Denis Villemonais

Michel Benaïm

Pierre Collet

Bénédicte Haas

Tony Lelièvre

Sylvie Méléard

Convergence exponentielle vers une distribution quasi-stationnaire et applications

I Criteria for the exponential convergence to a quasi-stationary distribution

Résumé

Ce manuscrit est le mémoire de mon dossier de candidature à l'habilitation à diriger des recherches. À ce titre, il contient un exposé représentatif de mes travaux de recherche depuis la fin de ma thèse. Son objectif est également de présenter sous une forme plus accessible les outils théoriques principaux développés dans ces travaux, ainsi que leur application à divers modèles.

Mes recherches, depuis l'obtention de ma thèse en novembre 2011, portent principalement sur l'étude du comportement quasi-stationnaire de processus avec absorption et de leurs applications. En particulier, je me suis attaché à développer des critères suffisants pour l'existence et la convergence des lois conditionnelles d'un processus vers une distribution quasi-stationnaire. Ces travaux sont exposés dans les première et seconde parties de ce manuscrit, qui contiennent respectivement les critères abstraits et leurs application à des modèles classiques. Ils correspondent aux publications et pré-publications suivantes 1 , écrites dans leur plus grande partie dans le cadre d'une collaboration fructueuse avec Nicolas Champagnat à l'Institut Élie Cartan de Lorraine à Nancy.

En premier lieu, je tiens à remercier Bénédicte Haas, Pierre Collet et Ross Pinsky qui ont tous trois pris le temps et fait l'effort de rapporter ce manuscrit. C'est un grand plaisir et un grand honneur d'avoir pu profiter de leurs précieux conseils et remarques. Merci également à Michel Benaïm, Jean Bertoin, Anne Gégout-Petit, Tony Lelièvre et Sylvie Méléard pour avoir accepté de faire partie du jury. Merci également à Nicolas Champagnat, qui m'a fait l'honneur de parrainer cette habilitation. Merci enfin à David Steinsaltz pour sa participation à ce projet.

La théorie des distributions quasi-stationnaires, que j'ai étudiée et continue d'étudier avec passion, m'a été dévoilée par Sylvie Méléard durant ma thèse. Je la remercie de m'avoir initié a ce sujet et pour son soutien toutes ses années. Merci également à Pierre Del Moral, Servet Martínez et Jaime San Martín pour leurs explications sur ce sujet alors que je terminais tout juste ma thèse il y a 8 ans. Ce document résume des travaux qui ont pour une large part été réalisés en collaboration avec Nicolas Champagnat, que je remercie pour sa présence et les heures innombrables que nous avons passées à discuter de mathématiques et, souvent, de distributions quasi-stationnaires. Je le remercie donc pour tout ce qu'il m'a apporté et continue de m'apporter, car il est certain que sa présence a été un facteur prépondérant dans le bon déroulement de ma recherche en mathématique. Je souhaite également remercier mon collaborateur William Oçafrain, le premier étudiant avec lequel j'ai travaillé pendant et suite à son parcours recherche à l'École des Mines de Nancy, et dont le travail de recherche aujourd'hui confirme les dispositions mathématiques que j'avais cru déceler chez lui il y a quelques années. Merci également à Camille Coron qui m'a patiemment expliqué certains aspects des probabilités génétiques. Merci aux membres de l'université Bath, dont Cécile Mailler qui m'a introduit aux modèles d'urnes, à Alexandre Cox, Andreas Kyprianou et Emma Horton qui m'ont accueilli dans leur exploration de l'équation de transport neutronal : ils m'ont apporté une ouverture nouvelle vers les processus de branchement et Emma m'a fait le plaisir de me rejoindre cette année en post-doc à Nancy. Merci aussi à Michel Benaïm pour la générosité dont il a fait preuve lors de nos rencontres à Neuchâtel et dans les Alpes suisses, pour m'avoir patiemment expliqué le fonctionnement des algorithmes d'approximation stochastique et, enfin et surtout, pour m'avoir recommandé son brillant étudiant Édouard Strickler, qui nous a fait le plaisir de venir collaborer avec Nicolas et moi dans le cadre d'un post-doctorat à l'IECL. Merci également à Koléhè Coulibaly-Pasquier pour sa collaboration et son aide en probabilités, et pour son éternelle bonne humeur qui éclaire nos journées à l'IECL. Merci également à René Schott, pour m'avoir introduit aux algorithmes distribués et aux applications possibles des distributions quasi-stationnaires dans ce domaine. Ces dernières années ont également été pour moi l'occasion de me pencher sur des problématiques nouvelles, en statistique et médecine notamment, en rejoignant un projet de travail sur des données de longueurs de télomères chez l'être humain. C'est une expérience qui a été et continue d'être enrichissante, je remercie pour cela Samy Tindel et Éliane Albuisson, sans qui je n'aurais jamais eu connaissance de cette problématique, Athanasios Benetos et Simon Toupance, qui ont pris le temps de discuter pendant de nombreuses heures de leurs données et de la mécanique des télomères, Anne Gégout-Petit, qui a porté le projet tout en lui v vi REMERCIEMENTS apportant sa précieuse expérience en statistique, et enfin nos étudiants et post-docs : Daphné Germain, étudiante de l'École des Mines de Nancy avec qui ce projet a commencé et qui a su oeuvrer pour le rapprochement des deux communautés, Sylvain Chabanet, étudiant de l'École des Mines de Nancy qui a abordé l'étude d'une cohorte plus nombreuse pour tenter d'étayer nos résultats avec des méthodes pointues qu'il a brillament mises en oeuvre, Olivier Coudray, stagiaire en Master 2 qui a sû très vite s'adapter et a pu mettre en place des outils théoriques essentiels pour l'étude de modèles de populations branchantes, et Lionel Lenotre, qui nous a accompagné pendant une courte année avant d'accepter un poste de maître de conférences à Mulhouse et qui nous a apporté son expertise dans le domaine de l'analyse des semi-groupes positifs.

Je tiens également à remercier les membres de l'équipe INRIA TOSCA. Merci en premier lieu à Madalina Deaconu pour la direction de cette équipe qu'elle assurait déjà avant mon arrivée à Nancy. Sa présence au sein de ce qui a été ma première structure d'accueil après ma thèse, m'a aidé à déterminer mes premières directions de recherche de manière autonome, tout en me sentant entouré et soutenu. Je tiens également à remercier Antoine Le Jay, pour sa présence et son rôle tant au sein de cette équipe projet, qu'en tant que responsable de l'équipe de probabilités et statistiques à l'IECL, et enfin et surtout en tant que collaborateur pour le co-encadrement de mon premier doctorant Alexis Anagnostakis. Que ce dernier, qui subit ma supervision en recherche et en enseignement, soit ici remercié pour ses efforts, son endurance et ses progrès. Au sein de cette équipe, j'ai également pris un grand plaisir à échanger et travailler avec Oana Lopalescu, Khaled Salhi et Coralie Fritsch.

Le poste de maître de conférences que j'occupe à l'Institut Élie Cartan de Lorraine et à l'École des Mines de Nancy m'a été attribué en l'année 2012. Depuis, ce laboratoire et cette école m'ont apporté en continu tout le support et les encouragements que j'étais en droit d'espérer et qui m'ont permis de m'épanouir dans le métier de mathématicien, dans ses missions d'enseignement, d'encadrement et de recherche. Je tiens à ce titre à remercier tout son personnel, et en particulier leurs directeurs actuels Xavier Antoine et François Rousseau, ainsi que les personnels de soutien, dont le sérieux et l'efficacité ont permis le déroulement sans encombre de toute cette période d'activité. Mon métier n'aurait pas été le même sans la patience et l'aide de Nathalie Benito, Élodie Cunat, Didier Gemmerlé, Laurence Quirot et Paola Schneider à l'IECL, ou sans le support de Françoise Laurent et l'efficacité redoutable de Thierry Umoret à l'École des Mines de Nancy. Je les remercie ici pour tout leur travail et leur présence. Cette page est bien sûr pour moi l'occasion de remercier Antoine Henrot, pour sa direction généreuse et humaine du département GIMA, ainsi que Céline Lacaux, qui m'a guidé dans mes premiers pas à l'École des Mines de Nancy. Tous deux m'ont offert par leur présence des perspectives nouvelles tant dans mes missions d'enseignement que de recherche. Enfin, je remercie pour leur bienveillance et leur amabilité les membres de l'IECL et du département GIMA, qualités qui ont été, au jour le jour, au coeur de la satisfaction que m'a apporté le métier de mathématicien à Nancy. Qu'on me permette enfin de rendre hommage à toutes les mathématiciennes et tous les mathématiciens dont les travaux m'ont accompagnés et qui ont nourri de leurs idées originales mes travaux de recherches.

C'est alors que toutes ces collaborations et rencontres suivaient leur chemin, qu'un grand et heureux événement a changé le cours de ma vie le 26 mai 2017 : ma compagne Sophie a donné naissance à ma fille Léonore et toutes deux m'accompagnent depuis pour mon plus grand bonheur. Le rayonnement de leur amour et de leurs sourires m'a accompagné, entre autres, tout au long de la rédaction de ce manuscrit et de la préparation de cette habilitation.

• N. Champagnat and D. Villemonais. Exponential convergence to quasi-stationary distribution and Q-process. Probab. Theory Related Fields, 164(1):243-283, 2016.

• N. Champagnat and D. Villemonais. Uniform convergence of conditional distributions for absorbed one-dimensional diffusions. Adv. in Appl. Probab., 50(1):178-203, 2017.

• N. Champagnat and D. Villemonais. General criteria for the study of quasi-stationarity. ArXiv e-prints, Dec. 2017.

• N. Champagnat and D. Villemonais. Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes. ArXiv e-prints, Apr. 2017.

• N. Champagnat

ABSTRACT

The consequences of these results go beyond the theory of quasi-stationary distributions and naturally lead to the study of quasi-ergodic properties, of R-positive recurrence of unbounded semi-groups, to measure valued Pólya processes and to reinforced processes. These applications are developed in the second part of this manuscript and correspond to the following publications and pre-publications.

• N. Champagnat and D. Villemonais. Uniform convergence to the Q-process. Electron. Commun. Probab., 22:7 pp., 2017.

• N. Champagnat and D. Villemonais. Practical criteria for R-positive recurrence of unbounded semigroups. ArXiv e-prints, Apr. 2019.

• C. Mailler and D. Villemonais. Stochastic approximation on non-compact measure spaces and application to measure-valued Pólya processes. ArXiv e-prints, Sep. 2018.

• M. Benaïm, N. Champagnat, and D. Villemonais. Stochastic approximation of quasi-stationary distributions for diffusion processes in a bounded domain. ArXiv e-prints, Apr. 2019.

Then I present other subjects of study in the fourth part, some of which have been supported by or lead to long-term research internships and the co-supervising of a PhD-thesis and a postdoctorate. They correspond to the following publications. Some of my post-PhD thesis works are not covered in this manuscript. In particular, works on Fleming-Viot type processes as approximation tools for quasi-stationary distributions are not presented, because they are in the direct continuity of my PhD thesis. Other subjects relate to models whose mere definition would require a very technical introduction. This represents a substantial proportion of the articles cited above and all of the subsequent publications and pre-publications.

