedness of σ and κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 (GC) Assumption : Gaussian Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 (G V ) Assumption : | ϕpxq |ď C V,ϕ p1 `aV pxqq . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 (P β ) Assumption : (UE), (D p α ), || Dσ | 2 8 ď 2α 2p1`βq´p , (R 1,β ), and (Σ ) . . . 151 (S) Assumption : small steps are small enough . . . . . . . . . . . . . . . . . . . . . . . . . 63 (T β ) Assumption : regularity of the test function ϕ . . . . . . . . . . . . . . . . . . . . . 127 (U) Assumption : uniqueness of the invariant measure of the diffusion . . 63

  |α|"k rD α ψs β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 || ¨|| C k`β b pR m ,R q the non-homogeneous Hölder norm, || ¨|| C k`β b pR m ,R q "|| ¨|| C k`β pR m ,R q `|| ¨|| L 8 pR m ,R q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 || ¨|| C k`γ d || ¨|| C k`γ d "|| ¨|| C k`γ d pR nd ,R q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205|| ¨|| C k`β b,d pR m ,R q || ¨|| C k`β b,d pR m ,R q " ř n i"1 sup zPR nd || ψ i pz,¨q || . . . . . . . . . . . . . . . . . . . . . . . . . . 205 || ¨|| L 8 || ¨|| L 8 "|| ¨|| L 8 pR nd ,R q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 τ the time freezing parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Gm,ξ the Green kernel associated with the proxy, also noted Gm,ξ . . . . . . . 210 pξ pt, s, x, ¨q the multi-scale density of the proxy, also noted pm,pτ,ξq pt, s, x, ¨q . . . . 209 P ξ T,t the semi-group associated with the proxy, also noted P m,ξ T,t . . . . . . . . . 210 Kpτ,ξq v,t the covariance matrix of the proxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Rpτ,ξq pv, tq the resolvent associated with pDFpv, θ v,τ pξqqq vPrt,ss . . . . . . . . . . . . . . . . . 220 Xv the linearized processus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 a the matrix diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 C k`β pR m , R q the non-homogeneous Hölder space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .205 C k`β b pR m , R q the homogeneous Hölder space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 DF the subdiagonal of the Jacobian matrix DF . . . . . . . . . . . . . . . . . . . . . . . 209 d the dimension of each block of the degenerate chain . . . . . . . . . . . . . . . 199 f the source of the Kolmogorov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 g the terminal condition of the Kolmogorov equation . . . . . . . . . . . . . . . . 199 L t the spatial Kolmogorov operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 n the number of levels in the degenerate chain. . . . . . . . . . . . . . . . . . . . . . .199 T the time horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 (T β ) Assumption : pF i q iPrr1,pj`1q^nss are uniformly β j -Hölder continuous in the j th spatial variable with β j P p 2j´2 2j´1 , 1s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 (H η ) Assumption : D x i´1 F i is non-degenerate and η 2i´3 Hölder continuous w.r.t. x i´1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .284 (A) Assumption : Chapter 6 (UE), (H), (S) . . . . . . . . . . . . . . . . . . . . . . . . . 207 (A) Assumption : Chapter 7 (ML), (UE), (T β ), (H η ) . . . . . . . . . . . . . . . 284 (H) Assumption : Weak Hörmander like condition . . . . . . . . . . . . . . . . . . . . . 206 (ML) Assumption : F and σ are measurable in time and σ is uniformly Lipschitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 (S) Assumption : Smoothness of the Coefficient, a P L 8 pC γ b,d q, F 1 P L 8 pC γ d q,F i P L 8 pC 2i´3`γ b,d q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 (UE) Assumption : Uniform Ellipticity of the diffusion Coefficient . . . . . . . 206

  0, alors pour toute fonction Lipschitz ϕ P C 3 pR d , Rq telle que D 2 ϕ soit bornée, D 3 ϕ soit bornée et Lipschitzienne, et sup xPR d

Theorem 3

 3 Sous les mêmes hypothèses et notations que le Théorème 2, s'il existe ϑ de même régularité que ϕ tel que Aϑ " ~σ~2 ´νp~σ~2q, pour tout n ě 1, 0 ă a ď χ n ? Γn Γ p2q n où χ n ą 0 tel que χ n Ñ n 0 (arbitrairement lentement) P " | a Γ n ν n pAϕq| ě a ‰ ď 2 r C n exp ˆ´r c n

(C R )

 R Confluence et Régularité : on suppose que (D p α ) est vérifiée, que f et σ, b sont de classe C 3`β , et que l'hypothèse de confluence forte }Dσ} 2 8 ď 2α dp2p3`βq´pq est vraie. (C U E ) Confluence et Uniforme Ellipticité : on suppose que (D p α ) est vérifiée, que f et σ, b sont de classe C 1`β , que l'hypothèse de confluence }Dσ} 2 8 ď 2α

Figure 2 . 1 -

 21 Figure 2.1 -Dessin venant de [DM10].

n `8 and pn pΓ p2q n q 2

 2 Γn Ñ n 0 in order to have, ēn Ñ n 0, Cn Ñ n 1 with the indicated monotonicity for large enough n.

  39), the definition of r k,n in (3.40), and using as well that e n -The other terms in (3.39), corresponding to sub-Gaussian tails, give the remainder R n .

  (a) If b, σ and f are smooth, under suitable confluence like conditions stated below in (D p α ) with the condition on Dσ: }Dσ} 2 8 ď 2α 2p1`βq´p for given pα, pq P p0, `8q ˆr1, 2q, the probabilistic representation ϕpxq " ´ş`8 0 E

Figure 4 . 1 -

 41 Figure 4.1 -Plot of a Þ Ñ g n paq with ϕpxq " σpxq " cospxq.

Figure 4

 4 Figure 4.1 reveals that the asymptotic curve S is much less sharp with respect to the realizations g n than our main estimations P n and P n,0,8 . In fact, these latter are very close to the realization g n . This claim enhances the significance of controlling finely, non-asymptotically, the deviation of the empirical measure.In this plot, we can see that our pick of ρ for P n,0,8 , set in Lemma 7, is very close to the numerical optimization of P n over ρ. Nevertheless, observe that for a ą 0.5, P n,0,8 paq and P n paq slightly differ. It means that progressively the regime goes from Gaussian deviations (i.e. a

For a β -

 β Hölder continuous function f : R d Ñ R, we denote by rf s β :" sup x‰x 1 |f pxq ´f px 1 q| |x ´x1 | β ă `8, its Hölder modulus of continuity. Here, |x´x 1 | stands for the Euclidean norm of x´x 1 P R d .

( C0 )

 C0 The functions b : R d Ñ R d , σ : R d Ñ R d b R r and κ : R d Ñ R d b R r are globally Lipschitz continuous.

  .13) Hence plugging (3.4), (3.10), (3.11), (3.12) and (3.13) into (3.5):

Figure 5 . 1 -

 51 Figure 5.1 -Plot of a Þ Ñ g n paq, for θ " 1 3 , with ϕpxq " σpxq " cospxq.

(

  UE) Uniform Ellipticity of the diffusion Coefficient. There exists κ ě 1 s.t. for all pt, xq P R `ˆR nd , z P R d , κ ´1|z| 2 ď xapt, xqz, zy ď κ|z| 2 , where | ¨| again denotes the Euclidean norm and x¨, ¨y is the inner product.

P

  m,ξ T,t g m pxq :" ż R nd pm,ξ pt, T, x, yqg m pyqdy,as well asGm,ξ f m pt, xq :" pt, s, x, yqf m ps, yqdy, (2.10)

  .22) Gathering (5.21) and (5.22) gives the result.

  s.t c0 :" C 0 pc

p2πaq d 2

 2 exp `´|z| 2 2a ˘is the standard Gaussian density of R d with covariance matrix aI d , and: qczi,λ pt, s, x, py 1:i´1 , y i`1:n qq " ź jPrr1,nss,j‰i N cλ 2j´1 2 ps´tq 2j´1 `pθ s,t pxq ´yq j ˘.

  pxq| β j ps ´tq ´pl´1{2q´1{2 pC ´1 pt, s, β j pj´1{2q ˆps ´tq ´pl´1{2q´1{2 pC ´1 pt, s, x, yq

  in duality, see e.g. Proposition3.6 in Lemarié-Rieusset ([LR02]). Indeed, with the notations therein B 4.1.3 in Adams Hedberg[START_REF] Adams | Function spaces and Potential Theory[END_REF] for the density of S in B

3 2

 3 ´ηpi´3 2 q

  i D k upt, x 1:i´1 , ¨, x i`1:n q} 8 `sup v ‹ D i D k upt, x 1:i´1 , ¨, x i`1:n qpzq|.

( 3 .( 3

 33 23)corresponds to the difference of the previous expansions on the off-diagonal regime,`hv ‹ D x i D x k upt, x 1:i´1 , ¨, x i`1:n qpzq ˘|S c i i D z h v pz ´xi q

  0 |px 1 ´xq i | 2 2i´1 ds ´Iξ ,k ps, xq ´Iξ ,k ps, x 1 q ¯ˇˇˇď C|px 1 ´xq i | α k i }Du} 8

|I ξ 3 ,

 3 k ps, xq ´Iξ 3,k ps, x 1 q| ď 1: ´1,y `1:n q,pt,xq ,k,m p¨q ´Ψps,y 1: ´1,y `1:n q,pt,x 1 q m ups, z 1: ´1, ¨, z `1:n q

Lemma 5 ď

 5 Let k P rr2, nss, P rr2, kss and m P rrk, nss and let Ψ ps,y 1: ´1,y `1:n q,pt,xq ,pk,iq,m : R d Ñ R d be the function defined by (2.35). There exist C :" CppAqq ą 0 and γ m :" γ m ppAqq :" 1{2 `ηp ´3{2q ą 1{2 such that › › › › Ψ ps,y 1: ´1,y `1:n q,pt,xq ,k,m p¨q ´Ψps,y 1: ´1,y `1:n q,pt,x 1 q C qcz pt, s, x, py 1: ´1, y `1:n qqps ´tq´1´pi´1 2 qα k i `γm |px ´x1 q i | α k i ,with qcz pt, s, x, py 1: ´1, y `1:n qq as in (2.40). ‡ ‡. Recall indeed that what we are able to control is precisely the Hölder moduli of the derivatives D ym ups, ¨q w.r.t. variables ď m .

ď

  sup tPr0,T s } `Di D k u ˘ipt, ¨q} B Ct}Du} 8 `}DD x 1 u} 8 u, (3.54)which concludes the proof of Lemma 4.

( 3 .

 3 56)where from (H η ), 1 `αk i pi ´1{2q ´γm ă 1. Turning to the singular in time contribution of the thermic part of the Besov norm of (3.55) we decompose with the notations of Lemma 3 (see e.g. (3.6), (3.7) which exhibit an additional spatial derivative): L " Ψ ps,y 1: ´1,y `1:n q,pt,xq ,k,m p¨q ´Ψps,y 1: ´1,y `1:n q,pt,x 1 q

Title:

  Non-asymptotic estimates of invariant measure and regularisation by a degenerate noise for a chain of Ordinary Differential Equations. Keys words : Invariant distribution, Concentration inequalities, Poisson Equation, Kolmogorov Equation, Schauder estimates, Strong uniqueness.

Titre:

  Estimations non-asymptotiques de mesures invariantes et régularisation par un bruit dégénéré de chaînes d'Équations Différentielles Ordinaires. Mots Clefs : Mesure invariante, Inégalités de concentration, Équation de Poisson, Équation de Kolomogorov, Estimées de Schauder, Unicité forte. Résumé : Dans la première partie de cette thèse, nous chercherons à estimer la mesure invariante d'un processus ergodique dirigé par une Équation Différentielle Stochastique. Le théorème ergodique nous suggère de considérer la mesure empirique associée à un schéma d'approximation du processus sous-jacent qui peut se voir comme le pendant discret de la mesure d'occupation dudit processus. Dans [LP02], Lamberton et Pagès ont introduit un algorithme de discrétisation à pas décroissant qui assure la convergence de la mesure empirique du schéma vers la mesure invariante du processus considéré ainsi qu'un théorème central limite (TCL) quantifiant asymptotiquement l'écart entre ces deux mesures. Nous établissons des inégalités de concentration non-asymptotiques pour les déviations de la mesure empirique (cohérentes avec le TCL mentionné ci-avant), ainsi que des contrôles sur la solution de l'équation de Poisson associée, utiles pour ces inégalités. Dans une seconde partie, nous établissons des estimées de Schauder liées à des équations paraboliques associées à un système stochastique dégénéré, où la dérive est un champ de vecteurs vérifiant une condition de type Hörmander (faible) mais en cherchant la régularité Hölder minimale. Ce travail fait suite à l'article de Delarue et Menozzi [DM10]. Enfin, notre approche nous permet de montrer l'unicité forte du système stochastique considéré dans le cadre de coefficients Hölder, étendant ainsi le résultat obtenu en dimension 2 par Chaudru de Raynal [CdR17].

  Équation de Kolmogorov et chaîne dégénérée . . . . . . . . . . . . . 1.2 Présentation du modèle . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Propriété d'échelles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Résultats existants . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Estimées de Schauder établies . . . . . . . . . . . . . . . . . . . . . 1.6 Grandes lignes de l'analyse . . . . . . . . . . . . . . . . . . . . . . . 1.7 Contrôle de la norme uniforme . . . . . . . . . . . . . . . . . . . . . k q kě1 the step sequence of the scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pU k q kě1 the discretized Wiener increment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pX k q kě0 the discretized process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Liste des Symboles
	Partie III
	Table des figures
	I Introduction
	p q n

1 Estimations de mesures invariantes 1 Processus Ergodiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Mesure invariante . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Théorèmes Ergodiques . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Discrétisation du processus . . . . . . . . . . . . . . . . . . . . . . . 1.4 Inégalités de concentration . . . . . . . . . . . . . . . . . . . . . . . 1.5 De ν n pAϕq à l'estimation de νpf q pour une source f donnée . . . . 2 Premier résultat de concentration . . . . . . . . . . . . . . . . . . . . . . . 2.1 Améliorations du Théorème 2 . . . . . . . . . . . . . . . . . . . . . 2.2 Sur le problème de Poisson . . . . . . . . . . . . . . . . . . . . . . . 2.3 Résultat de type Slutsky . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Source Lipschitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Inégalité de concentration optimale . . . . . . . . . . . . . . . . . . . . . . 3.1 Source Lipschitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Extension aux EDS dirigées par un Lévy . . . . . . . . . . . . . . . . . . . 2 Régularisation par un bruit dégénéré 1 Régularisation par un bruit dégénéré . . . . . . . . . . . . . . . . . . . . . 1.1 1.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Unicité forte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pγ rf s β the usual Hölder modulus, rf s β :" sup x‰x 1 |f pxq´f px 1 q| |x´x 1 | β . . . . . . . . . . . . . . . . Γ n the current time of the scheme, Γ n :" ř n k"1 γ k . . . . . . . . . . . . . . . . . . . . . . Γ

  σ ˚∇ϕpxq | 2 q the "carré du champ " pour la diffusion Brownienne . . . . . . . . . . . . . . . . σσ ˚pxq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R d `| σ ˚∇ϕ | 2 pxq `şR d | ϕpx `κpxqyq ´ϕpxq | 2 πpdyq ˘νpdxq. . . . . .172 ~¨~any matrix norm dominating the operator norm . . . . . . . . . . . . . . . . . . . . 71 B α p,q pR d q the Besov space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 e n a generic sequence s.t. e n " e n ppAqq Ñ n 0 that may change from line to line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 h v pzq the usual heat kernel space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .213 V Lyapunov function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 (L V ) Assumption : Lyapunov assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72 (C UE ) Assumption : (mild) confluence and non-degeneracy, pD p α q, pR 1,β q, || Dσ || 2 8 ď 2α 2p1`βq´p , and Σ i,j pxq " Σ i,j px i^j , ¨¨¨, x d q . . . . . . . . . . . . . . . . . 73 (D p

	(C R )	Assumption : (strong) confluence and regularity, pD p α q, pR 3,β q, and
		|| Dσ || 2 8 ď	2α 2p3`βq´p
	ν n	the empirical measure of the scheme, ν n p¨q :"	ř n k"1 γ k δ X k´1 pωq p¨q ř n k"1 γ k	. . . . . . .
	Σ Σpxq :" σ 2

n a generic sequence s.t. R n " R n ppAqq Ñ n 1 that may change from line to line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . || σpxq || F the Fröbenius norm of σpxq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . || B α p,q pR d q || the Besov norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . || ¨|| 8 the uniform norm associated with Euclidian or operator norm . . . . . . || σpxq || the operator norm of σpxq, || σpxq ||" sup zPR r ,|z|ď1 | σpxqz | . . . . . . . . . . | ¨| the Euclidian norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ν the invariant measure of the diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . νp| ϕ the "carré du champ " associated with the diffusion with jumps, σ 2 ϕ :" ş α ) Assumption : confluence condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 (R k,β ) Assumption : smoothness of the coefficients and the source . . . . . . . . . 72 (UE) Assumption : uniform ellipticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

  . . . . . . . . . . . . . . . . . . . . . 151 sup x‰x 1 ,px,x 1 qPpR nd q 2 |ψpxq´ψpx 1 q| d γ px,x 1 q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 pξ C ´1 pt, s, x, yq pξ C ´1 pt, s, x, yq " exp `´C ´1ps ´tq | T ´1 s´t pm ξ s,t pxq ´yq | 2 ˘, also noted ppτ,ξq C ´1 pt, s, x, yq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 T t intrinsic scale matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 θ v,τ pξq the flow associated with the drift, 9 θ v,τ pξq " Fpv, θ v,τ pξqq, θ τ,τ pξq " ξ209 ξ the spatial freezing parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 D the full spatial gradient, D " pD x 1 , ¨¨¨, D xn q . . . . . . . . . . . . . . . . . . . . . . 199 xq the drift at the i th level, F i pt, xq " F i pt, x i´1:n q . . . . . . . . . . . . . . . . . . . . 199

		C
		ps´tq	n 2 d 2
	d	the parabolic pseudo-distance, d `x, y	˘" ř d i"1 | y i ´xi |	1 2i´1 . . . . . . . 204
	F i pt, m ξ s,t pxq	the mean of the proxy process, also noted m
	Partie IIII
	Π x i pψqpzq	Π x

i pψqpzq " ψpz 1 , ¨¨¨, z i `x, ¨¨¨, z n q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 ψ i pz, xq ψ i pz, xq " Π x i pψqpzq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 rψs γ d pτ,ξq s,t pxq . . . . . . . . . . . . . . . 209 x i:n x i´1:n :" px i´1 , ¨¨¨, x n q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 X t the diffusion associated with the Kolmogorov operator . . . . . . . . . . . . 200 | ¨|C k`β pR m ,R q the Euclidean norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

  Dessin venant de [DM10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1 Unbiased Case. Plot of a Þ Ñ g n,θ paq, for pθ k q kPrr1,5ss , with ϕpxq " σpxq " x `ε cospxq, ε " 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Biased Case. Plot of a Þ Ñ g n,θ paq, for θ 0 " 1 3 , with ϕpxq " σpxq " cospxq. . . 3.3 Plot of a Þ Ñ g n,θ paq, for pθ k q kPrr1,5ss , with ϕpxq " σpxq " cospxq. . . . . . . . . 3.4 Plot of a Þ Ñ g n paq with f pxq " |x| β 1`|x| β , β " .5. . . . . . . . . . . . . . . . . .

	Table des figures
	2.1

4.1 Plot of a Þ Ñ g n paq with ϕpxq " σpxq " cospxq. . . . . . . . . . . . . . . . . . 5.1 Plot of a Þ Ñ g n paq, for θ " 1 3 , with ϕpxq " σpxq " cospxq. . . . . . . . . . .