• W. Oçafrain and D. Villemonais. Convergence of a non-failable mean-field particle system.

Stoch. Anal. Appl., 35(4):587-603, 2017.

• N. Champagnat and D. Villemonais. Exponential convergence to quasi-stationary distribution for absorbed one-dimensional diffusions with killing. ALEA Lat. Am. J. Probab. Math. Stat., 14(1):177-199, 2017.

• N. Champagnat, R. Schott, and D. Villemonais. Probabilistic Non-asymptotic Analysis of Distributed Algorithms. ArXiv e-prints, Feb. 2018.

• N. Champagnat Finally, the following works have been developed entirely or for a large part during my PhD thesis and are therefore not exposed in detail in this memory. They correspond to the following publications.

• D. Villemonais. Interacting particle systems and yaglom limit approximation of diffusions with unbounded drift. Electron. J. Probab., 16:1663-1692, 2011.

• S. Méléard and D. Villemonais. Quasi-stationary distributions and population processes. Probab. Surv., 9:340-410, 2012.

• D. Villemonais. Uniform tightness for time-inhomogeneous particle systems and for conditional distributions of time-inhomogeneous diffusion processes. Markov Process. Related Fields, [START_REF] Bandyopadhyay | A New Approach to Pólya Urn Schemes and Its Infinite Color Generalization[END_REF](3):543-562, 2013.

• D. Villemonais. General approximation method for the distribution of Markov processes conditioned not to be killed. ESAIM Probab. Stat., 18:441-467, 2014.

• P. Del Moral and D. Villemonais. Exponential mixing properties for time inhomogeneous diffusion processes with killing. Bernoulli, 24(2):1010-1032, 2018.

To conclude, note that some of the results of this memoir are presented in a form that differs from their first publication, and some of them are even original. However, these are incremental developments and the proofs are not of particular interest. In order not to increase unnecessarily the size of the manuscript, the proofs of these results, as well as those already published, will be omitted.

Notations

We will use the following notations for the sets • Z = {. . . , -2, -1, 0, 1, 2, . . .} of integers,

• Z + = {0, 1, 2, 3, . . .} of non-negative integers,

• N = {1, 2, 3, . . .} of positive integers,

• R = (-∞, +∞) of real numbers,

• R + = [0, +∞) of non-negative real numbers.

Given a measurable set (E , E ), we denote by • B(E ) the set of measurable functions from E to R,

• B b (E ) the set of bounded measurable functions from E to R,

• B + (E ) the set of non-negative measurable functions from E to R + ,

• • ∞ the uniform norm on B b (E ),

• M (E ) the set of non-negative measures on E ,

• M 1 (E ) the set of probability measures on E ,

• µ( f ) the integral of f with respect to µ, defined for all µ ∈ E and all f ∈ B + (E ) or all f ∈ B(E ) such that f is integrable with respect to µ (i.e. µ(| f |) < +∞),

• • T V the total variation norm, defined for all µ 1 , µ 2 ∈ M 1 (E ) by

µ 1 -µ 2 T V = 2 sup A∈E |µ 1 (A) -µ 2 (A)| = sup |µ 1 ( f ) -µ 2 ( f )| : f ∈ B b (E ), f ∞ ≤ 1 .
Given a measurable set (E , E ) and a positive measurable function ψ : E → (0, +∞), we define

• M (ψ) := µ ∈ M 1 (E ) : µ(ψ) < +∞

• • M (ψ) the weighted total variation norm, defined for all µ 1 , µ 2 ∈ M (ψ) by

µ 1 -µ 2 M (ψ) := sup |µ 1 ( f ) -µ 2 ( f )| : | f | ≤ ψ . xv xvi NOTATIONS • L ∞ (ψ) := f ∈ B(E ) : | f /ψ| ∈ B b (E ) , • f L ∞ (ψ) := sup x∈E | f (x)| ψ(x) for all f ∈ L ∞ (ψ).
The sets M 1 (E ), • T V , M (ψ), • M (ψ) and L ∞ (ψ), • L ∞ (ψ) are complete spaces, and

B b (E ) = L ∞ (1 E ) and • ∞ = • L ∞ (1 E ) .
Given a Polish space (E , d ), we define

• M d (E ) := µ ∈ M (E ), d (x, y)µ(d y) < ∞ ,
• the Wasserstein distance W d between two probability measures µ and ν on E belonging to M d (E ), as

W d (µ, ν) = inf π E ×E d (x, y) π(d x, d y),
where the infimum is taken over all probability measures π on E × E such that π(•, E ) = µ(•) and π(E , •) = ν(•) (π is called a coupling measure for µ and ν),

• the Wasserstein distance W d between two measures in M d (E ) with the same mass: for all α > 0 and any probability measures µ, ν on E belonging to M d (E ), we set

W d (αµ, αν) = inf π E ×E d (x, y) π(d x, d y) = αW d (µ, ν),
where the infimum is taken over all measures π on E ×E with mass α and such that π(•, E N ) = µ(•) and π(E N , •) = ν(•). Note that if a coupling π realizes the minimum in the definition of W d (µ, ν), then απ realizes the minimum in the definition of W d (αµ, αν).

We emphasize that the state space P d (E N ), W d is a complete state space (see for instance Lemma 5.2 and Theorem 5.4 in [START_REF] Chen | From Markov chains to non-equilibrium particle systems[END_REF]).

Given a measurable set (E , E ), a time homogeneous Markov process with state space E is a family (Ω, (F t ) t ≥0 , (X t ) t ≥0 , (P t ) t ≥0 , (P x ) x∈E ∪{∂} ) satisfying the conditions of [START_REF] Rogers | Diffusions, Markov processes, and martingales[END_REF]Definition III.1.1], in the discrete or continuous time settings (i.e. t ∈ Z + or t ∈ R + ). We recall that P x (X 0 = x) = 1, that P t is the transition function at time t ≥ 0 of the process, and that the family (P t ) t ≥0 defines a semi-group of operators on the set B b (E ). For all µ ∈ M 1 (E ) and all f ∈ B + (E ), we will use the notations

P µ (•) := E ∪{∂} P x (•)µ(d x) and µP t f := E ∪∂ P t f (x) µ(d x), ∀t ≥ 0.
We shall denote by E x (resp. E µ ) the expectation corresponding to P x (resp. P µ ). We will say that the process is absorbed at a point ∂ if, for all all s ≥ 0, X s = ∂ implies X t = ∂ for all t ≥ s. This implies in particular that the absorption time, defined as

τ ∂ := inf{t ≥ 0, X t = ∂},
is a stopping time. In this work, we will often assume (although not systematically) that, for all t ≥ 0 and ∀x = ∂, P x (t < τ ∂ ) > 0.

Part I

Criteria for the exponential convergence to a quasi-stationary distribution

Chapter 1

A necessary and sufficient condition for uniform exponential convergence to a quasi-stationary distribution

This chapter is dedicated to the presentation of a necessary and sufficient condition for uniform exponential convergence to a quasi-stationary distribution in the total variation norm. We first recall the definition of a quasi-stationary distribution in Section 1.1, state the abstract results in Section 1.2 and give a first application to irreducible Markov chains on finite state spaces in Section 1.4.

Definition

Let (X t ) t ≥0 be a time-homogeneous Markov process with state space E ∪ {∂} which is absorbed at ∂ ∉ E , in discrete or continuous time settings. A quasi-stationary distribution is a probability measure ν QSD on E such that

P ν QSD (X t ∈ A | t < τ ∂ ) = ν QSD (A), ∀t ≥ 0, A ∈ E , (1.1)
where we recall that τ ∂ = inf{t ≥ 0, X t = ∂} is the absorption time of X . We refer the reader to the book [START_REF] Collet | Quasi-stationary distributions[END_REF] and the surveys [START_REF] Martínez | Some properties of quasi-stationary distributions for finite Markov chains[END_REF][START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF][START_REF] Van Doorn | Quasi-stationary distributions for discrete-state models[END_REF] for several properties, analysis and historical notes on the concept of quasi-stationary distributions. In particular, it is known that a probability measure ν QSD on E is a quasi-stationary distribution if and only if there exists a probability measure µ on E such that

ν QSD (A) = lim t →+∞ P µ (X t ∈ A | t < τ ∂ ), ∀A ∈ E . (1.2)
For a given quasi-stationary distribution ν QSD , the set of probability measures µ such that (1.2) holds is called the domain of attraction of ν QSD . It is non-empty since it contains at least ν QSD and may contains an infinite number of elements. In particular, when the limit in (1.2) exists for any µ = δ x , x ∈ E , and doesn't depend on the initial position x, then ν QSD is called the Yaglom limit or the minimal quasi-stationary distribution. Thus the minimal quasi-stationary distribution, when it exists, is the unique quasi-stationary distribution whose domain of attraction contains {δ x , x ∈ E }.

The study of quasi-stationary distributions can be traced back to the works of Yaglom [START_REF] Yaglom | Certain limit theorems of the theory of branching random processes[END_REF] on Galton-Watson processes. Later, birth and death processes have been studied in [START_REF] Karlin | The classification of birth and death processes[END_REF][START_REF] Seneta | On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states[END_REF], and finite state space processes in [START_REF] Darroch | On quasi-stationary distributions in absorbing discrete-time finite Markov chains[END_REF][START_REF] Darroch | On quasi-stationary distributions in absorbing continuous-time finite Markov chains[END_REF]. A L 2 spectral approach was developed in [START_REF] Pinsky | On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes[END_REF] for the study of multi-dimensional diffusion processes and in [START_REF] Collet | Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption[END_REF] for one-dimensional diffusion processes, whose principles were later used in a long range of successful papers studying quasi-stationary distributions for diffusion processes (see for instance [START_REF] Asselah | Regularity of quasi-stationary measures for simple exclusion in dimension d ≥ 5[END_REF][START_REF] Steinsaltz | Quasistationary distributions for one-dimensional diffusions with killing[END_REF][START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF][START_REF] Cattiaux | Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned to non-extinction[END_REF][START_REF] Kolb | Quasilimiting behavior for one-dimensional diffusions with killing[END_REF][START_REF] Littin | Uniqueness of quasistationary distributions and discrete spectra when ∞ is an entrance boundary and 0 is singular[END_REF][START_REF] Hening | Quasistationary distributions for one-dimensional diffusions with singular boundary points[END_REF]). New advances based on the theory of R-positive recurrent processes have been developed in [START_REF] Arjas | Semi-markov processes on a general state space: α-theory and quasi-stationarity[END_REF][START_REF] Niemi | On nonsingular renewal kernels with an application to a semigroup of transition kernels[END_REF][START_REF] Nummelin | General irreducible Markov chains and nonnegative operators[END_REF][START_REF] Tuominen | Exponential decay and ergodicity of general Markov processes and their discrete skeletons[END_REF], with a nice practical criterion exhibited later in [START_REF] Ferrari | R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata[END_REF]. Other approaches have been developed, such as the use of h-transforms in [START_REF] Gong | Killed diffusions and their conditioning[END_REF], an original renewal method in [START_REF] Ferrari | Existence of quasi-stationary distributions. A renewal dynamical approach[END_REF], the use of intrinsic ultra-contractivity properties in [START_REF] Kim | Intrinsic ultracontractivity of non-symmetric diffusion semigroups in bounded domains[END_REF][START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF]. Several other methods have been used to describe the quasi-stationary behaviour of stochastic models with absorption, see in particular [START_REF] Haas | Quasi-stationary distributions and Yaglom limits of self-similar Markov processes[END_REF] where quasi-stationary distributions with rescaling are considered and in [START_REF] Strickler | Persistance de processus de Markov déterministes par morceaux[END_REF] where small noise limits of quasi-stationary distributions are studied. We also refer the reader to the large bibliography established by Pollett [START_REF] Pollett | Quasi-stationary distributions : a bibliography[END_REF] for more than 450 references on the theory of quasi-stationary distributions, classified by topics. In the present manuscript, a different approach is exposed, inspired by the classical theory of stationary processes [START_REF] Meyn | Markov chains and stochastic stability[END_REF]: we will use in particular modified Doblin conditions see Section 1.2, and drift criteria based on Lyapunov type functions (see Chapter 2).