  Γn ˘avec p n une suite ě 1 telle que p n Ñ n Remarquons que la variance de la borne supérieure de la déviation est une majoration du carré du champ. En effet, νp|σ ˚∇ϕ| 2 q " ş |σ ˚∇ϕ| 2 pxqνpdxq ď }σ} 2 8 }∇ϕ} 2 8 , avec }σ} 8 :" sup xPR d ~σpxq~où ~¨~désigne une norme matricielle. On peut considérer typiquement la norme d'opérateur, i.e. ~σpxq~O :" sup ξPR d |σpxqξ| |ξ| , en notant par | ¨| la norme Euclidienne. Celle-ci est indépendante de la dimension mais potentiellement délicate à estimer. Un autre choix possible est la norme de Fröbenius, ~σpxq~F " ř 1ďiďd,1ďjďr σ 2 ij pxq, qui est facile à calculer mais dépend de la dimension. De façon similaire, on note }∇ϕ} 8 :" sup xPR d |∇ϕpxq|.

					ˆ´c n	2}σ} 2 8 }∇ϕ} 2 8 a 2	˙,
	où C n " exp	`rD 3 ϕs β }σ} 3`β 8 Er|U 1 | 3`β s p1`βqp2`βqp3`βq	Γ p n 3`β 2 q ? Γn `pn	Γ p2q n ?	`8
	et p n	Γ p2q n ? Γn Ñ n 0.

  On peut choisir p " ppnq Ñ n `8 tel que Erexpp λpRn Γn qs 1{p " R n Ñ n 1. Dans l'inégalité précédente, en écrivant Ere

				´λqMn Γn s " E	" e	´λqM n´1 Γn	Ere	λq∆npX n´1 ,Unq Γn	|F n´1 s ‰	et en utilisant
	(GC), il vient :
		P	" a	Γ n ν n pAϕq ě a	‰	ď R n expp´λ a ? Γ n	qErexpp´λ qM n Γ n	qs 1{q
	pGCq ď R n expp´λ a ? Γ n	`λ2 qγ n }σ} 2 8 }∇ϕ} 2 8 2Γ 2 n	qE "	expp´λ qM n´1 Γ n	q ‰ 1{q .
	Après itération, on obtient facilement que :
	P	" a Γ n ν n pAϕq| ě a	‰	ď expp´λ a ? Γ n	`qλ 2 }σ} 2 8 }∇ϕ} 2 8 2Γ n	q.
				expp´λ a ? Γ n	qErexpp´λ qM n Γ n	qs 1{q Erexpp	λpR n Γ n	qs 1{p .

  BB06]) permet d'avoir une borne du gradient. Néanmoins, sous ce jeu d'hypothèses il semble compliqué d'avoir des contrôles plus fins, en premier lieu le module Hölder du gradient. En outre, cette condition est difficile à vérifier en pratique, même pour le cas scalaire d " 1. Nous utiliserons par la suite une hypothèse de type confluence : (D α

	2 ξ, ξ E	`1 2	r ÿ j"1	´pp ´2q	|xDσ ¨j pxqξ, ξy| 2 |ξ| 2	`|Dσ

p ) Supposons qu'il existe α ą 0 et p P p1, 2s tel que pour tout x P R d , ξ P R d A Dbpxq `Dbpxq ¨j ξ| 2 ¯ď ´α|ξ| 2 ,

4

  Extension aux EDS dirigées par un Lévy On considère maintenant, au lieu de la diffusion (1.1), une EDS avec un processus de Lévy : dX t " bpX t qdt `σpX t qdW t `κpX t ´qdZ t , (4.28)

où pZ t q tě0 un processus de Lévy de carré intégrable à valeur dans R r (par souci de simplicité), et κ : R d Ñ R d b R r une fonction Lipschitz. L'existence de mesure invariante a été montrée par Panloup dans

[START_REF] Panloup | Recursive computation of the invariant measure of a stochastic differential equation driven by a lévy process[END_REF]

. L'existence et l'unicité ont été établies dans

[START_REF] Masuda | Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps[END_REF]

sous hypothèse d'irréductibilité. Les premiers à avoir proposé un schéma de discrétisation d'une EDS à saut furent Protter et Talay dans l'article

[START_REF] Protter | The Euler scheme for lévy driven stochastic differential equations[END_REF]

, où ils établirent l'erreur faible associée à un schéma d'Euler dit «exact» (avec les vrais accroissements de processus de Lévy) pour un algorithme de Monte Carlo. Par la suite, Panloup a proposé différents schémas qui permettent la simulation du processus dirigé par tout Lévy de carré intégrable.

  ) avec σ tel que a " σσ ˚. Le caractère dégénéré de la diffusion associée est d'autant plus évident sous cette forme. Nous noterons également le générateur infinitésimal associé : pL t q tPr0,T s , pour toute fonction ϕ P C 2 0 pR nd , Rq :

	L t ϕpxq " xFpt, xq, Dϕpxqy	`1 2	Tr `D2

x 1 ϕpxqapt, xq ˘. (1.5) Par la suite, nous supposerons des hypothèses de régularité (Hölder appropriée) et de non-dégénérescence (qui conduit à des hypothèses de type Hörmander faible) sur les composants a, F ce qui implique que le problème martingale associé à (1.5) est bien posé, voir par exemple Menozzi [Men11] (dérive Lipschitz et σ Hölder), [Men18] (σ continu et dérive Lipschitz), et Chaudru de Raynal et Menozzi [CdRM17] (dérive et σ Hölder à coefficient «sharp») et montre l'unicité faible de (1.33).

  C|Aζ| 2 . Cette échelle rend compte de l'homogénéité qu'il y a entre X 1 t et le mouvement Brownien, et X 2 t et l'intégrale du Brownien.Dans un cadre plus général, Lunardi[START_REF] Lunardi | Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in R n[END_REF] a montré des estimées de Schauder associées à (1.2), lorsque a a une régularité Hölder appropriée (notée (S) plus loin) et est uniformément elliptique. Notamment, elle a considéré une chaîne dégénérée complète avec pour tout n P N donné et une dérive linéaire Fpxq " Ax, avec

	A "	¨a1,1 ¨¨¨¨¨¨¨¨¨a 1,n a 2,1 ¨¨¨¨¨¨¨¨¨a 2,n 0 d,d a 3,2 ¨¨¨¨¨¨a 3,n . . . 0 d,d . . . . . .	‹ ‹ ‹ ‹ ‹ '
		0 d,d ¨¨¨0 d,d a n,n´1 a n,n
	X t " x ˙dW K `ż t 0 AX s ds `ż t 0 ˆσpX s q 0 0 0 1{2 t -t 1{2 T t " ¨t1{2 I d,d 0 d,d ¨¨¨0 d,d 0 d,d t 3{2 I d,d . . . . . . . . . . . . 0 d,d . . .	' ‹ ‹ ‹ .	(1.7)
		0 d,d	¨¨¨0 d,d t p2n´1q{2 I d,d

6)

Pour des matrices A, B P R nd b R nd , la notation A -B signifie qu'il existe une constante C ě 1 telle que pour tout ζ P R nd , on a C ´1|Aζ| 2 ď |Bζ| 2 ď , où pa i,j q ijPrr1,nss 2 P R d b R d sont tels que `ai,i´1 ˘P GL d pRq, i P rr2, nss 2 . En particulier, remarquons que la matrice A est compatible avec la structure de la dérive dans (1.3), ce qui implique que le système vérifie des hypothèses de type Hörmander faibles. Enfin, le processus s est contrôlé par un processus Gaussien en termes de densité pour les marginales (cf

[START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF]

,

[START_REF] Menozzi | Parametrix techniques and martingale problems for some degenerate Kolmogorov equations[END_REF]

), avec comme matrice de covariance K t vérifiant également la propriété de «bonne échelle». Autrement dit, on a : Insistons encore une fois sur le fait que cette matrice rend compte des échelles intrinsèques des composants du processus, tout comme pour l'exemple de Kolmogorov. Les coefficients diagonaux reflètent l'homogénéité en temps des intégrales itérées du mouvement Brownien.

Un modèle typique du cas considéré par Lunardi a pour opérateur correspondant dans l'équation (1.2) :

  UE) Uniforme Ellipticité du coefficient de diffusion. Il existe κ ě 1 tel que pour tout pt, xq P R `ˆR nd , z P R d , κ ´1|z| 2 ď xapt, xqz, zy ď κ|z| 2 . (H) Condition du type Hörmander faible. Pour tout P rr2, nss, et pt, xq

Hypothèses

Tout d'abord, présentons le cadre d'hypothèses qui nous permet d'avoir le pendant du Théorème 1 dans le cas d'une dérive plus générale présenté dans (1.2).

(

  Rpτ,ξq pv, uqBapu, θ u,τ pξqqB ˚R pτ,ξq pv, uq ˚du.Nous montrons qu'il existe C ą 0 tel que pour tout multi-indice α " pα 1 , ¨¨¨, α n q P N

	En particulier,	Xpτ,ξq
								ż v
				Kpτ,ξq v,t :"
									t
	La covariance satisfait la «bonne échelle»	Kpτ,ξq v,t -pv ´tq ´1T 2 v´t où T u est définie en (1.7).
	Le processus gelé,	Xpτ,ξq s	a une densité Gaussienne (multi-échelle) qui s'écrit :
			ppτ,ξq pt, s, x, yq
		:"	p2πq	nd 2 detp 1	Kpτ,ξq s,t q	1 2	exp	2 ˆ´1	A p	Kpτ,ξq s,t q ´1pm pτ,ξq s,t pxq ´yq, m pτ,ξq s,t pxq	´yE	˙.
												2 d	exp ´´C ´1ps ´tq ˇˇT ´1 s´t	`mpτ,ξq s,t pxq	´y˘ˇˇ2 ":
												2
				C ř n ps ´tq i"1 α i pi´1{2q	ppτ,ξq
		Xpτ,ξq
	Le contrôle des normes associées au semi-groupe	P pτ,ξq T,t	: }	P pτ,ξq T,t g} L 8 , }D x 1	P pτ,ξq T,t g} L 8 ,
	}D 2 x 1	P pτ,ξq T,t g} L	pτ,ξq v,t pxq	`ż v
	et m pτ,ξq v,t pxq correspond à la moyenne de	Xpτ,ξq

v " Rpτ,ξq pv, tqx `ż v t Rpτ,ξq pv, uq ´Fpu, θ u,τ pξqq ´DFpu, θ u,τ pξqqθ u,τ pξq ¯du `ż v t Rpτ,ξq pv, uqBσpu, θ u,τ pξqqdW u ": m t Rpτ,ξq pv, uqBσpu, θ u,τ pξqqdW u , où Rpτ,ξq pv, uq est la résolvante associée à DFpv, θ v,τ pξqq, c'est-à-dire : B v Rpτ,ξq pv, tq " DFpv, θ v,τ pξqq Rpτ,ξq pv, tq, Rpτ,ξq pt, tq " I ndˆnd , v qui est affine au point de départ x. Remarquons que finalement ce choix de proxy est naturel pour l'étude du terme de reste de (1.15). En effet, le flot, θ v,τ pξq rend compte du transport de ξ (qui sera pris égal à x) à y. v est un processus Gaussien dont la matrice de covariance est : n , |α| ď 3, avec la dérivée multi-indice D α x :" D α 1 x 1 ¨¨¨D αn xn , telle que : |D α x ppτ,ξq pt, s, x, yq| ď C ps ´tq

ř n i"1 α i pi´1 2 q`n C ´1 pt, s, x, yq.

(1.18)

Autrement dit, dériver la densité gelée induit des singularités en temps correspondant aux échelles intrinsèques des indices de dérivation. Le travail qui suit consiste principalement à rattraper ces singularités temporelles par la régularité spatiale.

1.7 Contrôle de la norme uniforme

Contrôle de la norme uniforme des premiers termes Après de potentielles dérivées suivant x, nous choisissons le point de gel pτ, ξq " pt, xq, ainsi ce choix ne change pas les dérivées intervenant dans (1.15). En particulier, nous pouvons voir grâce à la dynamique (1.17) qu'avec ce choix de gel m pt,xq v,t pxq " θ v,t pξq. 8 , et associées au noyau de Green Gpτ,ξq

  ,T sˆR nd ppτ,ξq pt, s, x, yqdy " 0, c'est ce qu'on appelle en effet le principe de cancellation. Or comme f P C γ d pR nd , Rq et par (1.18), on déduit : La dernière inégalité vient du fait que pour tout ζ P R `il existe C ζ ě 1 tel que pour tout 0 ď β ď ζ et x P R

	La deuxième égalité vient du faite que	ş R nd D 2 x 1
		}D 2 x 1	Gpτ,ξq f } L 8
		ď Crf s γ d	sup pt,xqPr0,T sˆR nd	|	ż T t	ds	ż	R nd	ppτ,ξq C ´1 pt, s, x, yq s ´t	d γ `y, m pτ,ξq s,t pxq ˘dy	ˇˇˇˇˇp τ,ξq"pt,xq
		ď Crf s γ d	sup pt,xqPr0,T sˆR nd	|	ż T t	ds	ż R nd	ppτ,ξq C ´1 pt, s, x, yq ps ´tq 1´γ 2	dy	ˇˇˇˇˇp τ,ξq"pt,xq
		" Crf s γ d pT ´tq	γ 2 .			(1.19)
					ˇˇż T t	ds	ż R nd	D 2 x 1	ppτ,ξq pt, s, x, yqf ps, yqdy	ˇˇˇˇˇp	τ,ξq"pt,xq
						ż T		ż
	"	sup pt,xqPr0,T sˆR nd	|	t		ds	R nd	D 2 x 1

ppτ,ξq pt, s, x, yqrf ps, yq ´f ps, m pτ,ξq s,t pxqsdy ˇˇˇˇˇp τ,ξq"pt,xq . `:

  Cela s'explique par le fait que la dérive F i , à chaque étage va être amené à jouer le rôle de source dans l'équation de Kolmogorov associé à la transformation deZvonkin (voir (1.36) plus loin) et que les contrôles que l'on souhaite obtenir soient sans échelle, i.e. du type |X s ´X1 s | comme dans (1.35). Autrement dit, l'étage de la dérive doit fournir la même régularité. Enfin, remarquons que l'on retrouve le même type de régularité supposée dans le cas des estimées de Schauder pour γ " 1 `η. Pour σ non-dégénéré et F avec les régularités précédemment supposées, il existe une unique solution forte au système (1.33). Soit U la solution au problème de Cauchy suivant : # B t Upt, xq `xFpt, xq, DUpt, xqy `1 2 Tr `D2x 1 Upt, xqapt, xq ˘" ´Fpt, xq, UpT, xq " 0, x " px 1 , ¨¨¨, x n q P R nd .

	L'idée principale de la transformation de Zvonkin est de supprimer le terme de dérive en écrivant la formule d'Itô : ż t 0 Fps, X s qds " ´Um p0, xq `Um pt, X t q ´ż t 0 DU m ps, X s qqBσpt, X t qdW s `Rm t pXq, (1.36) 0 0 Theorem 3 Transformation de Zvonkin avec R m t pXq :" ż t Fps, X s q ´Fm ps, X s qds ´ż t pL s ´Lm

s qU m ps, X s qds. Rappelons que l'indice m signifie que l'on a régularisé, de sorte que le terme de reste R m t pXq p.s. ÝÑ mÑ`8 0. Ainsi, écrivons : X s ´Um ps, X s q " ´Um p0, xq ´ż t 0 DU m ps, X s qqBσps, X s qdW s `Rm t pXq, et l'étude de l'équation de Kolmogorov (1.36), avec une étude similaire à celle menée pour (1.2) et détaillée dans la partie 1.9, permet d'établir le contrôle ||DU m || 8 `||DpDU m Bq|| 8 ď C T . (1.37) Notons toutefois que contrairement à (1.2) la source F n'est pas bornée. Ce qui ne gène pas ici étant donné que nous n'avons besoin que des bornes sup des dérivées de U. Ensuite, en reprenant X et X 1 satisfaisant (1.34) et par (1.36), on écrit

  .39) Notons que les inégalités voulues sont pour les dérivées de U m , ainsi le caractère non borné de la source F ne posera pas de problème à l'analyse. Notamment, les normes Besov qui seront considérées seront des normes non-homogènes. Ce qui rend les propriétés de dualité bien plus simple.Chaque composante u m i , i P rr1, nss de la solution U m de (1.36) s'écrit pour tout pt, xq P r0, T s ˆRnd

		ż T		ż			!
	u m i pt, xq "	t	ds	R nd	dy	´Fm i ps, yq `pL m s ´L m s qu m i ps, yq
	u i pt, xq "	´ż T	"	P pτ,ξq s,t F i ps,	¨qı	pxqds	`ż T	"	P pτ,ξq s,t	´pL s ´L ξ s qu i ¯ps,	¨qı	pxq,
				t									t
	et par définition des opérateurs Ls et L s on a :
	u i pt, xq "	´ż T	"	P pτ,ξq s,t F i ps,	¨qı	pxqds
					t
				`ż T	"	P pτ,ξq s,t					ı	pxq
					t
				`ż T t	"	P pτ,ξq s,t	ˆ1 2	Tr	"	aps, ¨q ´aps, θ s,t pξqqD 2 1 u i ps,	¨q‰	˙ pxqds (1.41)
				`ż T	«	P pτ,ξq s,t	˜n ÿ	pF
					t				k"2

)

ppτ,ξq pt, s, x, yq, (1.40) avec la même méthode de linéarisation que pour le chapitre précédent sur les estimées de Schauder.

Dit autrement, la formule de Duhamel (ou le développement parametrix du premier ordre) donne : ppF 1 ps, ¨q ´F1 ps, θ s,t pξqqqD 1 u i ps, ¨qq k ps, ¨q ´Fk ps, θ s,t pξqq ´Dx k´1 F k ps, θ s,t pξqqp¨´θ s,t pξqq k´1 qD k u i ps, ¨q¸ff pxqds,

On montre par des méthodes de cancellation (partielles) et par un choix de point de gel convenable (voir les estimées de Schauder) que

  En revanche, pour le terme correspondant aux dérivées D k u i , k P rr2, nss, l'analyse doit être plus subtile.On développe d'abord les l ´1 premières composantes des termes de dérive autour du point de gel θ s,t pξq. C'est-à-dire, on écrit : F k ps, ¨q ´Fk pθ s,t pξqq ´Dk´1 F k ps, θ s,t pξqqp¨´θ s,t pξqq k´1 " rF k ps, ¨q ´Fi ps, ¨1:l´1 , θ l:n s,t pξqqs `rF k ps, ¨1:l´1 , θ ˆ´F k ps, ¨1:l´1 , θ l:n s,t pξqq ´Fk ps, θ s,t pξqq ´Dx k´1 F k ps, θ s,t pξqq `¨´θ s,t pξq ˘k´1

	ξq P ξ ˆ1 2 s,t Dk Tr " aps, ¨q ´aps, θ s,t pξqqD 2 1 u i ps, ¨q‰ ˙ pxqds › › › 8 ď C T . "x D j u i ps, ¨1:l´1 , ¨l:n `λp¨l :n s,t " P ξ s,t `l ÿ " ˇˇD x l D x 1 " ˇˇˇˇD x l D x 1 P ξ s,t ˆl ÿ k"2 ż 1 ˇˇˇˇl ÿ k"2 n ÿ j"l ż 1 0 dλD x l D x 1 " ´θl:n s,t pξqqqp¨´θ s,t pξqq j ˙pxq ˇˇˇď l ÿ k"2 n ÿ j"l ˇˇˇˇż 1 0 dλ ż R pn´1qd dpy 1:i´1 , y i`1:n q # " dλD que : ˆżR d dy k ps,y 1:k´1 ,y k`1:n q,pt,xq Ψ k,pl,1q,j
	j"l	0

l:n s,t pξqq ´Fk pθ s,t pξqq ´Dk´1 F k ps, θ s,t pξqqp¨´θ s,t pξqq k´1 qs Le premier terme permet de rattraper la singularité de D l D 1 pξ pt, s, x, yq. Puis, formellement, on développe on fait un développement de Taylor à l'ordre 2 de u à l'aide d'une cancellation partielle :

k"2 pF k ps, ¨1:l´1 , θ l:n s,t pξqq ´Fk pθ s,t pξqq ´Dk´1 F k ps, θ s,t pξqqp¨´θ s,t pξqq k´1 qD k u i ps, ¨q˘ı pxq ˇˇˇˇˇˇξ pF k ps, ¨1:l´1 , θ l:n s,t pξqq´F k ps, θ s,t pξqq´D x k´1 F k pθ s,t pξqq `¨´θ s,t pξq ˘k´1 q ˆn ÿ j D k u i ps, ¨1:l´1 , ¨l:n `λp¨l :n ´θl:n s,t pξqqqp¨´θ s,t pξqq j ˙pxq ˇˇˇˇ.

(1.42)

Notons en effet, par la structure triangulaire de DF on a pour tout l P rr2, nss, k P rr2, lss, et j P rrl, nss :

D x l D x 1 "

P ξ s,t ˆpF k ps, ¨1:l´1 , θ l:n s,t pξqq ´Fk ps, θ s,t pξqq ´Dx k´1 F k pθ s,t pξqq `¨´θ s,t pξq ˘k´1 q ˆDj D k u i ps, ¨1:l´1 , θ l:n s,t pξqqq ˙pxq " 0 d,d,d,d .

On ne s'attend pas à ce que toutes les dérivées de u i existent, néanmoins l'identité (1.42) a un sens d'un point de vue de la dualité Besov. C'est-à-dire, que l'on montre maintenant

?

  Γn " 0 and E r|U 1 | 6 s ă `8, then, for any Lipschitz continuous function ϕ in C 3 pR d , Rq with D 2 ϕ and D 3 ϕ bounded, one has (with pLq denoting weak convergence)

	paq Fast decreasing step. If lim n	Γ p2q n
			a Γ n ν n pAϕq	pLq ÝÑ N ˆ0,	ż	|σ ˚∇ϕ| 2 dν	˙.
							R d
	pbq Critical and slowly decreasing step. If lim n	Γ ? p2q n Γn " r γ Ps0, `8s and if E r|U 1 | 8 s ă
	`8, then for every Lipschitz continuous function ϕ P C 4 pR d , Rq with pD i ϕq iPt2,3,4u
	bounded:
	a Γ n ν n pAϕq	pLq ÝÑ N ˆr γm,	ż	|σ ˚∇ϕ| 2 dν ˙if r γ ă `8, pcritical decreasing stepq
						R d
	Γ n Γ n p2q	ν n pAϕq	P ÝÑ m if r γ " `8, pslowly decreasing stepq,
	where	m :"	´żR d ´1 2	D 2 ϕpxqbpxq b2 `Φ4 pxq ¯νpdxq,
	with	Φ 4 pxq :"	ż R r ´1 2	xD 3 ϕpxqbpxq, pσpxquq b2 y	`1 24	D 4 ϕpxqpσpxquq b4 ¯µpduq

  3,β pR d , Rq, β P p0, 1q centered w.r.t. ν.

			Furthermore, the
	following gradient bound holds
	}∇ϕ} 8 ď	rf s 1 α	,
	and the mapping x Þ Ñ x∇ϕpxq, bpxqy is Lipschitz continuous.

  if the diffusion matrix is scalar diagonal, i.e. Σpxq " ςpxqI d , x P R d , where ς is real valued. In that case, it is then shown that (BE ρ ) holds if and only if:

			´1 2	xpM pxq `M ˚pxqqξ, ξy ď ´ρςpxq|ξ| 2 ,	(2.6)
	where
	M pxq "	1 2	´ςpxq∆ςpxq `xbpxq, ∇ςpxqy ´}∇ςpxq} 2	¯Id	`´1 2 ´d 4	¯∇ς b ∇ςpxq ´ςpxq 2 Dbpxq.

  2,n Γn . Also, for D 3 ϕ P C 1 , B n,1 " Ḡn`D2,n ?

	Γn	Ñ

n ´r γm introduced in Theorem 1. For β P p0, 1q and ϕ P C 3 pR d , Rq, rϕ p3q s β ă `8, the only term giving a bias is B n,β " E β n " Ḡn ? Γn .

  .1) In the above definition, for a bounded mapping ζ : R d Ñ R m , m P t1, d, d ˆru, we write }ζ} 8 :" sup xPR d }ζζ ˚pxq} with }ζpxq} " Tr pζζ ˚pxqq 1{2

  b) Critical decreasing step. If θ " 1 3 and if Er|U 1 | 8 s ă `8, then for all function ϕ P C 3,1 pR d , Rq X C 4 pR d , Rq, one gets: Er|U 1 | 8 s ă `8, then for all globally Lipschitz function ϕ P C 3,1 pR d , Rq X C 4 pR d , Rq, one gets: It is possible to relax the boundedness condition on σ in (C2), considering |σ ˚∇ϕpxq| 2

	where
	r γ :" lim nÑ`8	Γ p2q n ? Γ n	, m :"	´żR d	´Tr	`1 2	D 2 ϕpxqbpxq b2 ˘`Φ 4 pxq ¯νpdxq,
	Φ 4 pxq :"	ż R r	Tr	´1 2	D 3 ϕpxqbpxqpσpxquq b2 `1 24	D 4 ϕpxqpσpxquq b4 ¯µpduq,
	recalling that µ denotes the law of the i.i.d. innovations pU k q kě1	† .
	(c) Slowly decreasing step. If θ P p0, 1 3 q and if Γ n Γ n p2q ν n pAϕq
								V pxq	|x|Ñ`8 Ñ 0
	(strictly sublinear diffusion) in case (a) and sup xPR d	|σ ˚∇ϕpxq| 2
								ż
								|σ ˚∇ϕ| 2 dν	˘,
								R d

a Γ n ν n pAϕq L ÝÑ nÑ8 N `r γm, P ÝÑ m.

  )where x `" maxpx, 0q and cn :" ´rϕs 1 Observe that two regimes compete in the above bound. From now on, we refer to Gaussian deviations when a

	rϑs 1	¯2{3	νp}σ} 2 q}σ} ´2{3 8 čn with čn being an explicit non-
	negative sequence s.t. čn Ó n 1.
	Remark 4 ? Γn Ñ n	0. In this case, asymptotically the right hand side
	of the inequality (2.7) is 2 exp `´a 2 2νp}σ} 2 q}∇ϕ} 2 8

  A first approach in Chapter 3 in order to iterate the estimates involving the conditional expectations, consisted in bounding uniformly rψpX n´1 , ¨qs 1 ď ? γ n }σ} 8 }∇ϕ} 8 (which is easily deduced from (3.2)). Iterating the procedure led to the estimate

	Pr a Γ n ν n pAϕq ě as ď exp	`´aλ ? Γ n	˘exp	`qλ 2 2Γ n	}σ} 2 8 }∇ϕ} 2 8 ˘Rn .	(3.7)
	Optimizing over λ, letting as well q Ó n 1 in a suitable way, gives the deviation upper-bound a 2 C n expp´c n 2}σ} 2 8 }∇ϕ} 2 8
					n 2Γ 2	¨qs 2 1	˘ı.	(3.6)

  3 8 }∇ϕ} 8 }D 2 ϕ} 8 , C 2,(3.28) :" 2}σ} 4 8 }D 2 ϕ} 2 8 , and C (3.8) :" 6}σ} 2 8 }∇ϕ} 8 }D 2 ϕ} 8 . The last inequality above is a consequence of convexity inequality (i.e. for all px, yq P R 2 , px `yq 2 ď 2x 2 `2y 2 ).

  Lemma 4 (Choice of ρ for the Gaussian concentration regime) Bhouououououou For P min pa, Γ n , ρq as in (3.34), there is ρ :" ρpn, aq ą 1 s.t. If a

	? Γn Ñ n	0, taking ρ´1 -a ? Γn
	P min pa, Γ n , ρq "
		.34)
	Then, from (3.32):

P `aΓ n ν n pAϕq ě a ˘ď C n exp `Pmin pa, Γ n , ρq ˘,

(3.35)

which is exactly the same bound appearing in Remark 11 in Chapter 3, up to a modification of Ãn , containing here the expected carré du champ.

The optimization over λ leads to study how ρ should asymptotically behave. The following lemma indicates that, when a " op ? Γ n q, taking ρ ´1 -a ? Γn yields a Gaussian concentration inequality in (3.32) with the optimal constant. a ?

  3 2 2 4 f Ψ pξq Ñ Controls of the optimized parameters for Gaussian deviations)

							a ? Γn Ñ n	0	1 4 . This would indeed yield
	P pλ n q " a ? Γn Ñ n	0	´a2 4 Ãn	. From the definition of Ψ in (6.13), we have Ψ " 27 Bna 2 8 ÃnΓn	ÝÑ ? Γn Ñ0 a	0.
	Observe then, that, taking ξ going to infinity such that
							ξΨ Ñ ? Γn Ñ0 a	0	(4.15)
	yields Λpξq " 3 2 2 4	gpξq Ψξ`1	Ñ a ? Γn Ñ0	1 4 , noting from (4.11) and the above definition of g that
	gpξq Ñ ξÑ`8 `2 3 ˘2.
	Remark 9 (

  13 For super Gaussian deviations, we obtain a sharper bound than in Theorem 2. Nonetheless, asymptotically, this regime is less sharp than Theorem 2 in Chapter 3 which provides a Gaussian bound with deteriorated constants (see also the User's guide to the proof in Section 3.1 below). Even if, from a numerical point of view, the deviation bounds in Theorem 7 below yield sharper controls with respect to simulated empirical measures (see Figure4.1 in the numerical Section below).For "intermediate Gaussian deviations", i.e. for a the constants in the Gaussian bound deteriorate. So, it seems reasonable to see this situation like for the first regime a namely where there are constants C 8 ą 1, c 8 ă 1 such that lim n C n " C 8 , lim n c n " c 8 and

		? Γn Ñ
	?	Γn Ñ

n C ą 0, n 0,

  Lemma 7 (Choice of ρ for the concentration regime) For P min pa, Γ n , ρq as in Bhou

	Remark 14 For Gaussian deviations, ρ´1 " 1 2	B1{2 n a Ã3{2 n ? Γn	p1`op1qq -a ?
	a ? Γn Ñ n	`8.			a ? Γn Ñ n	0 and a super Gaussian one for
	(3.34), we have
	(a) If a ? Γn Ñ n	0, taking ρ :" ρpa, nq s.t. ρ ´1 " 1 2	B1{2 n a Ã3{2 n ? Γn	p1 `op1qq
			P min pa, Γ n , ρq " a ? Γn Ñ n	0	´a2 2νp|σ ˚∇ϕ| 2 q	p1 `op1qq.
	(b) If a ? Γn Ñ
			Γn Ñ`8	1{3 ´a4{3 Γ n 2}σ} 2{3 8 rϑs 1 2{3	p1 `op1qq.

n `8, taking ρ :" ρpa, nq s.t. ρ ´1 " 1 2 `op1q P min pa, Γ n , ρq " a ?

  C2) Defining for all x P R d , Σpxq :" σσ ˚pxq, Kpxq " κκ ˚pxq, we suppose that

	sup xPR d	TrpΣpxqq " sup xPR d	}σpxq} 2 ": }σ} 2 8 ă `8, sup xPR d	TrpKpxqq " sup xPR d	}κpxq} 2 ": }κ} 2 8 ă `8.

  Γn ˘for p n ě 1 such that p n Ñ n `8 and p n

	where σ 2 8 :" p1 `rq}κ} 2 8 }∇ϕ} 2 8 `}σ} 2 8 }∇ϕ} 2 8 and C n " exp	`rD 3 ϕs β }σ} 3`β 8 Er|U 1 | 3`β s p1`βqp2`βqp3`βq	Γ p n 3`β 2 q ? Γn	pn
	Γ p2q n ?		Γ p2q n ? Γn Ñ n 0.
	The proof of Theorem 2 is given in Section 2.3.
	Remark 6 Note that for all θ P p 1 2`β , 1s,	? Γn Γ p2q n	" `8, then we can choose χ n s.t. χ n	? Γn Γ p2q n	Ñ n
	`8. In other words, we can pick a " apnq Ñ n `8. We have unsurprisingly that σ 2 ϕ ď
	σ 2 8 , where σ 2

ϕ is the asymptotic variance of ?

Γ n ν n pAϕq defined in (1.14). Moreover, the difficulty to adapt a Gaussian Concentration result to compound Poisson process yields that the upper-bound variance σ 2 8 depends on the dimension.

  Now we formulate several propositions and lemmas that are used to control the components of the remainder term R ϕ n . The following proposition is the counterpart to the jumps diffusion of the useful Proposition 1 in Chapter 3.

	CHAPTER 5. EXTENSION FOR A POISSON COMPOUND PROCESS
							Λ Γn ă	ε 6}κ}8}∇ϕ}8ρprq , where ρprq is defined
	in (1.11) , we have
	E	" exp	ˆ´Λ Γ n	r ∆ ϕ n pX n´1 , Z n q ˙ˇˇF	n´1		ď exp ˆγn }κ} 2 8 }∇ϕ} 2 8 p1 `r `εq	n 2Γ 2 Λ 2	˙.179

  for n big enough λn Γn e n ď Cae n . If a ď 1, we take C n " R n exppCae n q Ñ

	Hence, the condition			Γn -a ? Γn ă χn Γ p2q n	Ñ 0.
							pλ n Γ n	-	pa ? Γ n	ă	pχ n Γ n p2q	ă C{Γ p2q n	(2.24)
	P	" a Γ n ν n pAϕq ě a ‰	ď R n exp	´´c n a 2 2 `p1 `rq}κ} 2 8 }∇ϕ} 2 8 `}σ} 2 8 }∇ϕ} 2 8	? Γ n ˘`λ n	e n	¯,
	with cn " p1`rq}κ} 2 8 }∇ϕ} 2 8 `}σ} 2 8 }∇ϕ} 2 8 q `p1`r`εq}κ} 2 8 }∇ϕ} 2 8 `}σ} 2 8 }∇ϕ} 2 8 ˘Ñ nÑ`8	1, and λn ?
								nÑ`8	1, otherwise if a ą 1 then we set
	c n " 1 ´Cen a 2	Ñ nÑ`8	1. In any case, we write the result:
		P	" a Γ n ν n pAϕq ě a ‰	ď C n exp	´´c n a 2 2 `p1 `rq}κ} 2 8 }∇ϕ} 2 8 `}σ} 2 8 }∇ϕ} 2

has to be satisfied. Let us calibrate p " ppnq Ñ 8 depending on χ n Ñ 0 s.t. lim sup n pχ n ă C. This pick of p yields (2.24). We can also choose, for C ą 0 large enough ε n " C χn Γ p2q n such that all conditions over λ n , p, ε are satisfied with these choices.

The inequality (2.22) yields then for λ " λ n :

  viewed as a typical model for the operator in (1.1). Introducing now, for λ ą 0, the dilation operatorδ λ : pt, xq P R `ˆR nd Þ Ñ δ λ pt, xq " `λ2 t, λx 1 , λ 3 x 2 , ¨¨¨, λ 2n´1 x n ˘P

	I d,d	¨¨¨¨¨¨¨¨¨0 d,d . . . ¨¨¨¨¨¨. . .	‹ ‹
	0 d,d I d,d . . . . . .	. . . . . . ¨¨¨. . . . . . . . .	‹ ' ‹ ‹ ‹ ,	(1.6)
	0 d,d ¨¨¨0 d,d I d,d 0 d,d
	which can be

R `ˆR nd , i.e. with a slight abuse of notation, `δλ pt, xq ˘0 :" λ 2 t and for all i P rr1, nss, `δλ pt, xq ˘i :" λ 2i´1 x i , we have that

  pτ,ξq pt, s, x, yq|

	where m m,pτ,ξq s,t	pxq stands for the mean of	Xm,pτ,ξq s
	ď	ps ´tq	C 2k´1 2 `jp2i´1q 2	2 `n2 d	exp ´´C ´1ps ´tq|T ´1 s´t pm s,t m,pτ,ξq	pxq ´yq| 2	":
		ps ´tq	C 2k´1 2	2 `jp2i´1q	ppτ,ξq C ´1 pt, s, x, yq,	(2.8)

  Concentrating on the last term, which turns out to be the most delicate, we see that the choice of pτ, ξq must be made in order to equilibrate the time singularities coming from D 2 x 1 pm,pτ,ξq pt, s, x, yq. Let us first consider the non-degenerate part coming from the difference pL m s ´L ps, ¨qs d,γ }D y 1 u m ps, ¨q} L 8 `1 2 raps, ¨qs d,γ }D 2 y 1 u m ps, ¨q} L 8 ¯dγ py, θ m s,τ pξqq. The contribution d γ py, θ m s,τ pξqq in the above r.h.s. must then equilibrate the time singularity in ps ´tq ´1 coming from D 2 x 1 pm,pτ,ξq pt, s, x, yq (see (2.8) and Proposition (2) below). This is possible if d γ py, θ m s,τ pξqq is compatible with the off-diagonal bound ps tq|T ´1 s´t pm

		m,pτ,ξq s	qu m ps, yq which explicitly writes from (2.1) and (2.6):
		∆ 1,Fm,σm pτ, s, y, θ m s,τ pξq, u m q
		" xF m,1 ps, yq ´Fm,1 ps, θ m s,τ pξqq, D y 1 u m ps, yqy
		`1 2	s,τ pξqqqD 2 y 1 u m ps, yq Tr ´pa m ps, yq ´am ps, θ m ¯,
	and can be upper-bounded from the Hölder continuity assumption (w.r.t. the underlying
	homogeneous metric d) on F 1 and a as:
	|∆ 1,Fm,σm pτ, s, y, θ m s,τ pξq, u m q|	(2.12)
	ď ´rF 1 m,pτ,ξq s,t	pxq ´yq| 2 in (2.8).
	This is precisely the case considering pτ, ξq " pt, xq which gives
					m m,pτ,ξq s,t	pxq| pτ,ξq"pt,xq " θ m s,t pxq,	(2.13)
	as it can readily be checked from (2.5), (2.6) (taking the expectation) and the Grönwall's
	Lemma. Therefore, observing precisely from the metric homogeneity (see equations (1.8)
	and (1.9)) that:
	d γ py, θ m s,t pxqq " ps ´tq	γ 2 d γ pps ´tq	1 2 T ´1 s´t y, ps ´tq	1 2 T ´1 s´t θ m s,t pxqq
				ď Cps ´tq	γ 2	´n ÿ	|ps ´tq	2i´1 2 py ´θm s,t pxqq i |	γ 2i´1
							i"1
		`ż T t	ds	ż R nd	D 2 x 1

T,t g m pxq `D2 x 1 Gm,pτ,ξq f m pt, xq pm,pτ,ξq pt, s, x, yqpL m s ´L m,pτ,ξq s qu m ps, yqdy.

  pτ,ξq pt, s, x, yq @ ∆ i,Fm pτ, s, θ m s,t pξq, yq, D y i u m ps, yq

						D ˇˇˇˇˇˇp
						τ,ξq"pt,xq
	ď	n ÿ i"2 ż T t ds ˇˇż R nd dyD y i ¨´`D 2 x 1		τ,ξq"pt,xq
	":	n ÿ i"2	ż T t	ds ˇˇż R nd	τ,ξq"pt,xq i,pt,xq ps, yq ˘um ps, yq ˇˇˇˇˇˇp dyD y i ¨`Θ m,α	,	(2.16)

pm,pτ,ξq pt, s, x, yq b ∆ i,Fm pτ, s, θ m s,t pξq, yq ˘¯u m ps, yq ˇˇˇˇˇˇp

  LR02]). Namely B αi 8,8 is the dual of the closure of the Schwartz class S in B ´α i 1,1 . But S is dense in B ´α i 1,1 (see for instance Theorem 4.1.3 in [AH96]

.18) that C αi b pR d , Rq " B αi 8,8 pR d , Rq. It is also well known that B αi 8,8 pR d , Rq and B ´α i 1,1 pR d , Rq are in duality (see e.g. Proposition 3.6 in [

  pτ,ξq pt, s, x `µpx 1 ´xq, yq|

	ď	C ps ´tq j´1 2 `1`n 2 d 2	expp´C ´1ps ´tq|T ´1 s´t pm m,pt,ξq s,t	px `µpx 1 ´xqq ´y| 2 q
	ď	C ps ´tq j´1 2 `1`n 2 d 2	exppC ´1ps ´tq|T ´1 s´t pm m,pt,ξq s,t	px ´x1 qq| 2 q
		ˆexpp´C 2 ´1	ps ´tq|T ´1 s´t pm m,pt,ξq s,t	pxqq ´y| 2 q
	ď	C ps ´tq j´1 2 `1`n 2 d 2	expp´C 2 ´1	ps ´tq|T ´1 s´t pθ m s,t pxqq ´y| 2 q,	(2.32)
	using for the last inequality that m m,pt,ξq s,t	pxq| ξ"x " θ m s,t pxq and the fact that, from the
	linear structure of ODE satisfied by m m,pt,ξq s,t	pxq (which can be read from system (2.6)
	taking the expectation), ps ´tq	1 2 |T ´1 s´t pm m,pt,ξq s,t	qpx ´x1 q| 2 ď Cps ´tq	1 2

  The proof of the above proposition readily follows from Proposition 3.3 and Lemma 3.6 in[START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF]. We now state some important density bounds for the linearized model.