Although there are similarities with the classical notion of stationary distributions (which is, in fact, a particular instance of quasi-stationary distribution), some important differences remain. For instance, the linear combination of quasi-stationary distributions is not necessarily a quasistationary distribution. Also, there may exist several (a continuous infinite number of) quasistationary distributions even for simple irreducible regular processes (this is the case for instance for linear birth and death processes [START_REF] Van Doorn | Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes[END_REF]).

It is well known (see for instance [START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF]) that when ν QSD is a quasi-stationary distribution, there exists λ 0 ≥ 0 such that, for all t ≥ 0, P ν QSD (t < τ ∂ ) = e -λ 0 t and e λ 0 t ν QSD P t = ν QSD .

(1.3)

An other remarkable fact is that the absorption times and the absorption position are independent under P ν QSD [START_REF] Collet | Quasi-stationary distributions[END_REF]. Because of these properties and others, the concept of quasi-stationary distribution has found a wide range of applications, see for instance [START_REF] Di Gesù | Jump Markov models and transition state theory: the quasi-stationary distribution approach[END_REF][START_REF] Huisinga | Phase transitions and metastability in Markovian and molecular systems[END_REF][START_REF] Haas | Asymptotic behavior of solutions of the fragmentation equation with shattering: An approach via self-similar markov processes[END_REF][START_REF] Bianchi | Metastable states, quasi-stationary distributions and soft measures[END_REF][START_REF] Wang | Theoretical Properties of Quasistationary Monte Carlo Methods[END_REF][START_REF] Wang | An Approximation Scheme for Quasistationary Distributions of Killed Diffusions[END_REF].

Main result

Let us consider a time-homogeneous Markov process X with state space E ∪ {∂} which is absorbed at ∂ ∉ E . We are interested in a necessary and sufficient condition for the existence of a unique quasi-stationary distribution ν QSD on E for the process (X t ) t ≥0 , where, in addition, the convergence in (1.2) is exponential and uniform with respect to µ and A.

Our base assumption is the following one.

Assumption A. There exists a probability measure ν on E such that A1. there exist t 0 , c 1 > 0 such that for all x ∈ E ,

P x (X t 0 ∈ • | t 0 < τ ∂ ) ≥ c 1 ν(•);
A2. there exist c 2 > 0 such that for all x ∈ E and t ≥ 0,

P ν (t < τ ∂ ) ≥ c 2 P x (t < τ ∂ ).
In Section 1.4, we will consider the finite state space case as a simple illustrative example and show that Assumption A is satisfied as soon as the process is irreducible. More advanced applications will be provided in following chapters, namely to birth and death processes, to one dimensional diffusion processes, and to multi-dimensional diffusion processes, in Chapters 3, 4 and 5 respectively.

For conservative Markov processes (i.e. when P x (τ ∂ = +∞) = 1), one recognises in Condition A1 a Doblin condition and in c 1 a Dobrushin coefficient. In this case, the following theorem is already well known (see for instance [START_REF] Douc | Markov chains[END_REF]Theorem 18.2.4.] and [START_REF] Douc | Markov chains[END_REF]Section 18.7]) and can be proved using coupling methods. Natural extension of these methods are presented in a pedagogical way in [START_REF] Levin | Markov chains and mixing times[END_REF]. Assumption A is thus an extension of these Doblin criteria to conditioned processes. As thus, it suffers similar drawbacks. Namely, it only applies to processes that come down from infinity [START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF][START_REF] Martínez | Existence and uniqueness of a quasistationary distribution for markov processes with fast return from infinity[END_REF]. In the next chapter, we provide a criterion inspired by Meyn and Tweedie's works [START_REF] Meyn | Stability of Markovian processes i: Criteria for discrete-time chains[END_REF][START_REF] Meyn | Stability of Markovian processes. II. Continuous-time processes and sampled chains[END_REF][START_REF] Meyn | Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes[END_REF]), which is sufficient for the non-uniform exponential convergence to a quasi-stationary distribution.

The following result is proved in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF] with additional refinements, including equivalent assertions comparable to those provided in [START_REF] Meyn | Markov chains and stochastic stability[END_REF]Chapter 16]. Its proof, sketched in Section 1.3, immediately extends to the time-inhomogeneous setting (which were natively handled in [START_REF] Moral | Feynman-Kac formulae. Probability and its Applications[END_REF][START_REF] Moral | Mean field simulation for Monte Carlo integration[END_REF][START_REF] Del Moral | Exponential mixing properties for time inhomogeneous diffusion processes with killing[END_REF]), as detailed in [START_REF] Champagnat | Uniform convergence of penalized timeinhomogeneous markov processes[END_REF][START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF]. Prior works implying the uniform exponential convergence of normalised semigroups can be found in the bibliography of Del Moral (see [START_REF] Moral | Feynman-Kac formulae. Probability and its Applications[END_REF][START_REF] Moral | Mean field simulation for Monte Carlo integration[END_REF] and references therein), as well as in my earlier works in [START_REF] Martínez | Existence and uniqueness of a quasistationary distribution for markov processes with fast return from infinity[END_REF] and [START_REF] Del Moral | Exponential mixing properties for time inhomogeneous diffusion processes with killing[END_REF] and in [START_REF] Birkhoff | Extensions of Jentzsch's theorem[END_REF][START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF].

Theorem 1.1. Assumption A implies the existence of a probability measure ν QSD on E such that, for any initial distribution µ ∈ M 1 (E ),

P µ (X t ∈ • | t < τ ∂ ) -ν QSD (•) T V ≤ 2(1 -c 1 c 2 ) t /t 0 , (1.4)
where • is the integer part function and • T V is the total variation norm. Conversely, if there exists a probability measure ν QSD and positive constants γ,C such that, for all probability measures µ on E ,

P µ (X t ∈ • | t < τ ∂ ) -ν QSD (•) T V ≤ C e -γt , ∀t ≥ 0, (1.5)
then Assumption A holds true. In this case, for all probability measures µ 1 , µ 2 on E , and for all t > 0,

P µ 1 (X t ∈ • | t < τ ∂ ) -P µ 2 (X t ∈ • | t < τ ∂ ) T V ≤ (1 -c 1 c 2 ) t /t 0 c 2 (µ 1 ) ∨ c 2 (µ 2 ) µ 1 -µ 2 T V ,
here c 2 (µ) is a positive constant which only depends on µ. 1The constants c 1 and c 2 may seem difficult to compute explicitly and it is not clear at first glance if the above quantitative rates are of practical interest (besides the fact that they provide an exponential speed of convergence). However, in [START_REF] Chazottes | On time scales and quasi-stationary distributions for multitype birth-and-death processes[END_REF], the authors succeed in proving appropriate time scales for the convergence to quasi-stationary distribution by estimating the parameters c 1 and c 2 . Similarly, the authors of [START_REF] Benaim | Stochastic approximation of quasi-stationary distributions on compact spaces and applications[END_REF] used a time inhomogeneous version of Assumption A and, estimating c 1 and c 2 in this situation, were able to prove a sub-exponential convergence rate toward a quasi-stationary distribution for a model with two communication classes.

Assumption A also has the following consequences.

Proposition 1.2. Assume that Assumption A holds true. Then there exists a non-negative function η on E ∪ {∂}, positive on E and vanishing on ∂, defined by

η(x) = lim t →∞ P x (t < τ ∂ ) P ν QSD (t < τ ∂ ) = lim t →+∞ e λ 0 t P x (t < τ ∂ ),
where the convergence holds for the uniform norm on E ∪ {∂} and ν QSD (η) = 1. More precisely, there exists a positive constant a 1 such that

e λ 0 t P x (t < τ ∂ ) -η(x) ≤ a 1 e λ 0 t P x (t < τ ∂ )(1 -c 1 c 2 ) t /t 0 , (1.6)
Furthermore, the function η is bounded, belongs to the domain of the infinitesimal generator L of the semi-group

(P t ) t ≥0 on (B b (E ∪ {∂}), • ∞ ) and Lη = -λ 0 η.
In the irreducible case, exponential ergodicity is known to be related to a spectral gap property (see for instance [START_REF] Ledoux | The concentration of measure phenomenon[END_REF]). Our results imply a similar property under Assumption A for the infinitesimal generator L of the semi-group on (B b (E ∪ {∂}), • ∞ ).

= ν QSD ( f )η, 3. or λ ≤ -λ 0 -γ, ν QSD ( f ) = 0 and f (∂) = 0.
We conclude this section with an original result concerning a refinement of the speed of convergence of the conditional distribution of the process toward its quasi-stationary distribution. Its proof is a simple adaptation of the proof of Theorem 1.1 and is omitted here. Note that ν(η)/ η ∞ ≥ c 2 and thus the rate is an improvement over (1.4). Proposition 1.4. Suppose that Assumption A holds. Then there exists a constant C > 0 such that

P µ (X t ∈ • | t < τ ∂ ) -ν QSD (•) T V ≤ C 1 -c 1 ν(η) η ∞ t /t 0 .

Sketch of the proof for the sufficient condition

In order to sketch the proof of the direct implication "Assumption A ⇒ exponential convergence", let us assume that X satisfies Assumption A with t 0 = 1 (the extension to any t 0 is immediate).

Following an idea that goes back at least to [START_REF] Moral | Mean field simulation for Monte Carlo integration[END_REF] and already used in [START_REF] Martínez | Existence and uniqueness of a quasistationary distribution for markov processes with fast return from infinity[END_REF] in the context of quasistationary distributions, we define, for all 0 ≤ s ≤ t ≤ T , the linear operator R T s,t by

R T s,t f (x) = E x ( f (X t -s ) | T -s < τ ∂ ) = E( f (X t ) | X s = x, T < τ ∂ ), ∀ f ∈ B b (E ),
by the Markov property. For any T > 0, the family (R T s,t ) 0≤s≤t ≤T is a Markov (time-inhomogeneous) semi-group: we have, for all 0

≤ u ≤ s ≤ t ≤ T and all f ∈ B b (E ), R T u,s (R T s,t f )(x) = R T u,t f (x).
The main idea of the proof is to check that this conservative semi-group satisfies a Doblin condition (see Step 1): for all T ≥ 1 and all 0 ≤ t ≤ T -1, there exists a probability measure ν T -t on E such that, for all measurable sets A ⊂ E and all x ∈ E ,

R T t ,t +1 (A) = P x (X 1 ∈ A | T -t < τ ∂ ) ≥ c 1 c 2 ν T -t (A). (1.7)
Once this is proved, one deduces (as in the classical time-uniform conservative case) a uniform mixing property for the conservative semi-groups R T and then for the conditional distributions:

P µ 1 (X T ∈ • | T < τ ∂ ) -P µ 2 (X T ∈ • | T < τ ∂ ) T V ≤ 2(1 -c 1 c 2 ) T , ∀µ 1 , µ 2 ∈ M 1 (E ). (1.8)
This immediately implies that there is at most one quasi-stationary distribution and implies in particular that the sequence

P µ 1 (X T ∈ • | T < τ ∂ ) T ≥0
is a Cauchy sequence and hence that it converges to some probability ν QSD (recall that the set of probability measures endowed with the total variation norm is complete). By [START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF], ν QSD is a quasi-stationary distribution.