	Proposition 2 (Density of the linearized dynamics) Under (A), we have that, for
	all s P pt, T s the random variable	Xpτ,ξq s	in (3.3) admits a Gaussian density ppτ,ξq pt, s, x,	¨q
	which writes for all y P R nd :
	ppτ,ξq pt, s, x, yq :"	p2πq	nd 2 detp 1	Kpτ,ξq s,t q	1 2	exp	2 ˆ´1	A p	Kpτ,ξq s,t q ´1pm pτ,ξq s,t pxq ´yq, m pτ,ξq s,t pxq´y	E ˙,
											(3.6)
	with	Kpτ,ξq s,t
								. . .		. . .	. . .	. . .	' ‹ ‹ ‹ .
							0 d,d		¨¨¨0 d,d u n I d,d

d 0 d,d . . .

  t. From the good-scaling property of Proposition 1, it is plain to derive that y K s,t 1 a non-degenerate bounded matrix, i.e. there exists Ĉ ě 1 s.t. for all ζ P R nd , p Ĉq ´1|ζ| 2 ď x y K s,t 1 ζ, ζy ď Ĉ|ζ| 2 . A similar rescaling argument yields on the deterministic system (3.2) of the resolvent yields that Rpτ,ξq ps, tq can also be written as: r Rpτ,ξq ps, tqs ˚" T ´1 s´t From the analysis performed in Lemma 5.1 in[START_REF] Huang | A Parametrix Approach for some Degenerate Stable Driven SDEs[END_REF] (see also the proof of Proposition 3.7 in[START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF]) one derives that there exists Ĉ1 s.t. for all ζ P R nd ,

	d `ps ´tq C β 1 2 T ´1 s´t m pτ,ξq s,t pxq, ps ´tq ps ´tq ř n i"1 α i pi´1 2 q´β 2 ppτ,ξq C ´1 β pt, s, x, yq, 1 2 T ´1 s´t y ˘|D α x i.e. equation (3.8) quantifies the regularizing effect of the scaled arguments in the quasi-ppτ,ξq pt, s, x, yq| ď (3.8) distance. Proof: Expression (3.6) readily follows from (3.2) and (3.3). Differentiating w.r.t. x recalling from (3.4) that x Þ Ñ m pτ,ξq s,t pxq is affine yields: D x j ppτ,ξq pt, s, x, yq " ´"" Rpτ,ξq ps, tq ‰ ˚p Kpτ,ξq s,t q ´1pm pτ,ξq s,t pxq ´yq ı j ppτ,ξq pt, s, x, yq. (3.9) The point is now to use scaling arguments. We can first rewrite " Rpτ,ξq ps, tq ‰ ˚p Kpτ,ξq s,t q ´1 " ps ´tq " Rpτ,ξq ps, tq ‰ ˚T´1 s´t p y K s,t 1 q ´1T ´1 s´t , (3.10) where y K s,t 1 is the covariance matrix of the rescaled process `ps ´tq 1 2 T ´1 s´t Xt,x t`vps´tq ˘vPr0,1s p Rpτ,ξq,s,t p1, 0q ı ˚Ts´t , (3.11) where again p Rpτ,ξq,s,t p1, 0q is the resolvent at time 1 of the rescaled system ´Ts´t r Rpτ,ξq pt `vps ´tq, tqs ˚T´1 s´t ¯vPr0,1s " ´" p Rpτ,ξq,s,t pv, 0q ‰ ˚¯vPr0,1s associated with (3.2). | " p Rpτ,ξq,s,t p1, 0q ‰ ˚ζ| ď Ĉ1 |ζ|. (3.12) Equations (3.9), (3.10) and (3.11) therefore yield: |D x From the explicit expression (3.6), Proposition 1 and the above equation, we eventually derive: |D x j ppτ,ξq pt, s, x, yq| ď C ps ´tq j´1 2 ´ps ´tq 1 2 |T ´1 s´t pm pτ,ξq s,t pxq ´yq| 1 ps ´tq n 2 d 2 exp ´´C ´1ps ´tq|T ´1 s´t pm pτ,ξq s,t pxq ´yq| 2 ď C at time 1. " ps ´tq j´1 2

j ppτ,ξq pt, s, x, yq| ď ps ´tq ´j`1 2 ˇˇˇˇˆ" p Rpτ,ξq,s,t p1, 0q ı ˚`y K s,t 1 ˘´1 `ps ´tq 1 2 T ´1 s´t pm pτ,ξq s,t pxq ´yq ˘˙j ˇˇˇˇp pτ,ξq pt, s, x, yq ď Cps ´tq ´j`1 2 ps ´tq 1 2 |T ´1 s´t pm pτ,ξq s,t pxq ´yq|p pτ,ξq pt, s, x, yq.

  pxqq b2 1 ¯dy " 0 d,d,d , k P rr1, nss, M P R d b R d .

						ξq pt, s, x, yqpy	´mpτ,ξq s,t pxqq b2 1 dy " r	Kpτ,ξq s,t s 1,1 ,	(3.13)
					ż
					R nd	D 2 x 1	ppτ,ξq pt, s, x, yqpy	´mpτ,ξq s,t pxqq 1 dy " 0 d ,	(3.14)
		ż
		R nd	D x k D 2 x 1	ppτ,ξq pt, s, x, yq ¨py	´mpτ,ξq s,t pxqq 1 dy " 0 d,d , k P rr1, nss,	(3.15)
		ż R nd	D 2 x 1	ppτ,ξq pt, s, x, yqTr ´Mpy	´mpτ,ξq s,t pxqq b2 1 ¯dy " 2M, M P R d b R d ,	(3.16)
	ż R nd	D x k D 2 x 1	ppτ,ξq pt, s, x, yqTr ´Mpy	´mpτ,ξq s,t

  which yields D x 1 rm pτ,ξq s,t pxqs 1 " r Rpτ,ξq ps, tqs 1,1 " I d,d ) gives (3.14). Iterating the differentiation w.r.t. D x k then yields (3.15) (observing again that D x k rm pτ,ξq s,t pxqs 1 " r Rpτ,ξq ps, tqs 1,k , i.e. D x k rm pτ,ξq s,t pxqs 1 q " I d,d if k " 1 and 0 d,d for k ą 1).

	Observe that D x 1	ş R nd Tr ´Mpy´m pτ,ξq s,t pxqq b2 1 ¯p pτ,ξq pt, s, x, yqdy " D x 1 Tr ´Mr	s,t s 1,1 Kpτ,ξq	¯"
	0

d . Differentiating again w.r.t. D x 1 , the Leibniz formula and identity D x 1 rm pτ,ξq s,t pxqs 1 " r Rpτ,ξq ps, tqs 1,1 " I d,d yield

  yqI sąt 0 pL s ´L ξ s qups, yq

	´ż T	ds	ż	dyp
	t		R nd

ξ1 pt, s, x 1 , yqI sąt 0 pL s ´L ξ1 s qups, yq. (5.27)

  xq `xFpt, xq, Du k pt, xqy `1 2 Tr `D2 x 1 u k pt, xqapt, xq ˘" ´f pt, xq, t, P rT p1 ´k N q, T p1 ´k´1

	N qq, u k pp1 ´k´1 N qT, xq " upp1 ´k´1 N qT, xq,
	we derive that

  `pL t q ˚¯ϕpt, xqpu m pt, xq ´upt, xqq ": pR 1

	with
		R m pT, uq
	:"	ż T 0	dt	ż	R nd	dx ´pL m t q ˚´L t ¯ϕpt, xqu m pt, xq	(6.30)
		`ż T 0	dt	ż R nd	dx ´´B t m	`R2 m qpT, uq,
	where L t is the formal adjoint of L t . Observe first that:
										ż T	ż
				R 2 m,0 pT, uq :"	0	dt	R nd	dxB t ϕpt, xq `um pt, xq ´upt, xq ˘ÝÑ m	0,
										xq ´Bt	`Lm t ¯um pt, xq.	(6.27)
	Indeed, both the solution and the coefficients are smooth. Integrating by parts yields:
	ż T 0	dt	ż	R nd	dxf m pt, xqϕpt, xq "	ż T 0	dt	ż R nd	dx ´´B t `pL m t q ˚¯ϕpt, xqu m pt, xq,
	where pL m t q ˚denotes the adjoint of L m t . Write now:
		ż T	ż					ż T	ż
			dt				dxf m pt, xqϕpt, xq "	dt	dxf pt, xqϕpt, xq
		0			R nd		0	R nd
										`ż T	dt	ż	dxpf m ´f qpt, xqϕpt, xq
										0	R nd
										ż T	ż
										":	dt	dxf pt, xqϕpt, xq `Rm pT, f q. (6.28)
										0	R nd
	It is clear that under (A), recall that f P L 8 pr0, T s, C γ b,d pR nd , Rqq, R m pT, f q ÝÑ m	0. On
	the other hand, we now decompose:
									ż T 0	dt	ż R nd	dx ´´B t `pL m t q ˚¯ϕpt, xqu m pt, xq
								"	ż T	dt	ż	dx ´´B t `pL t q ˚¯ϕpt, xqupt, xq `Rm pT, uq,	(6.29)
										0	R nd

  s´t ´Fpu, zq ´Fδ pu, zq ¯ˇˇď Cps ´tq ´1.

						(7.2)
	By the previous definition of F δ , identity (7.2) is equivalent to:
					n ÿ	ps ´tq	2i´3`γ 2i´1 i 2 ´iδ 1	ď Cps ´tq ´1.	(7.3)
					i"2
	Hence, we choose from now on, for all i P rr2, nss:
						δ i " ps ´tq pi´3 2 q 2i´1 2i´3`γ .	(7.4)
	Next, let us control the last components of the flow. By the definition of θ s,t in (2.5),
	we get:
	|pθ s,t pxq ´θs,t px 1 qq n |
	ď |px ´x1 q n |	`ż s	´|F δ n pv, θ v,t pxqq ´Fδ n pv, θ v,t px 1 qq|
		t
	`|F δ n pv, θ v,t pxqq ´Fn pv, θ v,t pxqq| `|F δ n pv, θ v,t px 1 qq ´Fn pv, θ v,t px 1 qq| ¯dv
	ď |px ´x1 q n |	`C ż s ´ˇ`θ v,t pxq ´θv,t px 1 qq ˘n´1 ˇˇ`δ n ´1`2 n´3`γ 2n´1	ˇˇ`θ	v,t pxq ´θv,t px 1 qq	˘nˇˇ¯d v
				t
	2n´3`γ
	`ps ´tqδ n	2n´1	.
	Hence by Grönwall"s Lemma, we get:
	|pθ s,t pxq ´θs,t px 1 qq n |	(7.5)
	ď C exp ´Cps ´tqδ n ´1`2 n´3`γ 2n´1	|px
	´x1 q n | `ps ´tqδ n 2n´3`γ 2n´1	`ż s	ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘n´1 ˇˇdv	ď
						t
	C exp ´Cps ´tq	γ 2 ¯´|px ´x1 q n | `ps ´tq n´1 2 `ż s	ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘n´1 ˇˇdv	¯,
						t
	using (7.

  that Young type controls are more appropriate. Write: |pθ s,t pxq ´θs,t px 1 qq n´1 |

	ď C exp ´Cps ´tqδ n´1 ´1`2 pn´1q´3`γ 2pn´1q´1	2pn´1q´3`γ 2pn´1q´1 n´1 ¯´|px ´x1 q n´1 | `ps ´tqδ
	`ż s	ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘n´2 ˇˇ`| `θv,t pxq ´θv,t px 1 qq	˘nˇˇ2pn´1q´3`γ 2n´1	(	dv	ď
	t
	C exppCps ´tq	γ 2 q ´|px ´x1 q n´1 | `ps ´tq n´3 2 `ż s	" ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘n´2	ˇ|px
								t
					2pn´1q´3`γ		2pn´1q´3`γ
	´x1 q n |	2n´1	`pv ´tq	2
	`´ż v	ˇˇ`θ	w,t pxq ´θw,t px 1 qq ˘n´1 ˇˇdw	¯2pn´1q´3`γ 2n´1	dv *	˙.
	t
								(7.6)

  In order to obtain the suitable time scale, we choose δn s.t.The point is now to take the supremum in s P rs, ts in the above equation. This yields: `θw,t pxq ´θw,t px 1 qq ˘n´1 ˇˇpv ´tq 1´4 ´γ 2 dv ¯˙.

	CHAPTER 6. SHARP SCHAUDER ESTIMATES
			sup	|pθ s,t pxq ´θs,t px 1 qq n´1 |
			sPrt,ss
		ď C exppCps ´tq	γ 2 q ´|px ´x1 q n´1 | `ps ´tq n´3 2 `ż s	ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘n´2 ˇˇdv
								t
			`|px ´x1 q n |	2pn´1q´3`γ 2n´1	ps ´tq `ps ´tq 1`2 pn´1q´3`γ 2
			`´ż s	sup
				t	wPrt,vs
	We get then by Grönwall's Lemma:
			|pθ s,t pxq ´θs,t px 1 qq n´1 |
		ď C exppCps ´tq	γ 2 q ´|px ´x1 q n´1 | `ps ´tq n´3 2 `ż s	ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘n´2 ˇˇdv
								t
			`|px ´x1 q n |	2pn´1q´3`γ 2n´1	ps ´tq ¯(7.7)
		ď C exppCps ´tq	γ 2 q ´|px ´x1 q n´1 | `ps ´tq n´3 2 `ż s	ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘n´2 ˇˇdv
								t
			`|px ´x1 q n |	2n´3 2n´1 ¯,	(7.8)
	using again the Young inequality |px ´x1 q n |	2pn´1q´3`γ 2n´1	ps ´tq ď Cpps ´tq n´3 2 `|px	x1
		2pn´1q´3`γ	2n´3
	q n |	2n´1	2n´5			t pxq ´θw,t px 1 qq ˘n´1 ˇˇdw	¯2pn´1q´3`γ 2n´1
				ď C	ˆ´ż v	ˇˇ`θ	w,t pxq ´θw,t px 1 qq ˘n´1 ˇˇdw ¯δ n´1,n 2pn´1q´3`γ ´2n´1	n´1,n `δ 4´γ 2n´1	˙.
								t
				δ 2n´1 4´γ n´1,n " pv ´tq	2pn´1q´3`γ 2	ðñ δn´1,n " pv ´tq	p2pn´1q´3`γqp4´γq 2p2n´1q	,
	which also yields that
	´ż v	ˇˇ`θ	w,t pxq´θ w,t px 1 qq ˘n´1 ˇˇdw ¯δ n´1,n ´2n´1 2pn´1q´3`γ	ď	´ż v	ˇˇ`θ	w,t pxq´θ w,t px 1 qq ˘n´1 ˇˇdw ¯pv ´tq ´4´γ 2 .
	t							t
	Hence we get from (7.6) and the previous controls that for all s P rt, ss:
			|pθ s,t pxq ´θs,t px 1 qq n´1 |
		ď C exppCps ´tq	γ 2 q ´|px ´x1 q n´1 | `ps ´tq n´3 2 `ż	s	" ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘n´2	ˇ|px
								t
								2pn´1q´3`γ	2pn´1q´3`γ
				´x1 q n |	2n´1	`pv ´tq	2
			`´ż v	ˇˇ`θ	w,t pxq ´θw,t px 1 qq ˘n´1 ˇˇdw ¯pv ´tq ´4´γ 2	dv *	˙269
				t

|

  t pxq ´θs,t px 1 qq n | Cp|px ´x1 q n | `ps ´tq n´1 2 q and |px ´x1 q n´1 |ps ´tq ď C `|px ´x1 q n´1 |

	ď C exp ´Cps ´tq	γ 2 ¯´|px ´x1 q n | `ps ´tq n´1 2 `|px ´x1 q n´1 |ps ´tq
	`|px ´x1 q n |	2n´3 2n´1 ps ´tq	`ż s	ż v	ˇˇ`θ	w,t pxq ´θw,t px 1 qq ˘n´2 ˇˇdwdv	ď
								t	t
	C exp ´Cps ´tq	γ 2 ¯´|px ´x1 q n | `ps ´tq n´1 2 `|px ´x1 q n´1 |	2n´1 2n´3
	`ż s	ż v	ˇˇ`θ	w,t pxq ´θw,t px 1 qq ˘n´2 ˇˇdwdv	¯,
	t	t
	2n´3 2n´1 ps ´tq ď 2n´1 using again the Young inequalities |px ´x1 q n | 2n´3 `ps ´tq n´1 2 ˘for the last bound. Iterating these
	computations, we obtain:
	|pθ s,t pxq ´θs,t px 1 qq n |				(7.9)
	ď C ˜ps ´tq n´1 2 `n ÿ	|px ´x1 q j |	2n´1 2j´1 `ż vn"s dv n´1 . . .
			j"2					t

  t pxq ´θv,t px 1 qq j |

	recalling that ps ´tq ď T is small, and using again Young inequalities for the last bound.
	Namely,
	ps ´tq 1`pj´1q γ 2j´1 sup	|pθ v,t pxq ´θv,t px 1 qq 1 |	γ 2j´1 ď Cps ´tq ´1 `sup	|pθ v,t pxq ´θv,t px 1 qq 1 |	¯,
		vPrt,ss					vPrt,ss
	and
		ps ´tq|px ´x1 q k
							γ 2j´1 dv,
	which in turn implies
	sup	|pθ s,t pxq ´θs,t px 1 qq 1 | ď |px ´x1 q 1 |	`C´p	s ´tq `sup	|pθ s,t pxq ´θs,t px 1 qq 1 |	˘γ
	sPrt,ss					vPrt,ss
			`n ÿ	ż s	|pθ v,t pxq ´θv,t px 1 qq j |	2j´1 dv γ	ď
			j"2	t
			|px ´x1 q 1 |	`Cˆp	s ´tq `sup	|pθ s,t pxq ´θs,t px 1 qq 1 |	˘γ
							vPrt,ss
			`n ÿ	C j ps ´tq ´ps ´tq j´1 2 `n ÿ	|px ´x1 q k |	2j´1 2k´1
			j"2			k"2
							ȧnd
			`ps ´tq j´1 sup	|pθ v,t pxq ´θv,t px 1 qq 1 |	¯γ 2j´1
							vPrt,ss
	yields that
	sup	|pθ s,t pxq ´θs,t px 1 qq 1 | ď |px ´x1 q 1 |	`Cˆp	s ´tq `sup	|pθ s,t pxq ´θs,t px 1 qq 1 |	˘γ
	sPrt,ss					vPrt,ss
			`n ÿ	C j ps ´tq ´ps ´tq	γ 2 `n ÿ	|px ´x1 q k |	γ 2k´1
			j"2			k"2
			`ps ´tq pj´1q γ 2j´1 sup	|pθ v,t pxq ´θv,t px 1 qq 1 |	γ 2j´1 ¯˙,
							vPrt,ss
	using as well (7.10), Remark 8 and convexity inequalities for the last bound. We now
	write,
	sup	|pθ s,t pxq ´θs,t px 1 qq 1 | ď C ˆ|px ´x1 q 1 | `ps ´tq 1`γ 2 `ps ´tq	n ÿ	|px ´x1 q k |	γ 2k´1
	sPrt,ss						k"2
							ď
			`n ÿ	ps ´tq 1`pj´1q γ 2j´1 sup	|pθ v,t pxq ´θv,t px 1 qq 1 |	γ 2j´1
			j"1			vPrt,ss
			Cp|px ´x1 q 1 | `ps ´tq	`n ÿ	|px ´x1 q k |	1 2k´1 q	(7.11)
							k"2

  ps ´tq From the scaling properties of the resolvent, see e.g. the proof of Proposition 2 or Lemma 6.2 in[START_REF] Menozzi | Martingale problems for some degenerate Kolmogorov equations[END_REF], we have that:´DFpv, θ v,t pxqq Rx pv, tq ´DFpv, θ v,t px 1 qq Rx 1 pv, tq ¯dvT s´t ˇď |T ´1 s´t DFpv, θ v,t pxqqT s´t ||T ´1 s´t p Rx ´R x 1 qpv, tqT s´t |dv ˇˇT ´1 s´t ´DFpv, θ v,t pxqq ´DFpv, θ v,t px 1 qq ¯Ts´t ˇˇ|T ´1 s´tps ´tq ´1|DFpv, θ v,t pxqq ´DFpv, θ v,t px 1 q|dv, using the Grönwall's Lemma and the structure of the resolvent for the last inequality.