Step 1: Doblin condition (1.7) Let us show that, for all t ≥ 1, there exists a probability measure ν t on E such that (1.7) holds true. First, one can check that Assumption A1 and Markov property imply that

P x (X 1 ∈ A and t < τ ∂ ) ≥ c 1 ν (1 A (•)P • (t -1 < τ ∂ )) P x (1 < τ ∂ ) .
Dividing both sides by P x (t < τ ∂ ), we deduce that

P x (X 1 ∈ A | t < τ ∂ ) ≥ c 1 ν (1 A (•)P • (t -1 < τ ∂ )) P x (1 < τ ∂ ) P x (t < τ ∂ ) .
But, using again the Markov property, we have

P x (t < τ ∂ ) ≤ P x (1 < τ ∂ ) sup y∈E P y (t -1 < τ ∂ ) , so that P x (X 1 ∈ A | t < τ ∂ ) ≥ c 1 ν (1 A (•)P • (t -1 < τ ∂ )) sup y∈E P y (t -1 < τ ∂ ) .
Now Assumption (A2) implies that the non-negative measure

B → ν (1 B (•)P • (t -1 < τ ∂ )) sup y∈E P y (t -1 < τ ∂ )
has a total mass greater than c 2 . Therefore (1.7) holds with the probability measure

ν t : B → ν (1 B (•)P • (t -1 < τ ∂ )) P ν (t -1 < τ ∂ )
Step 2: exponential contraction for the conditional distributions Using the semi-group property of (R T s,t ) s,t , we deduce that, for any x, y ∈ E and all 0 ≤ t ≤ T ,

δ x R T 0,t -δ y R T 0,t T V ≤ 2 (1 -c 1 c 2 ) t .
By definition of R T 0,T , this inequality immediately implies that

P x (X T ∈ • | T < τ ∂ ) -P y (X T ∈ • | T < τ ∂ ) T V ≤ 2(1 -c 1 c 2 ) T .
Since, in general, P µ (X T ∈ • | T < τ ∂ ) is not linear in µ, it is not immediate that this inequality extends to any pair of initial probability measures µ 1 , µ 2 on E . However, this is easily overcome by the following computations. Let µ 1 be a probability measure on E and x ∈ E . We have

P µ 1 (X T ∈ • | T < τ ∂ ) -P x (X T ∈ • | T < τ ∂ ) T V = 1 P µ 1 (T < τ ∂ ) P µ 1 (X T ∈ •) -P µ 1 (T < τ ∂ )P x (X T ∈ • | T < τ ∂ ) T V ≤ 1 P µ 1 (T < τ ∂ ) y∈E P y (X T ∈ •) -P y (T < τ ∂ )P x (X T ∈ • | T < τ ∂ ) T V d µ 1 (y) ≤ 1 P µ 1 (T < τ ∂ ) y∈E P y (T < τ ∂ ) P y (X T ∈ • | T < τ ∂ ) -P x (X T ∈ • | T < τ ∂ ) T V d µ 1 (y) ≤ 1 P µ 1 (T < τ ∂ ) y∈E P y (T < τ ∂ )2(1 -c 1 c 2 ) T d µ 1 (y) ≤ 2(1 -c 1 c 2 ) T .
The same computation, replacing δ x by any probability measure, leads to (1.8).

Using the fact that M 1 (E ) endowed with the total variation norm is a complete space, this easily leads to (1.4).

The finite state space case

The problem of existence and uniqueness of a quasi-stationary distribution in the finite state space setting has been studied by Darroch and Seneta [START_REF] Darroch | On quasi-stationary distributions in absorbing discrete-time finite Markov chains[END_REF][START_REF] Darroch | On quasi-stationary distributions in absorbing continuous-time finite Markov chains[END_REF]. They completely solved this problem in the irreducible state space case using Perron-Frobenius theorem, inspiring a long and rich lineage of developments for the study of quasi-stationary distributions based on spectral theoretical tools. The aim of this section is to give an application of Theorem 1.1 in a simple situation, recovering this classical result with additional explicit bounds on the rate of convergence.

Let (X t ) t ∈Z + be a discrete time Markov process on a finite state space E ∪ ∂, where ∂ ∉ E is absorbing. We say that X is irreducible and aperiodic if there exists t 0 ∈ N such that, for all x, y ∈ E , P x (X t 0 = y) > 0. Darroch and Seneta obtained in [START_REF] Darroch | On quasi-stationary distributions in absorbing discrete-time finite Markov chains[END_REF] (see [START_REF] Darroch | On quasi-stationary distributions in absorbing continuous-time finite Markov chains[END_REF] for its continuous time version) that there exist two positive constants such that the exponential convergence (1.5) holds true, with γ being the second spectral gap of the transition matrix.

The following convergence result is not focused toward optimality, but rather aims at illustrating how to check Assumption A in a simple case. We observe that, associated with Proposition 1.3, it provides an explicit lower bound for the second spectral gap of the matrix P . Proposition 1.5. Let X be an irreducible and aperiodic Markov chain on a finite state space E with transition matrix (P x,y ) x,y∈E . Let t 0 ∈ N be such that P . Then X satisfies Assumption A with the constants c 1 , c 2 and t 0 .

We refer the reader to [START_REF] Martínez | Some properties of quasi-stationary distributions for finite Markov chains[END_REF][START_REF] Van Doorn | Quasi-stationary distributions for discrete-state models[END_REF] for a survey on different properties of quasi-stationary distributions in this finite state space/discrete time setting. Extension of this result to reducible discrete time Markov chains on a finite state space is developed in [START_REF] Champagnat | On Dirichlet eigenvectors for neutral twodimensional Markov chains[END_REF][START_REF] Van Doorn | Quasi-stationary distributions for reducible absorbing Markov chains in discrete time[END_REF]. In [START_REF] Collet | Order relations of measures when avoiding decreasing sets[END_REF][START_REF] Diaconis | On times to quasi-stationarity for birth and death processes[END_REF], the authors consider the problem of the stochastic comparison between convergence toward a quasi-stationary distribution and convergence toward a stationary distribution for an ad hoc conservative process. Probabilistic representations of the Perron-Frobenius theorem are provided in [START_REF] Cerf | A Markov chain representation of the normalized Perron-Frobenius eigenvector[END_REF][START_REF] Cerf | A probabilistic proof of Perron's theorem[END_REF][START_REF] Glynn | A Probabilistic Proof of the Perron-Frobenius Theorem[END_REF] for finite state space Markov chains.

Since the aim of this section is to illustrate the application of Theorem 1.1, we detail the elementary proof of the above proposition.

Proof. We define the probability measure ν on E by

ν({y}) = inf x∈E P t 0 x,y c 1 z∈E P t 0 x,z , ∀y ∈ E .
We have for all x, y ∈ E ,

P x (X t 0 = y | t 0 < τ ∂ ) = P t 0 x,y z∈E P t 0 xz ≥ c 1 ν({y}),
which entails Assumption A1. Now, for all x ∈ E and all n ≥ 2,

P ν (n < τ ∂ ) ≥ ν({x})P x (n < τ ∂ ) ≥ c 2 P x (n < τ ∂ ),
which implies Assumption A2.

Chapter 2

Non-uniform convergence toward a quasi-stationary distribution

In this chapter, we present a sufficient criterion ensuring the exponential convergence of the conditional distribution of Markov processes toward a quasi-stationary distribution. Contrarily to the criteria of the previous part, we obtain non-uniform convergence with respect to the initial distribution. This allows to derive new existence and convergence results for a far larger class of processes, since it does not require the conditioned process to come back from infinity. In particular, this result applies to birth and death chains and Galton-Watson processes, which do not enter the general settings of Chapter 1. In general, our results also apply to processes admitting several quasi-stationary distributions, which is known to happen in a variety of specific cases, even for processes irreducible in E (including branching processes [START_REF] Seneta | On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states[END_REF][START_REF] Athreya | Branching processes[END_REF][START_REF] Lambert | Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct[END_REF][START_REF] Maillard | The λ-invariant measures of subcritical bienaymé-galton-watson processes[END_REF], one-dimensional birth and death processes [START_REF] Van Doorn | Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes[END_REF][START_REF] Ferrari | Some properties of quasi-stationary distributions in the birth and death chains: a dynamical approach[END_REF][START_REF] Ferrari | Existence of quasi-stationary distributions. A renewal dynamical approach[END_REF][START_REF] Villemonais | Minimal quasi-stationary distribution approximation for a birth and death process[END_REF] and one-dimensional diffusion processes [START_REF] Lladser | Domain of attraction of the quasi-stationary distributions for the Ornstein-Uhlenbeck process[END_REF][START_REF] Martínez | Classification of killed one-dimensional diffusions[END_REF]). The results presented below, but in Section 2.2, first appeared in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF].

Main results

We present here our main assumption and main results, in the discrete time and continuous time settings. In order to illustrate them, we develop in Section 2.2 a simple application to birth and death chains. More involved applications, including comparison to the results of [START_REF] Ferrari | R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata[END_REF] based on R-positive matrix theory, application to Galton-Watson processes, and application to perturbed dynamical systems are presented in Sections 2.3, 2.4 and 2.5 respectively. Applications to continuous time processes are presented in the next chapters.

Discrete time models

Let (X n ) n∈Z + be a Markov process in E ∪{∂} where E is a measurable space and ∂ ∈ E In this section, we study the sub-Markovian transition semigroup of X denoted (P n ) n∈Z + and defined as

P n f (x) = E x f (X n )1 n<τ ∂ , ∀n ∈ Z + ,
for all bounded or nonnegative measurable function f on E and all x ∈ E . We recall the notations

µP n f = E µ f (X n )1 n<τ ∂ = E P n f (x) µ(d x), 12CHAPTER 2. NON-UNIFORM CONVERGENCE TOWARD A QUASI-STATIONARY DISTRIBUTION
for all probability measures µ on E and all bounded measurable f . We make the following assumption 1 .

Assumption E. There exist a positive integer n 1 , positive real constants θ 1 , θ 2 , c 1 , c 2 , c 3 , two functions ϕ 1 , ϕ 2 : E → R + and a probability measure ν on a measurable subset K ⊂ E such that E1. (Local A1-A2). ∀x ∈ K ,

P x (X n 1 ∈ •) ≥ c 1 ν(• ∩ K ) and sup n∈Z + sup y∈K P y (n < τ ∂ ) inf y∈K P y (n < τ ∂ ) ≤ c 2 .
E2. (Global Lyapunov criterion). We have

θ 1 < θ 2 ≤ 1 and inf x∈E ϕ 1 (x) ≥ 1, sup x∈K ϕ 1 (x) < ∞ inf x∈K ϕ 2 (x) > 0, sup x∈E ϕ 2 (x) ≤ 1, P 1 ϕ 1 (x) ≤ θ 1 ϕ 1 (x) + c 3 1 K (x), ∀x ∈ E P 1 ϕ 2 (x) ≥ θ 2 ϕ 2 (x), ∀x ∈ E .
E3. (Aperiodicity). For all x ∈ K , there exists n 4 (x) such that, for all n ≥ n 4 (x),

P x (X n ∈ K ) > 0.
Remark 2.1. The construction of Lyapunov functions such as ϕ 1 is rather classical. On the contrary, finding functions such as ϕ 2 may seem at first more challenging. In fact, they are many ways to construct such a function, as illustrated by the numerous applications of the original paper. For instance, if Assumption E1 holds true and if there exists θ 2 ∈ (0, 1) such that θ -n 2 P x (X n ∈ K ) → +∞ when n → +∞, then, for any n 0 large enough, the function ϕ 2 (x) :=

n 0 k=0 θ -k 2 P x (X k ∈ K ) satisfies (up to renormalisation) condition E2.
In the rest of this section, we state our main results. We start with the exponential contraction in total variation of the conditional marginal distributions of the process given non-absorption (refinements and extensions of the following results are detailed in the original article [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]).

In the following result, M (ϕ 1 ), • M (ϕ 1 ) is the complete space defined p. xv. Theorem 2.1. Assume that Condition E holds true. Then there exist a constant C > 0, a constant α ∈ (0, 1), and a probability measure ν QSD on E such that

µP n µP n 1 E -ν QSD M (ϕ 1 ) ≤ C α n µ(ϕ 1 ) µ(ϕ 2 ) , (2.1)
for all probability measures µ on E such that µ(ϕ 1 ) < ∞ and µ(ϕ 2 ) > 0. Moreover, ν QSD is the unique quasi-stationary distribution of X such that ν QSD (ϕ 1 ) < ∞ and ν QSD (ϕ 2 ) > 0. In addition ν QSD (K ) > 0.

We define

E := {x ∈ E : ∃k ≥ 0 s.t. P k 1 K (x) > 0} = x ∈ E : ∃k ≥ 0 s.t. P k ϕ 2 (x) > 0 .
We refer the reader to the original article for the equality between the two sets above.

Corollary 2.2. Assume that Condition E holds true. Then the domain of attraction of ν QSD contains all the probability measures µ on E such that µ(E ) > 0 and µ(ϕ

1/p
1 ) < ∞ for some p < log θ 1/log θ 2 . In particular, if ϕ 1 is bounded and E = E , there exists a unique quasi-stationary distribution which attracts all the initial distributions.

We focus now on the asymptotic behaviour of the absorption probabilities and on the existence of an eigenfunction for P 1 associated to the eigenvalue θ 0 , where θ 0 ∈ (0, 1] is such that

P ν QSD (n < τ ∂ ) = θ n 0 , ∀n ∈ Z + .
We recall that the existence of θ 0 is a classical general result for quasi-stationary distributions [START_REF] Collet | Quasi-stationary distributions[END_REF][START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF][START_REF] Van Doorn | Quasi-stationary distributions for discrete-state models[END_REF]. In the following result, (L ∞ (ψ), • L ∞ (ψ) ) is the Banach space defined p. xvi.

Proposition 2.3. Assume that Condition E holds true. Then, there exists a function η :

E → R + such that η(x) = lim n→+∞ P x (n < τ ∂ ) P ν QSD (n < τ ∂ ) = lim n→+∞ θ -n 0 P x (n < τ ∂ ), ∀x ∈ E , (2.2)
where the convergence is geometric in L ∞ (ϕ

1/p 1 ) for all p ∈ [1, log θ 1 / log θ 0 ). In addition, inf y∈K η(y) > 0, E = {x ∈ E : η(x) > 0}, ν QSD (η) = 1, P 1 η = θ 0 η and θ 0 ≥ θ 2 > θ 1 .
Remark 2.2. The last result implies that, when η is bounded, one can actually take ϕ 2 = η/ η ∞ in Condition (E2). This property can be adapted to the case where η is not bounded, using for instance the approach of Chapter 7.

Remark 2.3. Similarly to Chapter 1, Assumption E also entails the existence of a spectral gap between θ 0 and the next eigenvalue. It also implies that η ∈ L ∞ ϕ log θ 0 / log θ 1 1 (see Corollary 2.6 in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]).

Continuous time models

We consider in this section an absorbed Markov process (X t ) t ∈R + in the continuous time setting. Assumption F. There exist positive real constants γ 1 , γ 2 , c 1 , c 2 , c 3 , t 1 and t 2 , a measurable function ψ 1 : E → [1, +∞), and a probability measure ν on a measurable subset L ⊂ E such that F0. (A strong Markov property). Defining

τ L := inf{t ∈ R + : X t ∈ L}, ( 2.3)
assume that for all x ∈ E , X τ L ∈ L, P x -almost surely on the event {τ L < ∞}, and, for all t > 0 and all f ∈ B b (E ∪ {∂}),

E x f (X t )1 τ L ≤t <τ ∂ = E x 1 τ L ≤t ∧τ ∂ E X τ L f (X t -u )1 t -u<τ ∂ u=τ L .

F1. (Local A1 and A2

). ∀x ∈ L,

P x (X t 1 ∈ •) ≥ c 1 ν(• ∩ L) and sup t ∈R + sup y∈L P y (t < τ ∂ ) inf y∈L P y (t < τ ∂ ) ≤ c 2 .
F2. (Global Lyapunov criterion). We have γ 1 < γ 2 and

E x (ψ 1 (X t 2 )1 t 2 <τ L ∧τ ∂ ) ≤ γ t 2 1 ψ 1 (x), ∀x ∈ E E x (ψ 1 (X t )1 t <τ ∂ ) ≤ c 3 , ∀x ∈ L, ∀t ∈ [0, t 2 ], γ -t 2 P x (X t ∈ L) ----→ t →+∞ +∞, ∀x ∈ L.
Theorem 2.4. Under Assumption F , (X t ) t ∈I admits a quasi-stationary distribution ν QSD , which is the unique one satisfying ν QSD (ψ 1 ) < ∞ and P ν QSD (X t ∈ L) > 0 for some t ∈ I . Moreover, there exist constants α ∈ (0, 1) and C > 0 such that, for all probability measures µ on E satisfying µ(ψ 1 ) < ∞ and µ(ψ 2 ) > 0,

P µ (X t ∈ • | t < τ ∂ ) -ν QSD T V ≤ C α t µ(ψ 1 ) µ(ψ 2 ) , ∀t ∈ I , (2.4
)

where ψ 2 (x) = n 0 k=0 γ -kt 2 2
P x (X kt 2 ∈ L) for n 0 ≥ 1 and t 2 ∈ R + large enough. In addition, there exists a constant λ 0 ≥ 0 such that λ 0 ≤ log(1/γ 2 ) < log(1/γ 1 ) and P ν QSD (t < τ ∂ ) = e -λ 0 t for all t ≥ 0, and there exists a function η such that

η(x) = lim t →+∞ e λ 0 t P x (t < τ ∂ ), ∀x ∈ E , (2.5)
where the convergence is exponential in L ∞ (ψ 1/p 1 ) for all p ∈ [1, log(1/γ 1 ) /λ 0 ), and P t η(x) = e -λ 0 t η(x) for all x ∈ E and t ∈ R + .

In particular, if η is bounded and setting η(∂) = 0, then the function η defined on E ∪{∂} belongs to the domain of the infinitesimal generator L of X and L η = -λ 0 η. Remark 2.4. The main point of the proof is to check that Assumption F entails Assumption E for the sub-Markovian semigroup (P n ) n≥0 of the absorbed Markov process (X nt 2 ) n∈Z + , with the functions

ϕ 1 = ψ 1 and ϕ 2 = γ -t 2 2 -1 γ -(n 0 +1)t 2 2 -1 ψ 2 , any θ 1 ∈ (γ t 2 1 , γ t 2 2 ), θ 2 = γ t 2
2 and the set

K = y ∈ E , P y (τ L ≤ t 2 )/ψ 1 (y) ≥ (θ 1 -γ t 2 1 )/c 2 ⊃ L.
Remark 2.5. The first two lines of F2 can be deduced by classical Foster-Lyapunov inequalities (cf. [START_REF] Meyn | Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes[END_REF]). Indeed, denoting by L the infinitesimal generator of the process X , if

L ψ 1 (x) ≤ -λ 1 ψ 1 (x) +C 1 L (x), ∀x ∈ E , (2.6)
then (formally, assuming one can apply Dynkin's formula)

E x [1 1≤τ L ∧τ ∂ ψ 1 (X 1 )] ≤ e -γ 1 ψ 1 (x) and E x [ψ 1 (X t )1 t <τ ∂ ] ≤ e C t ψ 1 (x)
. However, a function ψ 1 satisfying (2.6) does not necessarily belong to the domain of the infinitesimal generator L , so one needs to extend the notion of infinitesimal generator as in [START_REF] Meyn | Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes[END_REF][START_REF] Champagnat | Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes[END_REF]]. An other approach has been considered in [START_REF] Marguet | A law of large numbers for branching Markov processes by the ergodicity of ancestral lineages[END_REF], with conditions involving Lyapunov functions for the family of time-inhomogeneous processes, defined as the process X conditioned not to be absorbed before time t , where t runs over R + . We also refer the reader to [START_REF] Velleret | Unique Quasi-Stationary Distribution, with a possibly stabilizing extinction[END_REF], where the authors provide criteria based on hitting time controls. Such controls are of course related to the existence of Lyapunov functions, as explained in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]Lemma 3.6].

Birth and death Markov chains

Let (λ x ) x∈N and (µ x ) x∈N be families of positive numbers such that µ x + λ x = 1 for all x ∈ N. We consider the birth and death chain on E ∪ {∂}, E = N and ∂ = 0, with the following transition probabilities

P x (X 1 = y) =        λ x if y = x + 1, µ x if y = x -1, 0 otherwise.
These models are known as birth and death chains, and we refer the reader to [START_REF] Ferrari | Existence of quasi-stationary distributions. A renewal dynamical approach[END_REF][START_REF] Bean | The quasi-stationary behavior of quasi-birth-and-death processes[END_REF][START_REF] Coolen-Schrijner | Quasi-stationary distributions for a class of discrete-time Markov chains[END_REF][START_REF] Diaconis | On times to quasi-stationarity for birth and death processes[END_REF][START_REF] Artalejo | Quasi-stationary and ratio of expectations distributions: A comparative study[END_REF] and references therein for earlier studies of their quasi-stationary behaviour. We assume that the process X is aperiodic, that inf x≥1 µ x > 0, and that λ x → 0 when x → +∞. We define

ϕ 1 (x) = 1 x≥1 e ax with a > 0 such that e -a ≤ 1 -µ 1 4 and ϕ 2 (x) = 1 x≥1 , ∀x ≥ 0,
and show that Assumption E is satisfied in this case. Since X is assumed to be aperiodic, there exists x 0 ≥ 1 such that λ x 0 + µ x 0 < 1 (otherwise it would be 2-periodic) and this entails Condition E3. Since the birth and death coefficients (λ x ) x∈N and (µ x ) x∈N are positive, for any finite set K = {1, . . . , x K }, there exists n K such that inf x,y∈K P x (X n K = y) > 0. Similarly to the proof of (A1-A2) in Section 1.4 of Chapter 1, this entails Condition E1.