	Proof: Lemma 12
			Rx ps, tq " T s´t	p Rs,t,x 1	T ´1 s´t , Rx 1	ps, tq " T s´t	p Rs,t,x 1 1	T ´1 s´t ,	(8.1)
	where p Rs,t,x 1	, p Rs,t,x 1 1	are non-degenerate bounded matrices. We define then:
						∆ p Rs,t,x,x 1 1	:" p Rs,t,x 1	´p Rs,t,x 1 1	.	(8.2)
	Hence, from (8.1) and the definitions in (8.2):
	|∆ p Rs,t,x,x 1 1	| " | p Rs,t,x 1	´p Rs,t,x 1 1	| " |T ´1 s´t p Rx ´R x 1	qps, tqT s´t |
			" ˇˇT ´1 s´t	ż s
						t
			ż s
			t
			`ż s				Rx 1	pv, tqT s´t |dv
				t
			ż s
			ď C
				t
								γ 2 `dγ px, x 1 q ˘.

  Rξ ps, uqBapu, θ u,t pξqqB ˚R ξ ps, uq ˚du. Rξ ps, uqB∆ u apθ u,t pξq, θ u,t pξ 1 qqB ˚R ξ ps, uq ˚, ∆ u apθ u,t pξq, θ u,t pξ 1 qq :" apu, θ u,t pξqq ´apu, θ u,t pξ 1 qq, Bapθ u,t pξq, θ u,t pξ 1 qqB ˚Ts´t q i,j |du ď Cps ´tq i`j´2 ż s t dpθ u,t pξq, θ u,t pξ 1 qq γ du.We deduce by Lemma 2, that |p∆ ξ,ξ 1 1 ps, tqq i,j | ď Cps ´tq i`j´1 `ps ´tq

	Proof of Lemma 1 Let us first explicitly write the covariance matrices Kξ s,t :" ż s t So we have to control the term Kξ s,t ´K ξ 1 s,t " ∆ ξ,ξ 1 1 ps, tq `∆ξ,ξ 1 2 ps, tq, ∆ ξ,ξ 1 1 ps, tq :" ż s t du ∆ ξ,ξ 1 2 ps, tq :" ż s t du∆ Rξ,ξ 1 ps, uqBapu, θ u,t pξ 1 qqB ˚R ξ ps, uq ż (8.7) s t du Rξ 1 ps, uqBapu, θ u,t pξ 1 qqB ˚∆ Rξ,ξ 1 ps, uq ˚, ∆ Rξ,ξ 1 ps, uq " Rξ ps, uq ´R ξ 1 ps, uq. (8.8) Hence, from the scalings of (8.1) and the definitions in (8.8), for all 1 ď j ď i ď n: |p∆ ξ,ξ 1 1 ps, tqq i,j | ď Cps ´tq ´2 ż s 2 `dγ pξ, ξ 1 q ˘. (8.9) Still from (8.1) and the definitions in (8.8), write now that: |p∆ ξ,ξ 1 2 ps, tqq i,j | ď Cps ´tq ´2˜ż s t ´|pT s´t p p Rs,u,ξ 1 ´p Rs,u,ξ 1 1 qBapu, θ u,t pξ 1 qqB ˚p p Rs,u,ξ 1 1 q ˚Ts´t q i,j | `|pT s´t p Rs,u,ξ 1 1 Bapu, θ u,t pξ 1 qqB ˚p p Rs,u,ξ 1 1 ´p Rs,u,ξ 1 q ˚Ts´t q i,j | ¯du ¸. (8.10) Thanks to equation (8.6) in the proof of Lemma 12, we thus obtain: |p∆ ξ,ξ 1 2 ps, tqq i,j | ď Cps ´tq i`j´1 `ps ´tq γ 2 `dγ pξ, ξ 1 q ˘. (8.11) |pT s´t ∆ u γ Gathering (8.9) and (8.11) in (8.8) yields:
	t

  pt,xq ps, yq ˘dy i ,i ps, y 1:i´1 , z, y i`1:n q ´Fλ,i ps, θ λ s,t pξqq ´Dx i´1 F λ,i ps, θ λ s,t pξqqpy ´θλ s,t pξqq i´1

	"	ż R d	h v pz ´yi qD y i ¨´Θ α,λ i,pt,xq ps, yq ´Θα,λ i,pt,xq ps, y 1:i´1 , z, y i`1:n q ¯dy i
		ż
	"	R d	D α pξ λ pt, s, x, yq	@	F λ,i ps, yq ´Fλ,i ps, y 1:i´1 , z, y i`1:n q, D z h v pz ´yi q D	dy i
		`żR d	λ pt, s, x, yq ´Dα pξ λ pt, s, x, y 1:i´1 , z, y i`1:n q ´Dα pξ Ā`F

λ

  t pxq ˘`λ ´1 2 T λ py ´θs,t pxqq ˘i´1 ˇˇďWe have for all µ P r0, 1s, |z ´θs,t pxq i | ď µ|z ´yi | `|z `µpy i ´zq ´pθ s,t pxqq i | ˆλ´1{2 T λ θ s,t pxq, λ ´1{2 T λ ´y1:i´1 , z `µpy i ´zq, y i`1:n q

	we thus derive
	|T λ,2 `v, t, s, x, py 1:i´1 , z, y i`1:n q	˘|
	ď Cλ ř n j"1 α j pj´1 2 q	ż R d	dy i h cv pz ´yi q	ż 1 0	dµ	qc,λ pt, s, x, y 1:i´1 , z `µpy i ´zq, y i`1:n q ps ´tq ř n j"1 α j pj´1 2 q`2 i´1 2
			2i´3`γ			2i´3`γ
	ˆˆλ	2	|y i ´z|	2i´1
			`d2i´3`γ ¯ď
	Cλ ř n j"1 α j pj´1 2 q	ż	dy i h cv pz ´yi q	ż 1	dµq c,λ pt, s, x, y 1:i´1 , z `µpy i ´zq, y i`1:n q
								R d	0
	ˆ´λ	2i´3`γ 2	v	2i´3`γ 2p2i´1q
	Cλ ř n j"1 α j pj´1 2 q	ż	R d	dy i h cv pz ´yi q	ż 1 0	dµ	qc,λ pt, s, x, y 1:i´1 , z `µpy i ´zq, y i`1:n q ps ´tq ř n j"1 α j pj´1 2 q`2 i´1 2
	ˆ´pλ	2i´1 2 |z ´θs,t pxq i |q	2i´3`γ 2i´1	`pλ	2pi´1q´1 2	|pθ s,t pxq ´yq i´1 |q 1`γ 2pi´1q´1
	`n ÿ	pλ	2 |pθ s,t pxq ´yq k |q 2j´1	2k´1 2i´3`γ	¯.
	k"i`1

  ´yi qN

				cλ	2i´1 2 ps´tq 2i´1 pz `µpy i ´zq ´pθ s,t pxqq i qdy i
	ˆ´λ	2i´3`γ 2	v	2i´3`γ 2p2i´1q

  pt, xq, ¨¨¨, Ψ n pt, xq ˘˚where for each i P rr1, nss, Ψ i is R d valued, we denote by }DΨ} 8 :" |||DΨ i pt, xq|||, }DpD 1 Ψq} 8 :" ||| ¨||| stands for a tensor norm in the appropriate corresponding dimension. Precisely, DΨ i pt, xq P R nd b R d and DpD 1

	n		n
	ÿ		ÿ
		sup		sup	|||DpD 1 Ψ i qpt, xq|||,
	i"1	pt,xqPr0,T sˆR nd	i"1	pt,xqPr0,T sˆR nd
					(1.3)
	where in the above equation

  t the degenerate component. Introducing the embedding matrix B from R d into R nd , i.e. B " pI d,d , 0 d,d , . . . , 0 d,d loooooomoooooon

				q ˚"
			pn´1q times
	L t ϕpxq " xFpt, xq, Dϕpt, xqy	`1 2	Tr ´apt, xqD 2 x 1 ϕpxq ¯,	(1.4)

pI d,d , 0 d,pn´1qd q ˚, where "˚" stands for the transpose, we rewrite (1.1) in the shortened form dX t " Fpt, X t qdt `Bσpt, X t qdW t , where F " pF 1 , . . . , F n q is an R nd -valued function. For all ϕ P C 2 0 pR nd , Rq and pt, xq P r0, T s ˆRnd let

  The representation (1.[START_REF]Appendix: Proof of technical results[END_REF]) is what is usually called the Zvonkin Transform. The main thing in this representation is that the bad drift can be, up to a remainder, rewritten in term of the solution of the system of PDEs (1.6) for which the source term is precisely a mollified version of the bad drift itself.

	where
	R m t pXq :"	ż t	Fps, X s q ´Fm ps, X s qds	´ż t	pL s	´Lm
		0		0

ż t 0 Fps, X s qds " ´Um p0, xq `Um pt, X t q ´ż t 0 DU m ps, X s qBσps, X s qdW s `Rm t pXq, (1.7) s qU m ps, X s qds.

d

  Xt " A t Xt dt `BdW t , (1.10) where A t is the nd ˆnd matrix with sub-diagonal block a i,i´1 ptq of size d ˆd and 0 d,d elsewhere. In particular, The entries pa i,i´1 ptqq iPrr2,nss are uniformly in time non-degenerate elements of R d b R d (which expresses the weak Hörmander condition). The corresponding generator Lt writes for all ϕ P C 2 0 pR nd , Rq: Rs,u BB ˚R s,u du. From (1.14), the density at time v " s and at the spatial point y therefore writes: Precisely, for a given T ą 0 there exists C :" C `pA v q vPr0,T s , T ˘ě 1 s.t. for all ξ P R nd , C ´1ps ´tq ´1|T s´t ξ| 2 ď x Ks,t ξ, ξy ď Cps ´tq ´1|T s´t ξ| 2 , (1.16)where for all u ą 0, we denote by T u the intrinsic scale matrix:Importantly, the good scaling property stated in (1.16) indicates that, for a given initial time t and for all i P rr1, nss, each R d -valued component Xi s has typical fluctuations of order ps ´tq i´1 2 which corresponds to those of the pi ´1q th iterated integrals of the Brownian motion. Accordingly, we derive that the frozen density p also satisfies the bound

		A t "	¨0d,d a 2,1 ptq 0 d,d ¨¨¨¨¨¨¨¨¨0 d,d ¨¨¨¨¨¨0 d,d 0 d,d a 3,2 ptq 0 d,d ¨¨¨0 d,d . . . 0 d,d . . . 0 d,d ¨¨¨0 d,d a n,n´1 ptq 0 d,d 0 d,d ¨¨¨0 d,d u n I d,d . . . T u " ¨uI d,d 0 d,d ¨¨¨0 d,d 0 d,d u 2 I d,d 0 d,d . . . . . . . . . . . . ' . . . ‹ ‹ ‹ .	' ‹ ‹ ‹ ‹ ‹ .	(1.11) (1.17)
	ppt, s, x, yq ď	C ps ´tq	Lt ϕpxq " xA t x, Dϕpxqy n 2 d 2	`1 2	∆ x 1 ϕpxq.
				Xv " Rv,t x	`ż v	Rv,u BdW u .	(1.14)
					t
	ş s t ppt, s, x, yq " Hence, the covariance between given times t ă s writes Ks,t :" 1 p2πq nd 2 detp Ks´t q 1 2 exp ˆ´1 2 A p Ks´t q ´1`R s,t x ´y˘, Rs,t x	´yE ˙. (1.15)
	Note that the resolvent also appears in (1.14) and in the density. Since the drift in (1.10)

In such a case, each component ũi , i P rr1, nss of the solution Ũ of the corresponding system of PDEs # pB t `L t q Ũpt, xq " Fpt, xq, pt, xq P r0, T q ˆRnd , ŨpT, xq " 0 nd , (1.12)

where F is a non-linear (non-mollified) source satisfying (T β ), writes

ũi pt, xq " ´ż T t ds ż R nd dyF i ps, yqppt, s, x, yq. (1.13)

Above, p stands for the transition density of the Gaussian process pX v q vě0 with dynamics (1.10). Using the resolvent associated with pA v q vPrt,ss , i.e. B s Rs,t " A s Rs,t , Rt,t " I nd,nd , the above equation can be explicitly integrated. Precisely, for a fixed starting point x at time t: in unbounded, the term Rs,t x actually corresponds to the transport of the initial condition x through the associated deterministic and linear differential system. It is well known, see e.g.

[START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF]

and Section 2.1 below, that the covariance Ks,t enjoys what we will call a good scaling property.

exp ´´C ´1ps ´tq|T ´1 s´t p Rs,t x ´yq| 2 ¯": C pC ´1 pt, s, x, yq,

  t. x l : this is what will be called a cancellation (or centering) argument in the following. Precisely, denoting for conciseness by θ s,t pxq " Rs,t x (which is coherent with the notation below when handling non-linear flows), we write:

	ż	ż
	dy 1:l´1 F i pt, y 1 , ¨¨¨, y l´1 , θ l s,t pxq, ¨¨¨, θ n s,t pxqqD x 1 D x l
	R pl´1qd	R pn´pl´1qqd

  ¨¨¨0 d,d D z n´1 F n pv, z n´1 , z n q 0 d,d Density of the linearized dynamics) Under (A), we have that, for all s P pt, T s the random variable Xpτ,ξq s in (2.6) admits a Gaussian density ppτ,ξq pt, s, x, ¨q which writes for all y P R nd :

	Proposition 2 (ppτ,ξq pt, s, x, yq :"	p2πq	nd 2 detp 1	Kpτ,ξq s,t q	1 2	exp	2 ˆ´1	A p	Kpτ,ξq s,t q ´1pm pτ,ξq s,t pxq ´yq, m pτ,ξq s,t pxq	´yE ˙,
											(2.9)
	with	Kpτ,ξq s,t
				. . .			0 d,d		. . .	d,d . . .	. . .	‹ ‹ ‹ ' ‹ ‹ .
				0 d,d

  Ĉ|ζ| 2 . A similar rescaling argument yields on the deterministic system (2.5) of the resolvent yields that Rpτ,ξq ps, tq can also be written as: r Rpτ,ξq ps, tqs ˚" T ´1 s´t Rpτ,ξq,s,t p1, 0q is the resolvant at time 1 of the rescaled system T s´t r Rpτ,ξq pt vps For all k P rr1, nss, 0 ď t ď s ď T , px, ξq P R nd ˆRnd , and M P R d the following identity hold: ,t pxqq k dy " 0 d . Differentiating this expression w.r.t. x k and using the Leibniz formula (recalling as well the identity (2.7) which yields D x k rm pτ,ξq s,t pxqs k " p Rpτ,ξq ps, tqq k,k " I d,d ) gives (2.14).Fix t P r0, T s, ξ P R nd . With the notations of the previous paragraph, we introduce the following inhomogeneous semi-group associated with (2.3) for τ " t. Namely, for all s P pt, T s, g P B lin pR nd , Rq (space of measurable functions with linear growth), x P R nd :

	From the explicit expression (2.9), Proposition 1 and the above equation, we eventually
	derive:
	|D x j	ppτ,ξq pt, s, x, yq| ď	C ps ´tq j´1 2	´ps ´tq	1 2 |T ´1 s´t pm s,t pxq ´yq| pτ,ξq	1
					ps ´tq	2 n 2 d	exp ´´C ´1ps ´tq|T ´1 s´t pm s,t pxq ´yq| 2 pτ,ξq	ď
					C ps ´tq j´1 2	ppτ,ξq C ´1 pt, s, x, yq,
	d up to a modification of C, which gives the statement for one partial derivative. The controls 2 exp ´´C ´1ps ´tq ˇˇT ´1 s´t `mpτ,ξq s,t pxq ´y˘ˇˇ2 ": on the higher order derivatives are obtained similarly (see e.g. the proof of Lemma 5.5 of
	C ř n i"1 α i pi´1 2 q ppτ,ξq pt, s, x, yq). ppτ,ξq C ´1 pt, s, x, yq. let us specify a useful control involving the previous Gaussian kernel which will [DM10] for the bounds on D 2 x 1 Now, (2.10) ps ´tq be exploited in some cancellation techniques.
	Proof: Expression (2.9) readily follows from (2.5). The control (2.10) in then a direct consequence of Proposition 1 for α " 0. Differentiating w.r.t. x recalling from (2.7) that x Þ Ñ m pτ,ξq s,t pxq is affine yields: D x j ppτ,ξq pt, s, x, yq " ´"" Rpτ,ξq ps, tq ‰ ˚p Kpτ,ξq s,t q ´1pm pτ,ξq s,t pxq ´yq ı " Rpτ,ξq ps, tq ‰ ˚p Kpτ,ξq s,t q ´1 " ps ´tq " Rpτ,ξq ps, tq ‰ ˚T´1 s´t p y K s,t 1 q ´1T ´1 s´t , (2.12) Proposition 3 ż R nd D x k ppτ,ξq pt, s, x, yq A M, py´m pτ,ξq s,t pxqq Associated
	where y K s,t 1 is the covariance matrix of the rescaled process `ps ´tq at time 1. From the good-scaling property of Proposition 1, it is plain to derive that 1 2 T ´1 s´t Xt,x t`vps´tq ˘vPr0,1s inhomogeneous semi-group
	y K s,t
						ż
				P ξ s,t gpxq :"	R nd	" dyp pt,ξq pt, s, x, yqgpyq. p Rpτ,ξq,s,t p1, 0q ı ˚Ts´t ,	(2.13) (2.15)
	where again p ´tq, tqs ˚T´1 s´t " ´" p Rpτ,ξq,s,t pv, 0q	‰ ˚¯vPr0,1s	associated with (2.5). From the analysis per-
	formed in Lemma 5.1 in [HM16] (see also the proof of Proposition 3.7 in [DM10]) one
	derives that there exists Ĉ1 s.t. for all ζ P R nd , |	"	p Rpτ,ξq,s,t p1, 0q ‰ ˚ζ| ď Ĉ1 |ζ|. Equations
	(2.11), (2.12) and (2.13) therefore yield:
	|D x j	ppτ,ξq pt, s, x, yq|
	ď ps ´tq ´j`1 2 ˇˇˇˇˆ"	p Rpτ,ξq,s,t p1, 0q ı ˚`y K s,t 1	˘´1 `ps ´tq	1 2 T ´1 s´t pm

j ppτ,ξq pt, s, x, yq. (2.11) The point is now to use scaling arguments. We can first rewrite 1 is a non-degenerate bounded matrix, i.e. there exists Ĉ ě 1 s.t. for all ζ P R nd , Ĉ´1 |ζ| 2 ď x y K s,t 1 ζ, ζy ď pτ,ξq s,t pxq ´yq ˘˙j ˇˇˇˇp pτ,ξq pt, s, x, yq ď Cps ´tq ´j`1 2 ps ´tq 1 2 |T ´1 s´t pm pτ,ξq s,t pxq ´yq|p pτ,ξq pt, s, x, yq. k E dy " xM, 1 d y, 1 d " p1, ¨¨¨, 1q P R d . (2.14) Proof: From Proposition 2, we have ş R nd ppτ,ξq pt, s, x, yqpy ´mpτ,ξq s