It only remains to check that E2 holds true, for some finite set K as above. Set θ 2 = λ 1 and θ 1 = θ 2 /2. For all x ≥ 1, we have

E x (ϕ 2 (X 1 )) = λ 1 = θ 2 ϕ 2 (x).
and

E x (ϕ 1 (X 1 )) = λ x e a(x+1) + µ x e a(x-1) = ϕ 1 (x) λ x e a + µ x e -a + (1 -λ x -µ x ) ≤ ϕ 1 (x) λ x e a + θ 1 /2 ≤ θ 1 ϕ 1 (x) +C 1 x≤n ,
where C and n are chosen large enough. This concludes the proof.

Theory of R-positive matrices

We consider a Markov chain (X n ) n∈Z + in a countable state space E ∪ {∂} absorbed at ∂ ∈ E and with irreducible transition probabilities in E , i.e. such that for all x, y ∈ E , there exists n = n(x, y) ≥ 1 such that P x (X n = y) > 0. In this case, one of the most general criterion for existence and convergence to a quasi-stationary distribution is provided in [START_REF] Ferrari | R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata[END_REF]. In this paper, the authors obtain a convergence result similar to the one of Theorem 2.1 restricted to Dirac initial distributions, and the pointwise convergence to η (as defined in Proposition 2.3), using the powerful theory of R-positive matrices (see [START_REF] Andjel | Convergence in distribution for subcritical 2D oriented percolation seen from its rightmost point[END_REF][START_REF] Andjel | Subcritical contact process seen from the edge: convergence to quasi-equilibrium[END_REF][START_REF] Ferrari | Tumor growth, R-positivity, Multitype branching and Quasistationarity[END_REF] for recent applications). In this section, we show how our criterion allows to recover these results, providing in addition the characterisation of a non-trivial subset of the domain of attraction and a stronger convergence to η. We assume that the absorption time τ ∂ is almost surely finite. Without loss of generality, we will assume that the process is aperiodic, meaning that P x (X n = y) > 0 for all x, y ∈ E provided n is large enough; the extension to general periodic processes is routine, as observed in [START_REF] Ferrari | R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata[END_REF].

/R = lim n→∞ P x (X n = y) 1/n , (2.7)
for some and hence any value of x, y ∈ E (see e.g. [START_REF] Vere-Jones | Ergodic properties of nonnegative matrices[END_REF]).

Theorem 2.5 (Theorem 1, [START_REF] Ferrari | R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata[END_REF]). Assume that the Markov chain satisfies the following conditions:

(a) there exist a nonempty set U 1 ⊂ E and two positive constants ε 0 ,C 1 such that, for all x ∈ U 1 and all n ≥ 0,

P x (τ ∂ > n, but X ∉ U 1 for all 1 ≤ ≤ n) ≤ C 1 (R + ε 0 ) -n ,
(b) there exist a state x 0 ∈ U 1 and a positive constant C 2 such that, for all x ∈ U 1 and n ≥ 0,

P x (n < τ ∂ ) ≤ C 2 P x 0 (n < τ ∂ ),
(c) there exist a finite set U 2 ⊂ E and constants 0 ≤ n 0 < ∞, C 3 > 0, such that for all x ∈ U 1 ,

P x (X n ∈ U 2 for some n ≤ n 0 ) ≥ C 3 .
Then X is R-positive-recurrent and there exists a probability measure ν QSD on E such that

lim n→+∞ P x (X n = y | n < τ ∂ ) = ν QSD (y), ∀x, y ∈ E
and a positive function η on E such that

lim n→+∞ R n P x (n < τ ∂ ) = η(x), ∀x ∈ E .
The main result of this section is the following Proposition 2.6. The assumptions of [START_REF] Ferrari | R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata[END_REF]Theorem 1] imply Assumption E.

In the settings of this section, Assumption E is actually equivalent to the conditions of [START_REF] Ferrari | R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata[END_REF]Theorem 1]. Besides the additional properties provided in Chapter 2, one of our main contribution in this particular setting is to provide a different, sometimes more tractable criterion, through the use of Lyapunov functions. This is illustrated in the next subsection, with an application to population processes, extending to the multi-dimensional case some models studied in [START_REF] Gosselin | Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology[END_REF]. The application of [START_REF] Ferrari | R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata[END_REF] is not "impractical for such models of biological population extinction" as claimed in [132, p. 262], but it would be a little bit more involved.

Application to the extinction of biological populations dominated by Galton-Watson processes

A Markov process (Z n ) n∈Z + evolving in Z d + = E ∪ {∂} absorbed at ∂ = 0 is called a Galton-Watson process with d types if, for all n ≥ 0 and all i ∈ {1, . . . , d },

Z i n+1 = d k=1 Z k n =1 ζ (n, ) k,i , (2.8)
where the random variables (ζ (n, ) k,1 , . . . , ζ (n, ) k,d ) n, ,k in Z + are assumed independent and such that, for all k ∈ {1, . . . , d }, (ζ (n, ) k,1 , . . . , ζ (n, ) k,d ) n, is an i.i.d. family. We define the matrix M = (M k,i ) 1≤k,i ≤d of mean offspring as

M k,i = E(ζ (n, )
k,i ), ∀k, i ∈ {1, . . . , d }, and assume that M k,i < +∞ and that there exists

n ≥ 1 such that [M n ] k,i > 0 for all k, i ∈ {1, . . . , d }.
Using the classical formalism of [START_REF] Harris | The theory of branching processes[END_REF], we consider a positive right eigenvector v of the matrix M of mean offspring and we denote by ρ(M ) its spectral radius. The sub-critical case corresponds to ρ(M ) < 1. It is well-known [START_REF] Joffe | On multitype branching processes with ρ ≤ 1[END_REF] (see also [START_REF] Heathcote | A refinement of two theorems in the theory of branching processes[END_REF][START_REF] Athreya | Branching processes[END_REF]) that this implies the existence of a quasi-stationary distribution whose domain of attraction contains all Dirac measures (a so-called Yaglom limit or minimal quasi-stationary distribution). The authors also prove that

ν QSD (| • |) < ∞ if and only if E[|Z 1 | log(|Z 1 |) | Z 0 = (1, . . . , 1)] < ∞. While the following result makes the stronger assumption that E[|Z 1 | q 0 | Z 0 = (1, . . . , 1)
] < ∞ for some q 0 > 1, we obtain a stronger form of convergence (in total variation norm with exponential speed), a non-trivial subset of the domain of attraction of the minimal quasi-stationary distribution and stronger moment properties for this quasi-stationary distribution.

Proposition 2.7. If (Z n ) n∈Z + is a d -type irreducible, aperiodic sub-critical Galton-Watson process, and if, for some q

0 > 1, E[|Z 1 | q 0 | Z 0 = (1, . . . , 1)] < ∞,
then Condition (E) holds true with ϕ 1 (z) = |z| q for any q ∈ (1, q 0 ]. In particular, the domain of attraction of ν QSD contains all the probability measures such that µ(| • | q ) < ∞ for some q > 1.

We focus now on population processes dominated by population-dependent Galton-Watson processes. More precisely, we consider an aperiodic and irreducible Markov population process (Z n ) n∈N on Z d + = E ∪ {∂} absorbed at ∂ = 0 such that, for all n ≥ 0,

Z n+1 ≤ |Z n | i =1 ξ (Z n ) i ,n , (2.9)
where • is a norm on R d and |z| = z 1 + . . . + z d for all z ∈ Z d + and, for all n ≥ 0, the nonnegative random variables (ξ

(Z n ) i ,n , 1 ≤ i ≤ |Z n |) are assumed independent (but not necessarily identically distributed) given Z n .
We assume that

E |z| i =1 ξ (z) i ,n ≤ m z , ∀z ∈ Z d + such that |z| ≥ n 0 , (2.10)
for some m < 1 and n 0 ∈ N. This means that the population size has a tendency to decrease (in mean) when it is large. This also implies that τ ∂ < ∞ a.s.

In the following theorem, R > 0 is the limiting value defined in (2.7).

Proposition 2.8. Assume that (Z n ) n∈Z + is aperiodic irreducible, that it satisfies the assumptions (2.9) and (2.10) and that, for some q

0 > log R log(1/m) ∨ 1, sup n≥0, z∈Z d + , 1≤i ≤|z| E[(ξ (z) i ,n ) q 0 ] < ∞,
Then Condition E holds true with ϕ 1 (x) = |x| q , for all q

∈ log R log(1/m) ∨ 1, q 0 . 18CHAPTER 2. NON-UNIFORM CONVERGENCE TOWARD A QUASI-STATIONARY DISTRIBUTION Remark 2.6. This result easily applies if sup n≥0, z∈Z d + , 1≤i ≤|z| E[(ξ (z) i ,n ) q ] < ∞ for all q > 0.
In other cases, one needs a upper bound for R > 0 in order to check the validity of the assumptions of Proposition 2.8. For instance, one may use the fact that R ≤ 1/ sup z∈Z d

+ P z (Z 1 = z).
The above theorem applies for instance when Z is obtained from a Galton-Watson multi-type process with an additional population-dependent death rate. Typically, one can assume that additional death events may affect a fraction of the population, modelling global death events. Note that, in this case and contrary to the Galton-Watson case, the independence between the progeny of individuals breaks down and the classical approach based on generating functions is rendered helpless.

Another situation covered by the above result is the case where the domain of absorption of Z is a larger set than 0, for example the process may be absorbed when it reaches one edge of Z d + (i.e. when one type disappears).

Another typical application of Proposition 2.8 is the case of population-dependent Galton-Watson processes, i.e. of processes such that, given Z n , Z n+1 is the sum of |Z n | independent random variables whose law may depend on Z n . In this situation, Proposition 2.8 and its consequences stated in Chapter 2 generalise the results of [START_REF] Gosselin | Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology[END_REF] to the multi-type models and provides finer results on the domain of attraction of the minimal quasi-stationary distribution. The reducible cases considered in [START_REF] Gosselin | Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology[END_REF] can also be recovered using the approach of Section 2.6. Of course, the above specifications may be combined.