  R nd dy l:n pξ pt, s, x, yq does not depend on x l anymore. This gives the statement (2.18).Let us now prove (2.19). The idea is to use first a centering argument w.r.t. to the variables l to n. Namely, rf j ps, ¨qqs β j |p¨´θ s,t pξqq j | β j ¸ff pxq, where for a function g P B lin pR d , Rq we denote, ´1 pt, s, x, ¨q :" pξ C ´1 pt, s, x, ¨q is the density appearing in Proposition 2 * * . That is, P ξ s,t can somehow be viewed as the pseudo semi-group associated with the density ppt,ξq

	where appearing in Proposition 2. The dependence in C in P ξ ppt,ξq C C s,t is then omitted for notational ´1
	simplicity.
	In the current regularized setting, it is plain to observe that taking ξ " x, m pt,xq s,t pxq "
	θ s,t pxq. Therefore, from (2.10):
	ˇˇD x 1 D x l	"	P ξ s,t f ps,	¨qı	pxq ˇˇˇˇξ	"x	ď C	n ÿ j"l	ż R nd	rf j ps, ¨qs β j dy	|py ´θs,t pxqq j | β j ps ´tq l	px C ´1 pt, s, x, yq
										ď C	n ÿ j"l	rf j ps, ¨qs β j	ż R nd dy	|py	´mpt,xq s,t pxqq j | β j ps ´tq l	px C ´1 pt, s, x, yq
													n
													ÿ
										ď C
													j"l
													R nd	dyp ξ pt, s, x, yqf py 1,k´1 , θ k:n s,t pξqq
													ż	ż
										" D x l	R nd	dy 1:l´1 f py 1,k´1 , θ k:n s,t pξqq	R nd	dy l:n	pξ pt, s, x, yq.
			ˇˇD x 1 D x l	"	P ξ s,t f ps,	¨qı	pxq ˇ"
			ˇˇD x 1 D x l	"	P ξ s,t ´f ps, ¨q ´f ps, ¨1:l´1 , θ l:n s,t pξqq ¯ı pxq ˇď
			Cps ´tq ´pl´1{2q´1{2	«	s,t P ξ	˜n ÿ
													j"l
						ż							ż
			P ξ s,t gpxq "	R nd		dyp	pt,ξq C ´1 pt, s, x, yqgpyq :"	R nd	dyp ξ C ´1 pt, s, x, yqgpyq,

t ´f p¨1 :k´1 , θ k:n s,t pξqq ¯pxq " D x l ż Now, the structure of the linearized dynamics (2.3) yields that the variable x l only appears in p Rpt,ξq ps, tqxq l:n from its l th to its n d-dimensional block. Therefore, setting e.g. ỹl:n " y l:n ´p Rpt,ξq ps, tqxq l:n , the integrated quantity ş

  @pF i ps, ¨q ´Fi ps, θ s,t pξqq ´Dx i´1 F i ps, θ s,t pξqqp¨´θ s,t pξqq i´1 q, D i ups, ¨qD

	upt, xq "	ż T	"	P ξ s,t f ps,	¨qı	pxqds	`ż T	"	P ξ s,t ´pL s ´L ξ s qu ¯ps,	¨qı	pxq
		t										t
	"	ż T	"	P ξ s,t f ps,	¨qı	pxqds	`ż T	"	P ξ s,t `@F 1 ps, ¨q ´F1 ps, θ s,t pξqq, D 1 ups,	¨qD˘ı pxq
		t										t
		`ż T t	"	P ξ s,t ˆ1 2	Tr	"	aps, ¨q ´aps, θ s,t pξqqD 2 1 ups,	¨q‰	˙ pxqds
		`ż T	«	s,t P ξ	˜n ÿ
			t				i"2
												¸ffpxqds,
												(2.20)

  @ pF i ps, ¨q ´Fi ps, θ s,t pξqq ´Dx i´1 F i ps, θ s,t pξqqp¨´θ s,t pξqq i´1 q, D i ups, ¨qD

									ps,	¨qı	pxqds	`ż T	D x l	"	P ξ s,t ´pL s ´L ξ s qu ¯ps,	¨qı	pxqds
										t
	"	#	ż T	D x l	"	P ξ s,t f ps,	¨qı	pxqds
				t
		`ż T	D x l	"	P ξ s,t `@pF 1 ps, ¨q ´F1 ps, θ s,t pξqqq, D 1 ups,	¨qD˘ı pxqds
				t
		t `ż T	D x l	"	P ξ s,t ˆ1 2	Tr	"´a	ps, ¨q ´aps, θ s,t pξqq ¯D2 1 ups,	¨qı ˙ pxqds	+
		`# ż T	D x l	«	s,t P ξ	ˆn ÿ
					t					i"2
										+
										˙ffpxqds
	":	ż T	dsH ξ l ps, xq	`ż T	dsI ξ l ps, xq.	(2.21)
		t								t

  3.Similarly, from (2.20), (2.21), for the drift associated with the non-degenerate part, we first rewrite from the centering properties of Lemma 1:|D x 1 H l,2 ps, xq| :" ˇˇD x l D x 1 "P ξ s,t `@F 1 ps, ¨q ´F1 ps, θ s,t pξqq, D 1 ups, ¨qD˘ı pxqds ˇˇˇˇˇˇξ

					"x
	ď ˇˇD x l D x 1	"	P ξ s,t `@F 1 ps, ¨q ´F1 ps, ¨1:l´1 , θ l:n s,t pξqq, D 1 ups,	¨qD˘ı pxq ˇˇˇˇˇˇξ
					"x
	`ˇˇD x l D x 1	"	P ξ s,t `@F 1 ps, ¨1:l´1 , θ l:n s,t pξqq ´F1 ps, θ s,t pξqq,
	s,t pξqq `D1 ups, ¨q ´D1 ups, ¨1:l´1 , θ l:n	˘D˘‰	pxq ˇˇˇˇˇˇξ
					"x

  : |D x 1 H l,31 ps, xq| `|D x 1 H l,32 ps, xq| . (2.26) The term D x 1 H l,31 ps, xq is already centered at the appropriate scales, i.e. from variables l to n. We thus readily derive, similarly to the previous computations thanks to Lemma 1 and recalling that a is Lipschitz continuous, that: |D x 1 H l,31 ps, xq| ď C}D 2 1 u} 8 raps, ¨qs 1 ps ´tq ´l`pj´1 2 q ď C}D 2 1 u} 8 raps, ¨qs 1 ps ´tq ´1 2 , (2.27) which does not give a critical contribution w.r.t. the previously exhibited thresholds in (2.24) and (T β ). For the contribution |D x 1 H l,32 ps, xq| we use, in the same spirit as for |D x 1 H l,2 ps, xq|, a centering argument and an integration by parts to obtain:

		|D x 1 H l,32 ps, xq|
		ˇˇˇˇD
	"		x l D x 1	"	P ξ s,t ´1 2	Tr	"`a ps, ¨1:l´1 , θ l:n s,t pξqq ´aps, θ s,t pξqq	pD
			2 1 ups, ¨q ´D2 1 ups, ¨1:l´1 , θ l:n s,t pξqqq ‰ ¯ıpxq ˇˇˇˇˇˇˇˇξ
									"x
	"	ˇˇˇˇ1 2	D x l D x 1	˜d ÿ j"1	ż R nd	dy	A B y j 1	´p ξ pt, s, x, yq `aj¨p s, y 1:l´1 , θ l:n s,t pξqq ´aj¨p s, θ s,t pξqq ˘¯,
		s,t pξqq ´D1 ups, yq ´D1 ups, y 1:l´1 , θ l:n	¯E¸ˇˇˇˇˇˇˇˇˇξ
									"x
					"	P ξ s,t ˆ1 2	Tr	"`a ps, ¨q ´aps, θ s,t pξqq ˘D2 1 ups,	¨q‰	˙ pxq ˇˇˇˇˇˇˇˇξ
									"x
	" ˇˇˇD x l D x 1	"	P ξ s,t ˆ1 2	Tr	"`a ps, ¨q ´aps, ¨1:l´1 , θ l:n s,t pξqq ˘D2 1 ups,	¨q‰	˙ pxq ˇˇˇˇˇˇˇˇξ
									"x
		`ˇˇˇD x l D x 1	"	P ξ s,t ˆ1 2	Tr	"`a ps, ¨1:l´1 , θ l:n s,t pξqq ´aps, θ s,t pξqq ˘D2 1 ups,	¨q‰	˙ pxq ˇˇˇˇˇˇˇˇξ
									"x
									n
									ÿ
									j"l

3 ps, xq| :" ˇˇˇD x l D x 1 "

  32 ps, xq| ď C}DD 1 ups, ¨q} 8 H l ps, xq ˇˇˇˇď CT δ p}D 1 u} 8 `}DD 1 u} 8 q.Degenerate part of the operator: estimates for D x 1 I l ps, xq. These are the most delicate terms to handle. Restarting from (2.21), we first write for all l P rr2, nss: @ pF i ps, ¨q ´Fi ps, θ s,t pξqq ´Di´1 F i ps, θ s,t pξqqp¨´θ s,t pξqq i´1 q, @ pF i ps, ¨q ´Fi ps, θ s,t pξqq ´Di´1 F i ps, θ s,t pξqqp¨´θ s,t pξqq i´1 q, @ pF i ps, ¨q ´Fi ps, θ s,t pξqq ´Dx i´1 F i ps, θ s,t pξqqp¨´θ s,t pξqq i´1 q,

	which readily yields that
	|D x 1 I l ps, xq|
	ď ˇˇˇD x l D x 1	"	s,t P ξ	ˆn ÿ
							i"l`1
							˙pxq ˇˇˇˇˇˇˇˇξ
					D i ups,	¨qD
							"x
	`ˇˇˇD x l D x 1	"		P ξ s,t	ˆl ÿ	@	F i ps, ¨q ´Fi ps, ¨1:l´1 , θ l:n s,t pξqq,
							i"2
							˙pxq ˇˇˇˇˇˇˇˇξ
					D i ups,	¨qD
							"x
	`ˇˇˇD x l D x 1	"		P ξ s,t	ˆl ÿ	@	pF i ps, ¨1:l´1 , θ l:n s,t pξqq ´Fi pθ s,t pξqq
							i"2
	´Di´1 F " s,t P ξ `n ÿ	ż R nd ˆ´|py ´θs,t pξqq 1:l´1 | ps ´tq 1 2 ď C}DD 1 ups, ¨q} 8 ps ´tq ´1 2 . dy ps ´tq l `1¯r aps, ¨qs 1 |py ´θs,t pξqq l:n | px C ´1 pt, s, x, yq	(2.28)
	With the notations of (2.21), plugging (2.27), (2.28) into (2.26) and together with (2.25), i"l`1
	(2.23), we eventually derive that there exists δ :" δppAqq ą 0: D i ups, ¨qD˘ı pxq ˇˇˇˇˇˇξ
							ˇˇˇˇż	T	"x
							dsD x 1 (2.29)
							t
	|D x 1 I l ps, xq|
	" ˇˇˇD x l D x 1	"	s,t P ξ	ˆn ÿ
							i"2
							˙pxq ˇˇˇˇˇˇˇˇξ
					D i ups,	¨qD	,
							"x

i ps, θ s,t pξqqp¨´θ s,t pξqq i´1 q, D i ups, ¨qD

˙pxq ˇˇˇˇˇˇˇˇξ "x ": |D x 1 I 1,l ps, xq| `|D x 1 I 2,l ps, xq| `|D x 1 I 3,l ps, xq|.

(2.30)

We emphasize that the integrands D x 1 I 1,l ps, xq and D x 1 I 2,l ps, xq are already designed to smoothen the time singularities generated by the cross differentiation of the inhomogeneous semi-group w.r.t. the variables x l and x 1 . Indeed, on the one hand

|D x 1 I 1,l ps, xq| " ˇˇD x l D x 1

  l ps, xq|

	ˇˇˇˇD
	"	x l D x 1	"	P ξ s,t	ˆl ÿ	B pF i ps, ¨1:l´1 , θ l:n s,t pξqq ´Fi ps, θ s,t pξqq
								i"2
			´Di´1 F i pθ s,t pξqq `¨´θ s,t pξq ˘i´1 q,
						n ÿ	ż 1	dλD k D i u `s, ¨1:l´1 , θ l:n s,t pξq `λp¨l :n	´θl:n s,t pξqq ˘p¨´θ s,t pξqq k	pxq F˙ ˇˇˇ"
						k"l	0
	ˇˇˇˇl ÿ	n ÿ	ż 1	dλD x l D x 1	"	P ξ s,t	ˆB´F i ps, ¨1:l´1 , θ l:n s,t pξqq ´Fi ps, θ s,t pξqq
	i"2	k"l		0
								¯,
			´Dx i´1 F i ps, θ s,t pξqq `¨´θ s,t pξq ˘i´1
								ˇˇˇˇ,
								F˙
	D i D k ups, ¨1:l´1 , θ l:n s,t pξq `λp¨l :n	´θl:n s,t pξqqqp¨´θ s,t pξqq k	pxq	(2.34)

  py 1:i´1 , y i`1:n qq, denoting as well for a ą 0, z P R d , by N a pzq " 1

	where, with the notations of Proposition 2,
		ż			n
	qczi pt, s, x, py 1:i´1 , y i`1:n qq :"	R d	px c pt, s, x, yqdy i "	jPrr1,nss,j‰i ź	N cps´tq 2j´1 `pθ s,t pxq ´yq j	˘,
						(2.40)
			p2πaq	d 2	exp ´´|z| 2 2a ¯the standard Gaussian
	density of R

d with covariance matrix aI d,d . In the above control c " C ´1.

  y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k

							ı
	"	ż 1 ps´tq	ρ i,k	dvv	α k i 2 ´1	}B v h v ‹ Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k	p¨q} L 1 pR d ,Rq
	"	ż 1 ps´tq	ρ i,k	dvv	α k i 2 ´1	ż R d	dz ˇˇż R d	B v h v pz ´yi qΨ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k	py i qdy i ˇˇ.

  ! |pθ s,t pxq ´yq j | β j ) `|pθ s,t pxq ´yq i´1 | 1`η

				ps,y 1:i´1 ,y i`1:n q,pt,xq	ı
				i,pl,1q,k
	ď C	ż 1 ps´tq	ρ i,k	dvv	α k i 2 ´1	ż R d	dz	ż R d	dy i	h cv pz ´yi q v	qc pt, s, x, yq ps ´tq pl´1 2 q`1 2	|pθ s,t pxq ´yq k |
	ÿ ˆ# l´1											+
		j"i
	ď C	ż 1 ps´tq	ρ i,k	dvv	α k i 2 ´1	ż R d	dz	ż R d	dy i	h cv pz ´yi q v	qc pt, s, x, yq 1 ps ´tq 2
	ˆ#l´1 ÿ ps ´tq β j pj´1{2q `ps ´tq p1`ηqpi´3{2q
		j"i
									ż 1
										ps´tq ρ i,k	dvv ´3 2	`αk i 2 ps ´tq ´1{2
	ˆ#l´1 ÿ

+

ď C qczi pt, s, x, py 1:i´1 , y i`1:n qq j"i ps ´tq β j pj´1{2q `ps ´tq p1`ηqpi´3{2q + ď C qczi pt, s, x, py 1:i´1 , y i`1:n qqps ´tq r´1 2 `αk i 2 sρ i,k ´1 2 `ps ´tq β i pi´1{2q `ps ´tq p1`ηqpi´3{2q

  Since this condition should hold for any β i P `2i´2 2i´1 , 1 ‰ and since the parameter η P p0, 1q is small (see (H η )) we have β i pi ´1{2q ^p1 `ηqpi ´3{2q " p1 `ηqpi ´3{2q. The above condition rewrites:Our global integrability constraint associated with the i th variable in the k th derivative writes:

	´1	´1 2	`γk i " r´1 2	`αk i 2	sρ i,k	´1 2	`p1 `ηqpi ´3{2q ą ´1,	(3.2)
	which gives
						ρ i,k ă	p1 `ηqp2i ´3q 1 ´αk	`1
	´1 2	`γk i " r´1 2	`αk 2 i	sρ i,k	´1 2	`´β i pi ´1{2q ^p1 `ηqpi ´3{2q	¯.
		´1	´1 2	`γk i " r´1 2	`αk i 2	sρ i,k	´1 2	`p1 `ηqpi ´3{2q.

i

.

  where for the second inequality, we used that for k ě l ą i, |py ´θs,t pxqq k |ps ´tq ´pl´1{2q ď |py ´θs,t pxqq k |ps ´tq ´pk´1{2q which can be absorbed by the k th variables of qc .Writing now for any λ P r0, 1s, |z ´θs,t pxq i | ď λ|z ´yi | `|z `λpy i ´zq ´pθ s,t pxqq i |,

	we thus derive
	ps,y 1:i´1 ,y i`1:n q,pt,xq 2,i,pl,1q,k |T

+

,

  Consider now T 1 , |z ´yi | β i |py ´θs,t pxqq k |dy i From (3.8) and (3.9) we derive, with the notation introduced in (2.40):}B v h v ‹ Ψ ps,y 1:i´1 ,y i`1:n q,pt,xq i,pl,1q,k } L 1 pR d ,Rqď C qczi pt, s, x, py 1:i´1 , y i`1:n qq dy i h cv pz ´yi qN cps´tq 2i´1 pz `λpy i ´zq ´pθ s,t pxqq i q + ď C qczi pt, s, x, py 1:i´1 , y i`1:n qq ˜1 v 1´β i

	|T 1,i,pl,1q,k ps,y 1:i´1 ,y i`1:n q,pt,xq	`v, z ˘| ď C	ż R d	h cv pz ´yi q v	qc pt, s, x, yq ps ´tq pl´1 2 q`1 2
			ď C	ż R d	h cv pz ´yi q v 1´β i 2	qc pt, s, x, yq 2 ps ´tq 1	dy i .	(3.9)
					#	v 1´β i 2 ps ´tq 1	1 2	`˜v ps ´tq i 2 ´1`β i	ps ´tq i´p1`ηqpi´3{2q `v´1 2	ż
	1	ż	ż
	dλ	dz
	0	R d	R d
						2 ps ´tq	1 2	`v ´1`β i 2 ps ´tq i	`v´1 2 ps ´tq i´p1`ηqpi´3{2q

  Let us check condition (3.10) is satisfied. Actually, the first two terms in (3.11) yield negligible contributions. Recall indeed from the statement of the lemma that the parameter α k i must be chosen so that α k i ă choice (recall indeed that we chose η ă inf jPrr2,nss tβ j ´2j´2 2j´1 u in assumption (H η )). It gives in particular that α k i `βi ą which already provides a regularizing term in time for the first term in the r.h.s. of(3.11). Now from (3.3), ρ i,k ă p1`ηqp2i´3q`1 is an admissible choice. We therefore get for the exponent of the second term in (3.11):

									1´p1´β k qpk´1 2 q i´1 2	:" ᾱk i . Since β k P p 2k´2 2k´1 , 1s,
	ᾱk i " 2´p1´β k qp2k´1q 2i´1	ą 3´2k`p2k´2q 2i´1	" 1 2i´1 . Thus,
									α k i "	1 `η 4 2i ´1 ,	(3.12)
	is an admissible 1`η 4 `p2i´2q 2i´1	" 1 `η 4 2i´1 . Hence,
							ρ i,k	α k i `βi 2	´1	´1 2	ą	´1 2
				1´α k i		": ρi,k . The previous choice for α k i gives
	ρi,k "	2i ´2 `ηp2i ´3q 2i´1´p1`η 4 q 2i´1	" p2i ´1q	2i ´2 `ηp2i ´3q 2i ´2 ´η 4	ą p2i ´1q,
	and
									ρ i,k " 2i ´1	(3.13)
			ρ i,k	α k i `βi 2	´ią	1 2	´ρi,k p1 `η 4 2i ´1 q ´2i ¯ą	´1 2	.
									.10)
	Write:
			ż ps´tq ρ i,k	˜v´3 2	2 i `βi `αk	2 i `βi k	2 k i	ḑ
	B 1,ρ i,k pt, sq ď		0	dv			ps ´tq	1 2	`v´1`α ps ´tq i	`v´1`α ps ´tq 3 2 ´ηpi´3 2 q
			C ´ps ´tq ρ i,k	α k i `βi 2	´1	´1 2 `ps ´tq ρ i,k	α k i `βi 2	´i `ps ´tq ρ i,k	α k 2 ´3 2 `ηpi´3 2 q i	¯.
									(3.11)

  x 1:i´1 , ¨, x i`1:n qpzq| (3.19) dx i h v pz ´xi qD x i D x k upt, x 1:i´1 , x i , x i`1:n q

	ˇˇˇˇż	ˇˇˇ"
	"
	R d
	ˇˇˇˇż
	R d

  x 1 , yqf ps, yq. (3.21) Differentiating (3.19) w.r.t.x k and setting then ξ 1 " x 1 , ξ1 " x, we rewrite:h v ‹ D x i D x k upt, x 1:i´1 , ¨, x i`1:n qpzq ": `hv ‹ D x i D x k upt, x 1:i´1 , ¨, x i`1:n qpzq ˘|S i ``h v ‹ D x i D x k upt, x 1:i´1 , ¨, x i`1:n qpzq ˘|S c i ``h v ‹ D x i D x k upt, x 1:i´1 , ¨, x i`1:n qpzq ˘|BS i ,(3.22)where, thanks to (2.21), (3.20) and then (3.19): `hv ‹ D x i D x k upt, x 1:i´1 , ¨, x i`1:n qpzq ˘|S i

	ż	ż
	:"	ds	dx
	S i	R d

i D z h v pz ´xi q

  x 1:i´1 , ¨, x i`1:n qpzq| ˇˇS Those terms can then be handled following the previous analysis performed in Theorem 3 and Lemma 3, observing here that, w.r.t. the previous proofs, the above terms are not differentiated w.r.t. x 1 . This improves the exponents of the time singularities of 1{2. Similarly to (2.23), (2.25), (2.27) and (2.28) this therefore yields for the terms H ξ k ps, xq:ˇˇH `ps ´tq ´1`δ k i `}DD 1 u} 8 ˘,(3.27)with 1 ąδ k i ą 1{2. Reproducing the arguments that led to equations (2.31), (2.32), (2.39) and the statement of Lemma 3, exploiting again that there is now no differentiation w.r.t. x 1 we get with the notations of (2.35)}D i D k u i ps, ¨q} } `Dx i D x k u ˘ips, ¨q} C|x i ´z| α k i T δ 1 ´}Du} 8 `}DD 1 u} 8 `sup } `Dx i D x k u ˘ips,¨q} ,T s,z j , jPrr1,nss,j‰i }D x i D x k u i ps, z 1,i´1 , ¨, z i`1,n q}

	for some δ 1 :" δ 1 ppAqq ą 0, denoting by
	sup sPr0,T s								B 8,8 α k i ´1	:"	sup	B 8,8 α k i ´1	,
										(3.29)
				R d	dx i D z h v pz ´xi q	`żS i	ds	" H ξ k ps, xq ´Hξ 1 k ps, x 1 q ı
			`żS i	ds	" I ξ k ps, xq ´Iξ 1 k ps, x 1 q ı	˘ˇˇˇˇˇˇp	ξ,ξ 1 q"px,x 1 q
		ď	ż R d	dx i	h cv pz ´xi q v 1 2	ż S i	ds ˆˇH ξ k ps, xq ˇˇˇˇˇξ	"x	`ˇH ξ 1 k ps, x 1 q ˇˇˇˇˇξ	1 "x 1
										˙.
										`ˇI ξ k ps, xq	ˇˇˇˇˇξ	"x	`ˇI ξ 1 k ps, x 1 q ˇˇˇˇˇξ	1 "x 1	(3.26)
		ˇˇI k ps, xq ˇˇˇˇˇˇξ ξ	"x	ď Cps ´tq ´1`γ k i `}Du} 8 `sup sPr0,T s	B 8,8 α k i ´1	q.
	Similar bounds hold for ˇˇH k ps, x 1 q ˇˇˇˇˇˇξ ξ 1	and ˇˇI k ps, x 1 q ˇˇˇˇˇˇξ ξ 1	. Hence, from (3.27) and
										1 "x 1	1 "x 1
	(3.28),
		˜ˇˇHξ					ḑ
	ż	ds		k ps, xq ˇˇˇˇˇˇξ	`ˇˇHξ 1 k ps, x 1 q ˇˇˇˇˇˇξ	`ˇˇIξ k ps, xq ˇˇˇˇˇˇξ	`ˇˇIξ 1 k ps, x 1 q ˇˇˇˇˇˇξ
	S i								"x	1 "x 1	"x	1 "x 1
									2
	C	´ż pt`c 0 |x i ´z|	2i´1 q^T	ds ´`ps ´tq ´1`δ k i }Du} 8 `}DD 1 u} 8	ps
		t
		i p}Du} 8 `sup sPr0,T s ´tq ´1`γ k	B	α k 8,8 i ´1	q ď
	C ´|x i ´z|	2δ k i 2i´1 ^pT ´tq δ k i p}Du} 8 `}DD 1 u} 8 q
	`|x i ´z|	2γ k i 2i´1 ^pT ´tq γ k i p}Du} 8 `sup sPr0,T s	B 8,8 α k i ´1	q ď
										¯,
										B	α k 8,8 i ´1	(3.28)

i :" ˇˇˇż ξ k ps, xq ˇˇˇˇˇˇξ "x ď C } `Dx i D x k u ˘ips, ¨q} sPr0,T s } `Dx i D x k u ˘ips, ¨q}

sPr0

  x 1:i´1 , ¨, x i`1:n qpzq| ˇˇS i ď CT δ 1 ´}Du} 8 `}DD 1 u} 8 `sup sPr0,T s } `Dx i D x k u ˘ips, ¨q} CT δ 1 ´}Du} 8 `}DD 1 u} 8 `sup sPr0,T s } `Dx i D x k u ˘ips, ¨q}

	B 8,8 ¯żR d α k i ´1	dx i	h cv pz ´xi q 2 v 1	|z ´xi | α k i
	ď B 8,8 α k i ´1	¯v α k i 2 ´1

  s, x, yq ´Dx k pξ pt, s, x 1 , yq " ´ż 1 0 dλD x i ´Dx k pξ pt, s, x `λpx 1 ´xq, yq ¯¨px 1 ´xq i .|D x k pξ pt, s, x, yq ´Dx k pξ pt, s, x 1 , yq| ď C|px 1 ´xq i |ps ´tq pi´1 2 q`pk´1 2 q ´1 pt, s, x `λpx 1 ´xq, yq.(3.32) Now, from the definition of pC ´1 in Proposition 2, recalling as well from (2.7) that x Þ Ñ m ξ s,t pxq is affine, we get:|p ξ C ´1 pt, s, x `λpx 1 ´xq, yq|

			(3.31)
	From Lemma 1 we thus derive:
			ż 1
	0 C ď dλp ξ C 2 ps ´tq n 2 d expp´cps ´tq|T ´1 s´t pm ξ s,t px `λpx 1 ´xqq ´yq| 2 q
	ď	C ps ´tq	n 2 d 2

1 2

 1 |T ´1 s´t m ξ s,t px ´x1 q| ď Cps ´tq 1 2 |T ´1 s´t px ´x1 q| " Cps ´tq ´i`1{2 |px 1 ´xq i | ď C, from the very definition of S c i . Hence, |p ξ C ´1 pt, s, x `λpx 1 ´xq, yq| ď ´1 s´t pm ξ s,t pxq ´yq| 2 q, so that, from (3.32) and recalling that on S c i , |px 1 ´xq i |{ps ´t ď Cp|px 1 ´xq i |{ps ´tqq α k i , the following important control holds: |D x k pξ pt, s, x, yq ´Dx k pξ pt, s, x 1 , yq| ď C|px 1 ´xq i | α k |h v ‹ D x i D x k upt, x 1:i´1 , ¨, x i`1:n qpzq| ˇˇS

	C ps ´tq Write now from (3.24), recalling that c n 2 d 2 ξ1 " ξ, expp´c 2 ps ´tq α k i pi´1 2 q`pk´1 2 q i :" ˇˇˇˇż R d dx i D z h v pz ´xi q ´żS c i ds " H ξ k ps, xq ´Hξ k ps, x 1 q ı `żS c i ds pξ C ´1 pt, s, x, yq. " I ξ k ps, xq ´Iξ k ps, x 1 q (3.33) ı ¯ˇˇˇˇ, ps ´tq|T i (3.34)

  k ps, xq ´Iξ 3,k ps, x 1 q

	k	ż
	ÿ
	"
	"2	R nd

dy ´F ps, y 1:k´1 , θ k:n s,t pξqq ´F ps, θ s,t pξqq ´D ´1F ps, θ s,t pξqq `y ´θs,t pξq ˘ ´1D y ups, yq ´Dy ups, y 1:k´1 , θ k:n s,t pξqq ¯`D x k pξ pt, s, x, yq ´Dx k pξ pt, s, x 1 , yq ˘.

  k ps, xq ´Iξ 3,k ps, x 1 q y 1:k´1 , θ k:n s,t pξqq ´F ps, θ s,t pξqq ´D ´1F ps, θ s,t pξqq `y ´θs,t pξq pξ pt, s, x, yq ´Dx k pξ pt, s, x 1 , yq ˘py ´θs,t pξqq m  D y D ym u `s, y 1:k´1 , θ k:n s,t pξq `λpy ´θs,t pξqq k:n

	n ÿ	k ÿ	ż 1		ż
	"			dλ		dy
	m"k	"2	0		R nd
	"
	´F
		ps, ˘	´1D
		x k
							":
	n ÿ m"k	k ÿ "2	ż 1 0	dλ	ż R nd	dy	" Ψ ps,y 1: ´1,y `1:n q,pt,xq ,k,m

  C|px 1 ´xq i | α k i ˜}Du} 8 `sup mPrrk,nss, Prr1,kss,sPr0,T s Now, the term H ξ k ps, xq´H ξ k ps, x 1 q in (3.34) (non-degenerate variables) can be handled reproducing the same previous arguments for I ξ k ps, xq ´Iξ k ps, x 1 q, exploiting (3.33) and following the computations performed for H k in the proof of Theorem 3 (see e.g. (2.29)).C|px 1 ´xq i | α k i `}DD x 1 u} 8 `}Du} 8 C|px 1 ´xq i | α k i `}DD x 1 u} 8 `}Du} 8 ˘T δ ,(3.39)

											B	8,8 ´1`α m	t ż T	dsps ´tq ´1´pi´1 2 qα k i	`γm	¸|px ´x1 q i | α k i
	ď CT δ	sup mPrrk,nss, Prr1,kss,sPr0,T s	}	`D	D m u	˘	ps, ¨q} B	8,8 α m ´1	|px ´x1 q i | α k i ,	(3.37)
	for some δ :" 3η{8 ą 0. Combining this estimate together with (3.35) we eventually derive
	ˇˇˇˇż								ˇˇˇď
		i S c	ds	" I ξ k ps, xq ´Iξ k ps, x 1 q	ı
											¸.(3.38)
	T δ }	`D	D m u	˘	ps, ¨q} B 8,8 α m ´1
	From the definition of S c i we obtain:
			ˇˇˇˇż t`c 0 |px 1 ´xq i | T	2i´1 2	ds ´Hξ
											˘ż T	dsps ´tq ´1´α k i pi´1 2 q`δ k i
											t
		ď

k ps, xq ´Hξ k ps, x 1 q ¯ˇˇˇď

  x 1:i´1 , ¨, x i`1:n qpzq| ˇˇS ´}DD x 1 u} 8 `}Du} 8 `sup Discontinuity term associated with the regime time change: control of the term (3.25). Here, we aim at handling`hv ‹ D x i D x k upt, x 1:i´1 , ¨, x i`1:n qpzq ˘|BS i

			ż
	"
				R d
						c
						i
	ď CT δ	ż R d	dx i	h cv pz ´xi q 2 v 1	|x i ´z| α k i
									ď
	ˆ´}DD x 1 u} 8 `}Du} 8 `sup 2ď ďmďn,sPr0,T s	} `D	˘ D m u	ps, ¨q}	α m ´1 8,8 B
	CT δ v	α k i 2 ´1		2ď ďmďn,sPr0,T s	} `D	D m u ˘	ps, ¨q}	B 8,8 α m ´1	¯. (3.40)

  t 0 , x 1 , yq `p ξ C ´1 pt, t 0 , x 1 , yq rpD k uq j pt 0 , ¨qs α k j pt 0 ´tq α k rpD k uq j pt 0 , ¨qs α k j pt 0 ´tq α k rpD k uq j pt 0 , ¨qs α k j ¯pt 0 ´tq1 2 `η 8 ,(3.50)with again the notation of (1.2). Thus, plugging estimates (3.45), (3.46), (3.49) and (3.50) in (3.41) we deduce that|D x k P ξ 1 t 0 ,t upt 0 , x 1 q ´Dx k P ξ1 t 0 ,t upt 0 , x 1 q| ď Cpt 0 ´tq 1 2 `η 8 rpD k uq k pt 0 , ¨qs α k k `C}Du} 8 rpD k uq j pt 0 , ¨qs α k j ¯pt 0 ´tqThe last inequality is a consequence of Proposition 4, stated in Appendix 5.2, which allows to dominate the Hölder norms ´rpD k uq j pt 0 , ¨qs α k j ¯2ďjďkďn in terms of the Besov norms ´}`D j D k u ˘jpt 0 , ¨qq}

								) k´1 ÿ	j pj´1 2 q
									j"1
	`C ż R nd	dyp ξ 1 C ´1 pt, t 0 , x 1 , yq	k j"1 ÿ				j pj´1 2 q
	´k ÿ
	ď C
	j"1
								! pt 0 ´tq	`cϑ´pk´1 2 q 0	i |px ´x1 q i | α k	)
	`C´k ÿ					1 2 `η 8
	j"1
	ď Cpt 0 ´tq	1 2 `η 8 } `D2		B 8,8 α k k ´1	`C}Du} 8	! pt 0 ´tq	`cϑ´pk´1 2 q 0	i |px ´x1 q i | α k	)
	`C´k ÿ j"1		B	8,8 α k j ´1	¯pt 0 ´tq	1 2 `η 8 .	(3.51)
			B 8,8 ¯2ďjďkďn α k j ´1

k u ˘kpt 0 , ¨q} } `Dj D k u ˘jpt 0 , ¨q}

  Thus, from (3.25), there exists δ :" δppAqqP p0, 1q such thatˇˇ`h v ‹ D x i D x k upt, x 1:i´1 , ¨, x i`1:n qpzq ˘|BS i ˇďConclusion: control of (3.22). Plugging (3.30), (3.40) and (3.52) into (3.22), (3.19), we eventually derive that for some positive δ :" δppAqq ą 0:}D x i D x k upt, x 1:i´1 , ¨, x i`1,n q}

									B	α k i 8,8 ´1
											x 1 q|
		#												+
	ď C	c	1 2 `η 8 0	´k ÿ j"1	} `Dj D k u ˘jpt 0 , ¨q} B 8,8 α k j ´1	¯`pc	ϑ´pk´1 2 q 0	`c 1 2 `η 8 0	q}Du} 8	|px ´x1 q i | α k i .
				v	α k i 2 ´1	Cpc ´δ 0	´1	}Du} 8	`cδ 0 max 2ď ďmďn	} `D	D m u	˘	pt 0 , ¨q}	8,8 B α m ´1	q.	(3.52)

  k:n s,t pξqq ˘, py ´θs,t pξqq m B w h w pz ´y q « ´Dx k pξ pt, s, x, yq ´Dx k pξ pt, s, x, y 1: ´1, z, y `1:n q Dx k pξ pt, s, x 1 , yq ´Dx k pξ pt, s, x 1 , y 1: ´1, z, y `1:n q ¯ff A F ps, y 1: ´1, z, y `1:k´1 , θ k:n s,t pξqq ´F ps, θ s,t pξqq ´Dx ´1 F ps, θ s,t pξqqpy ´θs,t pξqq ´1,

				D	dy ,
	with a slight abuse of notation when " k ´1 and
	T 2, ,k,m ps,y 1: ´1,y `1:n q,pt,xq	`w,	z˘( 3.58)
	ż
	"
	R d
				E
			py ´θs,t pξqq m	dy .

  C qcz pt, s, x, py 1: ´1, y `1:n qq|px 1 ´xq i | α k By the previous definition of F δ ,identity (4.2) is equivalent to:Next, let us control the last components of the flow. By the definition of θ s,t in (2.4), we get:|pθ s,t pxq ´θs,t px 1 qq n | ď |px ´x1 q n | `ż s t ´|F δ n pv, θ v,t pxqq ´Fδ n pv, θ v,t px 1 qq| `|F δ n pv, θ v,t pxqq ´Fn pv, θ v,t pxqq| `|F δ n pv, θ v,t px 1 qq ´Fn pv, θ v,t px 1 qq| ¯dv ď |px ´x1 q n | `C ż s t´ˇ`θ v,t pxq ´θv,t px 1 qq ˘n´1 ˇˇ`δ ) for the last inequality. For the pn ´1q th component, the situation is quiet different in the sens that we have to handle the non-Lipschitz continuity of F δ n´1 in its n th variable.

					n ÿ	ps ´tq	2i´2 2i´1 i 2 ´iδ 1	ď Cps ´tq ´1.	(4.3)
					i"2
	Hence, we choose from now on, for all i P rr2, nss:
							δ i " ps ´tq pi´3 2 q 2i´1 2i´2 .	(4.4)
							´1`2 n´2 2n´1 n	ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘nˇˇ¯d v
			2n´2
	`ps ´tqδ	2n´1
				´1`2 n´2 2n´1 n	¯´|px ´x1 q n | `ps ´tqδ n 2n´2 2n´1	`ż s	ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘n´1 ˇˇdv	ď
							t
	C exp ´Cps ´tq	1 2 ¯´|px ´x1 q n | `ps ´tq n´1 2 `ż s	ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘n´1 ˇˇdv ¯,	(4.5)
							t
	using (4.3
							ż ps´tq ρ ,m	α m ´1
							i	dww	2
							0
	#	1 w 1´β 2 ps ´tq α k i pi´1 2 q	`w ´1`β 2 ps ´tq ´1 2 `αk i pi´1 2 q	`w´1 2 ps ´tq ´1 2 `αk

p¨q  ď i pi´1 2 q´p1`ηqp ´3{2q + ď C qcz pt, s, x, py 1: ´1, y `1:n qqps ´tq ´1´pi´1 2 qα k i `γm |px 1 ´xq i | α k i , n .