Perturbed dynamical systems

We consider the following perturbed dynamical system

X n+1 = f (X n ) + ξ n ,
where f : R d → R d is a measurable function and (ξ n ) n∈N is an i.i.d. sequence in R d . The quasistationary behaviour of such processes has been studied under different constraints in [START_REF] Faure | Quasi-stationary distributions for randomly perturbed dynamical systems[END_REF][START_REF] Berglund | Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model[END_REF][START_REF] Baudel | Spectral theory for random poincaré maps[END_REF][START_REF] Hinrichs | Persistence of one-dimensional AR(1)-sequences[END_REF]. We assume that the process evolves in a measurable set D of R d with positive Lebesgue measure, and that it is immediately sent to ∂ ∈ R d as soon as X n ∈ D. We consider here the case where the law of the random variables ξ n have support R d and, more precisely, admit a positive bounded density with respect to the Lebesgue measure2 . This includes the particular case of dynamical systems perturbed by a Gaussian noise, which is considered in Example 2.1 below. In this setting, the perturbed dynamical system X n+1 = f (X n ) + ξ n with (ξ i ) i ∈Z + i.i.d. Gaussian, absorbed when it leaves a given measurable set D of R d with positive Lebesgue measure, admits a quasi-stationary distribution as soon as |x| -| f (x)| → +∞ when |x| → +∞. Proposition 2.9. Assume that f is locally bounded, that the law of ξ n has a bounded density g (x) with respect to Lebesgue's measure, that

inf |x|≤R g (x) > 0, ∀R > 0,
and that there exists a locally bounded function ϕ :

R d → [1, +∞) such that x → E(ϕ(x +ξ 1 )) is locally bounded on R d and lim sup |x|→+∞, x∈D E(ϕ( f (x) + ξ 1 )) ϕ(x) = 0. (2.11)
Then Condition E is satisfied with ϕ 1 = ϕ and ϕ 2 positive on D.

Let us illustrate this proposition with three examples.

Example 2.1. If there exists α > 0 such that Ee α|ξ 1 | < +∞ and if |x| -| f (x)| → +∞ when |x| → +∞, then Proposition 2.9 applies. Indeed, choosing ϕ(x) = exp(α|x|), we have

Eϕ(| f (x) + ξ 1 |) ϕ(x) ≤ e α(| f (x)|-|x|) Ee α|ξ 1 | ------→ |x|→+∞ 0.
For instance, this covers the case of Gaussian perturbations.

Example 2.2. If there exists p > 0 such that E(ξ x) for some C > 0 and some ε(x) → 0 when |x| → +∞, then Proposition 2.9 applies. Indeed, choosing ϕ(x) = log(e + |x|), we have

p 1 ) < +∞ and if | f (x)| = o(|x|) when |x| → +∞, then Proposition 2.9 applies. Indeed, choosing ϕ(x) = (1 + |x|) p , we have Eϕ(| f (x) + ξ 1 |) ϕ(x) ≤ (1 + | f (x)|) p (1 + |x|) p E[(1 + |ξ 1 |) p ] ------→ |x|→+∞ 0. Example 2.3. If E log(1 + |ξ 1 |) < ∞ and | f (x)| ≤ C |x| ε(
Eϕ(| f (x) + ξ 1 |) ϕ(x) ≤ log(e +C ) + ε(x) log(e + |x|) log(1 + |x|) + E log(1 + |ξ 1 |) log(e + |x|) .
The condition on f is true for example if | f (x)| ≤ C exp log(1 + |x|) for some constant C .

Reducible models

The study of quasi-stationary behaviour of models with multiple communication classes has been conducted in [START_REF] Ogura | Asymptotic behavior of multitype Galton-Watson processes[END_REF][START_REF] Gosselin | Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology[END_REF][START_REF] Champagnat | Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions[END_REF][START_REF] Champagnat | On Dirichlet eigenvectors for neutral twodimensional Markov chains[END_REF][START_REF] Van Doorn | Quasi-stationary distributions for discrete-state models[END_REF][START_REF] Benaim | Stochastic approximation of quasi-stationary distributions on compact spaces and applications[END_REF]. Our criteria provide new practical tools to tackle this problem. In Subsection 2.6.1, we consider a general setting with three successive sets. In Subsection 2.6.2, we consider a birth and death process with a countable infinity of communication classes.

Three successive sets

Consider a discrete time Markov process (X n ) n∈Z + evolving in a measurable set E ∪ {∂} with absorption at ∂ ∉ E . We assume that the transition probabilities of X satisfy the structure displayed in Figure 2.1 : one can find a partition {D 1 , D 2 , D 3 } of E such that the process starting from D 1 can access D 1 ∪D 2 ∪D 3 ∪{∂}, the process starting from D 2 can only access D 2 ∪D 3 ∪{∂}, and the process starting from D 3 can only access D 3 ∪ {∂}. More formally, we assume that

P x (T D 3 ∧ τ ∂ < T D 1 ) = 1 for all x ∈ D 2 and that P x (τ ∂ < T D 1 ∪D 2 ) = 1 for all x ∈ D 3
, where we recall that, for any measurable set

A ⊂ E , T A = inf{n ∈ Z + , X n ∈ A}.
Our aim is to provide sufficient conditions ensuring that X satisfies Assumption E. In order to do so, we assume that Assumption E is satisfied by the process X before exiting D 2 . This corresponds to the following assumption.

Y n = X n if n < T D 1 ∪D 3 ∪{d } , ∂ if n ≥ T D 1 ∪D 3 ∪{d } ,
satisfies Assumption E. In what follows, we denote the objects related to Y with a superscript Y , for instance, the constants of Assumption E for Y are denoted by θ Y 1 > 0, θ Y 2 > 0. We also assume that the exit times from D 1 and D 3 for the process X admit exponential moments of sufficiently high order, as stated by the following assumption. Assumption H2. There exists a positive constant γ < θ Y 0 such that, for all x ∈ D 1 ,

E x γ -T D 2 ϕ Y 1 X T D 2 1 T D 2 <T D 3 ∧τ ∂ < +∞, E x γ -T D 3 ∧τ ∂ 1 T D 3 ∧τ ∂ <T D 2 < +∞,
and such that sup

x∈D 3 E x γ -τ ∂ < +∞.
We are now able to state the main result of this section.

Proposition 2.10. Under Assumptions H1 and H2, the process X satisfies Assumption E with K

= K Y , ϕ 1 (x) = E x γ -T K ∧τ ∂ and ϕ 2 (x) ≥ c1 x∈K , ∀x ∈ E .
In particular, it admits a unique quasi-stationary distribution ν QSD such that ν QSD (ϕ 1 ) < ∞ and ν QSD (ϕ 2 ) > 0. Moreover, there exist two constants C > 0 and α ∈ (0, 1) such that, for all probability measures µ on E such that µ(ϕ 1 ) < ∞ and µ(ϕ 2 ) > 0,

P µ (X n ∈ • | n < τ ∂ ) -ν QSD T V ≤ C α n µ(ϕ 1 ) µ(ϕ 2 ) .
Finally, θ 0 = θ Y 0 , ν QSD (D 1 ) = 0 and the function η of Proposition 2.3 vanishes on D 3 .

In particular, one deduces from the last property that E ⊂ D 1 ∪ D 2 , where we recall that

E = {x ∈ E : ∃n ∈ N, P n 1 K (x) > 0}. Remark 2.7.
1. The fact that there are three different sets D 1 , D 2 and D 3 in the decomposition of E is not restrictive on the number of communication classes. Indeed, the three sets can contain several communication classes.

2. A similar result can be obtained for continuous time processes, based on Assumption F instead of E, with the additional technical assumption that the exit times of D 1 and D 2 are stopping times.

3. Beside the exponential moment assumption, there is no additional requirement on the behaviour of the Markov process in D 1 and D 3 . In these sets, the process might be periodic or deterministic for instance.

4. The quasi-stationary distribution of this process may not be unique, for instance if the process restricted to D 3 also admits a quasi-stationary distribution.

Countably many communication classes

We consider now a particular case of a continuous time càdlàg Markov process (X t ) t ∈R + with a countable infinity of communication classes and we show that the process admits a quasi-stationary distribution.

More precisely, we assume that X evolves in the state space N × Z + (the first component is the index of the communication class and the second is the position of the process in this communication class) and, denoting N t ∈ N and Y t ∈ Z + the two components of X t for all t ∈ R + , that there exist three positive functions b, d , f :

N → (0, +∞) such that • N is a Poisson process with intensity 1, • Y is a process such that, at time t , Y jumps from Y t to y ∈ Z + with rate        f (N t ) b(Y t ) if y = Y t + 1 and Y t ≥ 1, f (N t ) d (Y t ) if y = Y t -1 and Y t ≥ 1, 0 otherwise.
The set N × {0} is absorbing for X and we are interested in the quasi-stationary behaviour of X conditioned to not hit this set. Note that, in this case, each set {n} × N is a communication class. This process can be used to model the evolution of the vitality of an individual (for example a bacterium) whose metabolic efficiency (for example its ability to consume resources) changes with time, due to ageing [START_REF] Steinsaltz | Markov mortality models: Implications of quasistationarity and varying initial conditions[END_REF]. Here Y is the vitality of the individual, who dies when its vitality hits 0, and f (N ) is the metabolic rate of the individual.

This process can also be used to model the accumulation of deleterious mutations in a population under the assumption that mutations do not overlap, i.e. that when a mutant succeeds to invade the population (either because they are advantaged or due to genetic drift for deleterious mutations), other types of mutants disappear rapidly. Here Y represents the size of the population and N the number of mutations (see e.g. [START_REF] Coron | Quantifying the mutational meltdown in diploid populations[END_REF][START_REF] Coron | Stochastic modeling and eco-evolution of a diploid population[END_REF]).

In both cases, it is relevant to assume that f is decreasing on {1, 2, . . . , n 0 } and increasing to +∞ on {n 0 , n 0 + 1, . . .}, which we do from now on. We also assume that (d (y)b(y))/y → +∞ when y → +∞ or that there exists δ > 1 such that d (y) -δ b(y) → +∞. Proposition 2.11. Under the above assumptions, the process X satisfies Assumption F and admits a quasi-stationary distribution ν QSD whose domain of attraction contains all Dirac measures δ n,y , with n ≤ n 0 and y ∈ N.

Part II

Application of the criteria of Part I to classical models

Chapter 3

Birth and death processes

In this chapter, we focus on the application of the results of Chapters 1 and 2 to birth and death processes. In Section 3.1, we recall the classical result on quasi-stationary distributions for birth and death processes. In Section 3.2, we consider the case of one-dimensional birth and death processes with entrance boundary at infinity, and, in Section 3.3, the case of one-dimensional birth and death processes which are λ 0 -positive recurrent. In Section 3.4, we consider the case of multidimensional birth and death processes.