Hence by Grönwall"s Lemma, we get:

|pθ s,t pxq ´θs,t px 1 qq n | ď C exp ´Cps ´tqδ

  t pxq ´θs,t px 1 qq n´1 | ď C exp ´Cps ´tqδ

					´1`2 pn´1q´2 2pn´1q´1 n´1	2pn´1q´2 2pn´1q´1 n´1 ¯´|px ´x1 q n´1 | `ps ´tqδ
	`ż s	ˇˇ`θ	˘nˇˇ2pn´1q´2 2pn´1q´1 ( v,t pxq ´θv,t px 1 qq ˘n´2 ˇˇ`| `θv,t pxq ´θv,t px 1 qq	dv	ď
	t
	C exppCps ´tq	1 2 q ´|px ´x1 q n´1 | `ps ´tq n´3 2 `ż s	" ˇˇ`θ	v,t pxq ´θv,t px 1 qq ˘n´2	ˇ|px
											t
		´x1 q n |	2pn´1q´2 2pn´1q´1	`pv ´tq	2pn´1q´2 2	`´ż v	ˇˇ`θ	w,t pxq ´θw,t px 1 qq ˘n´1 ˇˇdw	¯2pn´1q´2 2pn´1q´1	dv *	˙.
	looooooooomooooooooon				t
			2pn´1q´2
	ď|px´x 1 qn|	2n´1
											(4.6)

  `θ w,t pxq ´θw,t px 1 qq ˘n´1 ˇˇdwIn order to obtain the suitable time scale, we choose δn s.t. pxq ´θw,t px 1 qq ˘n´1 ˇˇdw ¯pv ´tq ´3 2 . From (4.6) and the previous controls, we deduce that for all s P rt, ss: |pθ s,t pxq ´θs,t px 1 qq n´1 | ď C exppCps ´tq 1 2 q ´|px ´x1 q n´1 | `ps ´tq n´3 2 `ż pxq ´θw,t px 1 qq ˘n´1 ˇˇdw ¯pv ´tq ´3 2 `θw,t pxq ´θw,t px 1 qq ˘n´1 ˇˇpv ´tq 1´3 2 dv ¯˙.Where we used for the last identity the following Young inequality : |px ´x1 q n | We explicitely see from (4.7) that each entry of the difference of the starting points appears at its intrinsic scale for the homogeneous distance d.Plugging the above inequality into (4.5) we derive:|pθ s,t pxq ´θs,t px 1 qq n | ď C exp ´Cps ´tq 1 2 ¯´|px ´x1 q n | `ps ´tq n´1 2 `|px ´x1 q n´1 |ps ´tq `|px ´x1 q n | Cp|px ´x1 q n | `ps ´tq n´1 2 q and |px ´x1 q n´1 |ps ´tq ď C `|px ´x1 q n´1 | 2n´1 2n´3 `ps ´tq n´1 2 ˘for the last inequality. We iterate the procedure, hence, we get:|pθ s,t pxq ´θs,t px 1 qq n | (4.8)ď C ˜ps ´tq n´1 2 `n ÿ Observe that equations (4.8) and (4.9) are available for any fixed time s and v P rt, T s.The first term, i.e. for i " 1 is controled slightly differently. In other words, for all s P rt, ss, write:|pθ s,t pxq ´θs,t px 1 qq 1 | ď |px ´x1 q 1 | `C n s,t pxq ´θs,t px 1 qq 1 | ď |px ´x1 q 1 | `Cˆp s ´tq sup vPrt,ss |pθ s,t pxq ´θs,t px 1 qq 1 | The last inequality is a consequence of (4.9), Remark 3 and convexity inequalities. Next, let us write, sup sPrt,ss |pθ s,t pxq ´θs,t px 1 qq 1 | ď C ˆ|px ´x1 q 1 | `ps ´tq 1`1 2 `ps ´tq

	which yields that				2pn´1q´2 2n´1	ps	tq
	ď C `|px ´x1 q n | sup	2n´3 2n´1	2n´3 2n´5 `ps ´tq n´3 2	˘.
	sPrt,ss
						`n ÿ	C j ps ´tq ´ps ´tq	1 2 `n ÿ	|px ´x1 q k |	1 2k´1
						j"2	¯2pn´1q´2 2pn´1q´1	k"2
		ď C	ˆ´ż v C exp ´Cps ´tq t ˇˇ`θ w,t pxq ´θw,t px 1 qq ˘n´1 ˇˇdw ¯δ 1 2 ¯´|px ´x1 q n | `ps ´tq n´1 2 `|px ´x1 q n´1 | ´2pn´1q´1 2pn´1q´2 n´1,n n´1,n t t `δ 2pn´1q´1 3 ˙. 2n´3 2n´1 ps ´tq `ż s ż v ˇˇ`θ vPrt,ss w,t pxq ´θw,t px 1 qq ˘n´2 ˇˇdwdv 2n´3 2n´1 ď `ps ´tq pj´1q 1 2j´1 sup |pθ v,t pxq ´θv,t px 1 qq 1 | 1 2j´1 ¯˙.
		δ 2pn´1q´1 3 n´1,n `ż s " pv ´tq t t ż v ˇˇ`θ	2pn´1q´2 2 w,t pxq ´θw,t px 1 qq ˘n´2 ˇˇdwdv ðñ δn´1,n " pv ´tq ¯, 3p2pn´1q´2q 2p2pn´1q´1q ,
	which also implies that ´ż v t ˇˇ`θ w,t pxq ´θw,t px 1 qq ˘n´1 ˇˇdw ¯δ n´1,n ´2pn´1q´1 2pn´1q´2 using again the Young inequalities |px ´x1 q n | ď `n ÿ j"1 ps ´tq 1`pj´1q 1 2j´1 sup ´ż v t ˇˇ`θ 2n´3 vPrt,ss |pθ v,t pxq ´θv,t px 1 qq 1 | w,t s " ˇˇ`θ v,t pxq ´θv,t px 1 qq ˘n´2 n ÿ k"2 |px ´x1 q k | 1 2k´1 1 2j´1 ď ˇ|px 2n´1 ps ´tq ď j"2 |px ´x1 q j | 2n´1 ż v 2 2j´1 `ż vn"s t dv n´1 . . . k"2 t dv 1 ˇˇ`θ Cp|px ´x1 q 1 | `ps ´tq `n ÿ |px ´x1 q k | 1 2k´1 q, (4.10)
						t
						2pn´1q´2	2pn´1q´2
	Namely,	´x1 q n |	2n´1	`pv ´tq	2
	`´ż v t ps ´tq 1`pj´1q 1 2j´1 sup ˇˇ`θ vPrt,ss and w,t 1 2 q ´|px ´x1 q n´1 | `ps ´tq n´3 2 `ż s j"2 |px ´x1 q j | 2i´1 2j´1 `ż v i "s t dv i´1 . . . ż v 2 ˇˇ`θ t dv 1 v,t pxq ´θv,t px 1 qq ˘n´2 ˇˇdv ˇˇ`θ |pθ v,t pxq ´θv,t px 1 qq 1 | 1 2j´1 ď Cps ´tq ´1 `sup vPrt,ss |pθ v,t pxq ´θv,t px 1 qq 1 | 1 1 ps ´tq|px ´x1 q k | 2k´1 ď |px ´x1 q k | 2k´1	¯,
						t
		`|px ´x1 q n | `´ż s t wPrt,vs sup	2pn´1q´2 2n´1	ps ´tq `ps ´tq 1`2 pn´1q´2 2 j"1 t ÿ ż s	|pθ v,t pxq ´θv,t px 1 qq j |	1 2j´1 dv,
	which in turn implies Next, Grönwall's Lemma yields: sPrt,ss |pθ s,t pxq ´θs,t px 1 qq n´1 | sup |pθ s,t pxq ´θs,t px 1 qq 1 |
	ď C exppCps ´tq 2pn´1q´2 1 2 q ´|px ´x1 q n´1 | `ps ´tq n´3 2 `ż s ď vPrt,ss j"2 t t ˇˇ`θ v,t pxq ´θv,t px 1 qq ˘n´2 ˇˇdv ď |px ´x1 q 1 | `C´p s´tq sup |pθ s,t pxq´θ s,t px 1 qq 1 |`n ÿ ż s |pθ v,t pxq ´θv,t px 1 qq j | 2j´1 dv 1 ď
	`|px ´x1 q n | |px ´x1 q 1 | `Cˆp s ´tq sup 2n´1 ps ´tq |pθ s,t pxq ´θs,t px 1 qq 1 |
		C exppCps ´tq	vPrt,ss	˙,
	j"2 `n ÿ				2n´3 2n´1 ¯.	(4.7) ¯1 2j´1

* dv

Ṫaking then the supremum in s P rs, ts in the above equation, we obtain:

sup sPrt,ss |pθ s,t pxq ´θs,t px 1 qq n´1 | ď C exppCps ´tq | 1 2 q ´|px ´x1 q n´1 | `ps ´tq n´3 2 `ż s t ˇˇ`θ v,t pxq ´θv,t px 1 qq ˘n´2 ˇˇdv `|px ´x1 q n | v 1 ,t px 1 q ´θv 1 ,t pxq ˘1ˇˇ¸.

Anagolously, for i P rr2, nss, we obtain:

|pθ s,t pxq ´θs,t px 1 qq i | (4.9)

ď C ˜ps ´tq i´1 2 `n ÿ v 1 ,t px 1 q ´θv 1 ,t pxq ˘1ˇˇ¸.

Remark 3

C j ps´tq ´ps ´tq j´1 2 `n ÿ k"2 |px´x 1 q k | 2j´1 2k´1

`ps´tq j´1 sup vPrt,ss |pθ v,t pxq ´θv,t px 1 qq 1 | |pθ recalling that ps ´tq ď T ă 1, and using again Young inequalities for the last bound.

  v k D j´1 F j pv j , θ v j ,t pxqq |F i pv, θ v,t pxqq ´Fi pv, θ v,t px 1 qq|dv

					dv k´1	i ź	ı rx 1 k´1 ´xk´1 s ˇˇż
					t	j"k
		s
		t
	ď C	´i´1 ÿ	ps ´tq i´k |x k	´x1 k |
		k"2
	`ż s	´n ÿ
		t		j"i

  8,8 ď }Df } B α´1 8,8 ď C}f } B α 8,8 . We write also that }f } B α 8,8 -}Df } B α´1 8,8 .

˘¯.

Note that r Ñ ρprq is increasing, hence the condition we need to put on λ in order to propagate the Gaussian concentration from Y P R r to Z became stronger if the dimension r increase.3 Exponential integrability of the square root of Lyapunov function.

† †. We could also observe from Proposition 4 that this term could also be directly bounded from the supremum norm of the gradient.

Remerciements

To begin we check that ? V satisfies assumption (L V ) iii). We have readily that:

with βV :" βV 2 ? v ˚. The first inequality is a consequence of Remark 3. Furthermore, for the purely jump part of the infinitesimal generator we write:

V px `κpxqyq ´aV pxqqπpdyq ď ? C V }κ} 8 πp| ¨|q 2 , using (3.1). The previous inequality and (3.3) implies that:

where β1

V " βV `?C V }κ}8πp|¨|q 2 . Next, let us decompose the Lyapunov function ?

V with a Taylor expansion similarly to Lemma 1. We again use a splitting between the deterministic contributions and those involving the innovation. We write for all n P N: ? V pX n q ´?V pX n´1 q " ? V pX n´1 `γn b n´1 `?γ n σ n´1 U n q ´?V pX n´1 q `?V pX n q ´?V pX n´1 `γn b n´1 `?γ n σ n´1 U n q " γ n A

´"γ n 2 Tr `D2 ? V pX n´1 qqΣ n´1 ˘ı `"? γ n σ n´1 U n ¨∇?

V pX n´1 `γn b n´1 q `γn ż 1 0 p1 ´tqTr ´D2 ? V pX n´1 `γn b n´1 `t?

´γn π `aV pX n´1 `κpX n´1 q¨q ´aV pX n´1 q

Controls for the frozen density

We explicitly integrate (3.1) to obtain for all v P rt, ss: and corresponds as well to the solution of (3.1) when σ " 0 and the starting point is x. We write:

Importantly, we point out that Those controls actually follow from the more general following result, which will again be useful for the Hölder norm in Section 5.

Lemma 14 (Scaled Besov Control Lemma) For all multi-index α " pα 1 , . . . , α n q P N d :

With Lemma 14 at hand, we readily derive (6.18) and (6.19) taking α " p2, 0, ¨¨¨, 0q `ek (where e k stands for the k th vector of the orthonormal basis) for each k P rr2, nss and α " p2, 0, ¨¨¨, 0q respectively. Let us now turn to the proof of the Lemma 14.

Proof: Lemma 14

The analysis of singularities is identitc to the ones in the proof of Lemma 5. However, here, we have to track the scalling coefficient λ through the identites. Note carefully, we write the upper-script/sub-script λ to mean that we manage the scaled variables. In particular, we write:

With these notations, we have:

The point is here again to control, for i P rr2, nss, the quantity }ϑ α,λ i,pt,xq ps,

, αi " 2`γ 2i´1 with the indicated bounds in the scaling parameter λ. Accordingly with what can be seen e.g. in (6.13), the previous analysis of the proof of the (non-scaled) Lemma Note that thanks to the particular structure of B one has DU m B " pD

Hence, thanks to Theorem 2 and Grönwall's lemma, there exists CT :" CT pC T , σ, n, d, T q satisfying CT Ñ 0 when T goes to 0 such that

(1.9)

Letting m Ñ `8 and choosing T small enough so that CT ď 1{2, we deduce that strong uniqueness holds on a sufficiently small time interval. Iterating this procedure in time gives the result on R `from usual Markov arguments involving the regular versions of conditional expectations, see e.g. [START_REF] Stroock | Multidimensional diffusion processes[END_REF].

1. 4 Regularization properties of the underlying PDE (1.5): strategy of proof and primer

As mentioned above, the regularization properties of the PDE (1.6) given by estimate (1.8) in Theorem 2 are the core of this work. Smoothing properties of linear partial differential operators of second order with non-degenerate diffusion matrix have been widely studied in the literature and, in that setting, the estimates of Theorem 2 are well known (see e.g. the book of Friedman [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] or of Bass [START_REF] Bass | Diffusions and Elliptic Operators[END_REF]). In our case, the story is rather different since the diffusion matrix Ba of the system is totally degenerate in the directions 2 to n. However, as we already emphasized, the non-degeneracy condition assumed on the family of Jacobians pD x i´1 F i q iPrr2,nss allows the noise to propagate in the mentioned directions thanks to the drift. It can be viewed as a weak type Hörmander condition. Under such a condition, the operator L m with mollified coefficients is said to be hypoelliptic § and it is well known that hypoelliptic differential operators also have some smoothing properties (see the seminal work of Hörmander [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF] or, for a probabilistic viewpoint, the ouvrage of Stroock [START_REF] Stroock | Partial differential equations for probabilists[END_REF]). The tricky point in our weak Hörmander setting is that the pointwise gradient estimates (1.8) of Theorem 2 had, to the best of our knowledge, not been established yet. Although such a setting has already been considered by several authors (see e.g. Delarue and Menozzi [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF] for density estimates, Menozzi [START_REF] Menozzi | Parametrix techniques and martingale problems for some degenerate Kolmogorov equations[END_REF], [START_REF] Menozzi | Martingale problems for some degenerate Kolmogorov equations[END_REF] and Priola [START_REF] Priola | On weak uniqueness for some degenerate SDEs by global L p estimates[END_REF] for the martingale problem and also Bramanti, Cupini, Lanconelli and Priola [START_REF] Bramanti | Global L p estimates for degenerate Ornstein-Uhlenbeck operators[END_REF], [START_REF] Bramanti | Global L p estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients[END_REF] for related L p estimates and Bramanti and Zhu [START_REF] Bramanti | L p and Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift[END_REF] for the VMO framework). We can mention the work of Lorenzi [START_REF] Lorenzi | Estimates of the derivatives for a class of parabolic degenerate operators with unbounded coefficients in R N[END_REF] which gives gradient estimates in the degenerate kinetic like case (n " 2 in our framework) when the diffusion coefficient is sufficiently smooth and the drift linear. We point out that our main estimate in Theorem 2 needs precisely to be uniform w.r.t. the mollification parameter and therefore does not depend on the smoothness of F m , a m , but only on known parameters appearing in (A). Again, this is what would also allow to transfer those bounds to equation (1.5) from a suitable compactness argument, extending well known results for non-degenerate diffusions with Hölder coefficients to the current degenerate setting.

To prove this result our main strategy rests upon the parametrix approach see e.g. the work of McKean and Singer [START_REF] Mckean | Curvature and the eigenvalues of the Laplacian[END_REF] or the book of Friedman [START_REF] Friedman | Partial differential equations of parabolic type[END_REF]. Roughly speaking, §. Pay attention that this is not the case for L whose coefficients do not have the required smoothness in (T β ) to compute the corresponding Lie brackets.

From the non-degeneracy of σ and Hörmander like condition, the Gaussian process defined by (2.5) admits a density ppτ,ξq pt, s, x, ¨q which is suitably controlled (see Proposition 2 below and for instance [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF], [START_REF] De Raynal | Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result[END_REF]).

We explicitly integrate (2.3) to obtain for all v P rt, ss: (2.5)

Note in particular that since the partial gradients are subdiagonal detp Rpτ,ξq pv, tqq " 1. Also, for v P rt, ss, we recall that m pτ,ξq v,t pxq stands for the mean of Xpτ,ξq v and corresponds as well to the solution of of (2.3) when σ " 0 when the starting point is x. We write:

Xpτ,ξq

Rpτ,ξq pv, uqBσpu, θ u,τ pξqqdW u , v P rt, ss.

(

Importantly, we point out that x P R nd Þ Ñ m pτ,ξq v,t pxq is affine w.r.t. the starting point x. Precisely, for x, x 1 P R nd : m pτ,ξq v,t px `x1 q " Rpτ,ξq pv, tqx 1 `mpτ,ξq v,t pxq.

(2.7)

We first give in the next proposition a key estimates on the covariance matrix associated with (2.6) and its properties w.r.t. a suitable scaling of the system.

Proposition 1 (Good Scaling Properties of the Covariance Matrix) The covariance matrix of Xpτ,ξq v in (2.6) writes:

Kpτ,ξq v,t :"

Rpτ,ξq pv, uqBapu, θ u,τ pξqqB ˚R pτ,ξq pv, uq ˚du.

Uniformly in pτ, ξq P r0, T s ˆRnd and s P r0, T s, it satisfies a good scaling property in the sense of Definition 3.2 in [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF] (see also Proposition 3.4 of that reference). That is, for all fixed T ą 0, there exists C 2.8 :" C 2.8 ppAq, T q ě 1 s.t. for all 0 ď t ă v ď s ď T , for all pτ, ξq P r0, T s ˆRnd :

where we again use the notation introduced in (1.17) fpr the scaling matrix T v´t .

The proof of the above proposition readily follows from Proposition 3.3 and Lemma 3.6 in [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF]. We now state some important density bounds for the linearized model.

where γ m " 1{2 `ηp ´3{2q. From (H η ) and the very definition of α k i we hence have 1 ´αk i pi ´1{2q `γm ă 1 which, together with (3.56), concludes the proof. 4

Proof of technical results of Lemma 6

In this section, we establish some control of the flows sensitivity. Our results are still available up to any mollification procedure.

First of all, let us state a control of the spatial regularity of the flow. Next, we will provide a proof of Lemma 6.

A first sensitivity result for the flow

Lemma 7 Under (A), there exists C :" CppAq, T q s.t. for all px, x 1 q P pR nd q 2 , 0 ď t ă s ď T and i P rr1, nss:

The flow, θ s,t is, somehow, "almost" Lipschitz continuous in space w.r.t. the homogeneous distance d, up to a time additive term. This time contribution is a consequence of the non-Lipschitz continuity of the drif F. The analysis which was already done for F Lipschitz continuous in Proposition 4.1 of [START_REF] Menozzi | Martingale problems for some degenerate Kolmogorov equations[END_REF], and in Chapter 6 with diferrent Hölder regularity of F. Actually, as we consider a smoother drift than in Chapter 6, the following lemma is a by-product of Lemma 11 therein. For the sake of completeness, we provide the analogous analysis below.

Proof: In the current analysis, we will assume that dpx, x 1 q ě 1, a general case can be directly deduced.

The analysis mainly come from a Grönwall's Lemma. Nevetheless, because F is not Lipschitz continuous, we have to mollify suitably the function. We denote by δ P R n , the vector entries δ i ą 0 for i P rr2, nss. and for all v P r0, T s, z P R nd , i P rr2, nss,

with ρ δ i pwq :" 1 δ d i ρ ´w δ i ¯where ρ : R d Ñ R `is a usual mollifier, namely ρ has compact support and ş R d ρpzqdz " 1. Eventually, we write F δ pv, zq :" pF 1 pv, zq, F δ 2 pv, zq, ¨¨¨, F δ n pv, zqq. With a slight abuse of notation in the previsous definitions, since the first component F 1 is not mollified. The sublinearity of F 1 is actually enough to obtain the aimed control.

To be at the good current time scale for the contributions associated with the mollification, we pick δ i in order to have C :" CppAq, T q ą 0 s.t. for all z P R nd , u P rt, ss: To establish this equivalence of norms we use the local means characterisation of the Besov space, see for instance 2.5.3 in [START_REF] Triebel | Theory of function spaces[END_REF]. In other words, for all function k, k 0 , k 0 P C 8 pR d q (infinitely differentiable) such that supppkq, supppk 0 q, supppk 0 q Ă Bp0 d,d , 1q (the unit ball centered at the origin) and for all x P R d : kpxq "

x i k 0 pxq.

We define furthermore the local means for all px, tq P R d ˆp0, `8q: It is then known that for all f P B α p,q pR d q:

}f } B α 8,8 -}k 0 p1, f qp¨q} 8 `sup tPp0,1q

`t´α }kpt, f qp¨q} 8 ˘.

We can consider similarly the sequences of functions, k i " B x i k, k i 0 " B x i k 0 ,i P rr1, dss, which satisfy the same assumptions as k and k 0 . Hence, we can write:

-}k i 0 p1, f qp¨q} 8 `sup tPp0,1q

`t´α }k i pt, f qp¨q} 8 -}k 0 p1, B x i f qp¨q} 8 `sup tPp0,1q

`t1´α }kpt, B x i f qp¨q} 8 -}B x i f } B α´1 8,8 . The penultimate identity is a consequence of an integration by parts over the convolution products, and the last equivalence is obtained again thanks to Theorem 2.5.3 in [START_REF] Triebel | Theory of function spaces[END_REF]. Iterating over each component yields the result.