The results of this chapter first appeared in the articles [START_REF] Martínez | Existence and uniqueness of a quasistationary distribution for markov processes with fast return from infinity[END_REF][START_REF] Villemonais | Minimal quasi-stationary distribution approximation for a birth and death process[END_REF][START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF][START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]

Quasi-stationary distributions for one-dimensional birth and death processes

Let (X t ) t ∈R + be a birth and death process on Z + with birth rates (b x ) x∈Z + and death rates (d x ) x∈Z + . We assume that b x > 0 and d x > 0 for any x ∈ N and b 0 = d 0 = 0. The stochastic process X is a Z + -valued pure jump process whose only absorption point is 0 and whose transition rates from any point x ≥ 1 are given by

x → x + 1 with rate b x , x → x -1 with rate d x , x → y with rate 0, if y ∉ {x -1, x + 1}.
It is well known (see e.g. [203, Theorem 10 and Proposition 12]) that X is stable, conservative and hits 0 in finite time almost surely (for any initial distribution) if and only if

∞ n=1 d 1 d 2 • • • d n b 1 b 2 • • • b n = +∞, (3.1)
which we shall assume from now on. Such processes are extensively studied because of their conceptual simplicity and pertinence as demographic models. Concerning the study of their quasi-stationary behaviour, see for instance [START_REF] Karlin | The classification of birth and death processes[END_REF][START_REF] Karlin | The differential equations of birth-and-death processes, and the Stieltjes moment problem[END_REF][START_REF] Good | The limiting behavior of transient birth and death processes conditioned on survival[END_REF][START_REF] Cavender | Quasi-stationary distributions of birth-and-death processes[END_REF][START_REF] Van Doorn | Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes[END_REF][START_REF] Ferrari | Some properties of quasi-stationary distributions in the birth and death chains: a dynamical approach[END_REF][START_REF] Ferrari | Existence of nontrivial quasi-stationary distributions in the birth-death chain[END_REF][START_REF] Kijima | Some results for quasi-stationary distributions of birth-death processes[END_REF][START_REF] Kijima | Limiting conditional distributions for birth-death processes[END_REF][START_REF] Van Doorn | On associated polynomials and decay rates for birth-death processes[END_REF][START_REF] Van Doorn | Birth-death processes and associated polynomials[END_REF][START_REF] Van Doorn | On the α-classification of birth-death and quasi-birth-death processes[END_REF].

From a demographic point of view, the study of the minimal quasi-stationary distribution of a birth and death process aims at answering the following question: knowing that a population isn't extinct after a long time t , what is the probability that its size is equal to n at time t ? For these processes, van Doorn [START_REF] Van Doorn | Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes[END_REF] gave the following picture of the situation: a birth and death process can have no quasi-stationary distribution, one unique quasi-stationary distribution or an infinity (in fact a continuum) of quasi-stationary distributions. In order to determine whether a birth and death process has 0, one or an infinity of quasi-stationary distributions, one define inductively the sequence of polynomials (Q n (x)) n≥0 for all x ∈ R by

Q 1 (x) = 1, b 1 Q 2 (x) = b 1 + d 1 -x and b n Q n+1 (x) = (b n + d n -x)Q n (x) -d n-1 Q n-1 (x), ∀n ≥ 2. (3.2)
As recalled in [246, eq. (2.13)], one can uniquely define the non-negative number λ 0 satisfying

x ≤ λ 0 ⇐⇒ Q n (x) > 0, ∀n ≥ 1. (3.3)
Also, the useful quantity

S := sup x≥1 E x (T 0 ), can be easily computed (see [5, Section 8.1]), since, for any z ≥ 1, sup x≥z E x (T z ) = k≥z+1 1 d k π k l ≥k π l , with π k = k-1 i =1 b i / k i =2 d i .
The following theorem answers the question of existence and uniqueness of a quasi-stationary distribution for birth and death processes. Theorem 3.1 (van Doorn, 1991 [START_REF] Van Doorn | Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes[END_REF]). Let X be a birth and death process satisfying (3.1).

1. If λ 0 = 0, there is no quasi-stationary distribution.

2. If S < +∞, then λ 0 > 0 and the Yaglom limit is the unique quasi-stationary distribution.

3. If S = +∞ and λ 0 > 0, then there is a continuum of quasi-stationary distributions, given by the one parameter family (ρ a ) 0<a≤λ 0 :

ρ a (x) = π x d 1 a Q x (a), ∀x ≥ 1,
and the minimal quasi-stationary distribution is given by ρ λ 0 .

Theorem 3.1 is quite remarkable since it describes completely the possible outcomes of the existence and uniqueness problem for quasi-stationary distributions. However, it only partially answers the crucial problem of finding the domain of attraction of the existing quasi-stationary distributions. The aim of this chapter is to show how the theory exposed in the two previous chapters entail new result for this class of processes. We first look at birth and death processes with entrance boundary at infinity (i.e. S < +∞) in Section 3.2 and then to λ 0 -positive birth and death processes in Section 3.3.

Birth and death processes with entrance boundary at infinity

Theorem 3.1 tells us that a birth and death process admits a unique quasi-stationary distribution if and only if S < +∞. Building on the methods and results of [START_REF] Van Doorn | Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes[END_REF], Zhang and Zhu [START_REF] Zhang | Domain of attraction of the quasistationary distribution for birth-anddeath processes[END_REF] proved that, in this setting, the limit (1.2) holds true for all probability measures µ on N. However, the spectral theory tools used in these publications are not well suited to study total variation convergence to the quasi-stationary distribution. The following result, first proved in [START_REF] Martínez | Existence and uniqueness of a quasistationary distribution for markov processes with fast return from infinity[END_REF], completes the picture offered in [START_REF] Van Doorn | Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes[END_REF] on the quasi-limiting behaviour of birth and death processes with an entrance boundary at infinity. It shows that S < +∞ if and only if Assumption A of Chapter 1 holds true. Proposition 3.2. A birth and death process X admits a unique quasi-stationary distribution if and only if there exist two constants C , γ > 0 and a probability measure ν QSD on N such that, for any initial distribution µ on N,

P µ (X t ∈ •|t < T 0 ) -ν QSD T V ≤ C e -γt , ∀t ≥ 0. (3.4)
In this case, ν QSD is the unique quasi-stationary distribution associated to X .

In [56, Section 4.1.1], we extended this result to birth and death processes with catastrophes. The existence of quasi-stationary distributions for similar processes was already studied in [START_REF] Van Doorn | Conditions for the existence of quasi-stationary distributions for birthdeath processes with killing[END_REF]. The settings are the following. Let X c be a birth and death process on Z + with birth rates (b x ) x≥0 and death rates (d x ) x≥0 with b 0 = d 0 = 0 and b x , d x > 0 for all x ≥ 1, and allow the process to jump to 0 from any state x ≥ 1 at rate a x ≥ 0. In particular, the jump rate from 1 to 0 is

a 1 + d 1 . This process is absorbed in ∂ = 0 at time T c 0 := inf{t ≥ 0, X c t = 0}.
Proposition 3.3. Assume that sup n≥1 a n < ∞. Then S < +∞ if and only if Assumption A is satisfied.

We conclude this section with an original extension of the above proposition to birth and death processes with (possibly large) negative jumps. These processes are called skip-free to the right in [START_REF] Kijima | Quasi-limiting distributions of markov chains that are skip-free to the left in continuous time[END_REF], where the quasi-stationary behaviour of skip-free to the left processes are studied. Let X s f be a process on Z + with transition rate matrix (Q(x, y)) x,y∈Z + of X given by (1) x , if x ≥ 2 and y = x -1, d (2) x , if x ≥ 3 and y = x -2, . . .

Q(x, y) =                                0, if x = 0 or y ≥ x + 2, b x , if x ≥ 1 and y = x + 1, d x = d
d (k) x , if x ≥ k and y = x -k, . . . d (x)
x , if y = 0.

Here we assume that the families (b x ) x≥1 and (d (k) x ) x≥1,k≥1 are positive and b = 0, so that the process is irreducible on N and with only absorbing state 0, whose hitting time is denoted by

T s f 0 := inf{t ≥ 0, X s f t = 0}.
The proof of this result can be obtained as a straightforward adaptation of the arguments of [56, Section 4.1.1] and is omitted here. Proposition 3.4. Assume that S < ∞ and that sup x d (x) x < ∞. Then Assumption A is satisfied.

The case of λ 0 -positive recurrent birth and death processes

We consider now the more involved situation where ∞ is not an entrance boundary. This case, where there may exist an infinity of quasi-stationary distributions, is trickier and can be partially solved, as we will show, when the birth and death process is λ 0 -positive recurrent, as defined below.

We refer the reader to [START_REF] Hart | The λ-classification of continuous-time birth-anddeath processes[END_REF][START_REF] Van Doorn | On associated polynomials and decay rates for birth-death processes[END_REF][START_REF] Van Doorn | Birth-death processes and associated polynomials[END_REF][START_REF] Van Doorn | On the α-classification of birth-death and quasi-birth-death processes[END_REF] for several properties and examples of λ 0 -positive birth and death processes.

The quantity λ 0 in Theorem 3.1 is equal to the decay parameter (see Theorem 3.3 in [START_REF] Van Doorn | Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process[END_REF]), usual to the theory of λ 0 -positive semigroups (see for instance [START_REF] Arjas | Semi-markov processes on a general state space: α-theory and quasi-stationarity[END_REF] and references therein) and defined as follows: for all x ∈ N,

λ 0 = inf λ > 0, s.t. lim inf t →+∞ e λt P x (X t = x) > 0 .
(3.5)

Definition 3.1. The birth and death process X is said to be λ 0 -positive recurrent if the decay parameter λ 0 is positive and if, for some x ∈ N and hence for all x ∈ N, we have

lim t →∞ e λ 0 t P x (X t = x) > 0.
In the following theorem (proved in [START_REF] Villemonais | Minimal quasi-stationary distribution approximation for a birth and death process[END_REF], we assume that the process is λ 0 -positive recurrent and we exhibit a subset of the domain of attraction for the minimal quasi-stationary distribution. Theorem 3.5. Let X be a λ 0 -positive recurrent birth and death process as in Section 3.1. Then the domain of attraction of the minimal quasi-stationary distribution of X contains the set D defined by

D = µ ∈ M 1 (N), ∞ i =1 µ i Q i (λ 0 ) < +∞ .
Assume moreover that there exist C > 0, λ 1 > λ 0 and ϕ :

Z + → [1, +∞) such that ϕ(i ) goes to infinity when i → ∞ and b i (ϕ(i + 1) -ϕ(i )) + d i (ϕ(i -1) -ϕ(i )) ≤ -λ 1 ϕ(i ) +C , ∀i ≥ 1. (3.6)
Then the domain of attraction of the minimal quasi-stationary distribution of X contains the set D ϕ defined by

D ϕ = µ ∈ M 1 (N), ∞ i =1 µ i ϕ(i ) < +∞ .
As shown in [START_REF] Villemonais | Minimal quasi-stationary distribution approximation for a birth and death process[END_REF], we have D ϕ ⊂ D for all function ϕ satisfying the assumptions of Theorem 3.5. However, Q • (λ 0 ) cannot be computed explicitly but in few situations. On the contrary, in many situations, it is possible to guess a function ϕ satisfying the Lyapunov criterion of the above theorem. In fact, in this situation, the results of Chapter 2 apply, as stated in the next proposition, and hence improve the description of the convergence.

Importantly, we do not assume that the process is λ 0 -positive recurrent in the next statement. This is a key step for the generalisation developed afterwards, since in these cases the classical theory is lacking practical criteria for of λ 0 -positive recurrence (for general considerations on these properties, we refer the reader to [START_REF] Kersting | Strong ratio limit property and R-recurrence of reversible Markov chains[END_REF][START_REF] Kersting | A note on R-recurrence of Markov chains[END_REF]). Proposition 3.6. Assume that X is a birth and death process as in Section 3.1 and that there exist C > 0, λ 1 > λ 0 and ϕ :

Z + → [1, +∞) such that b i (ϕ(i + 1) -ϕ(i )) + d i (ϕ(i -1) -ϕ(i )) ≤ -λ 1 ϕ(i ) +C , ∀i ≥ 1. (3.7)
Then there exist positive constants C , γ and a probability measure ν QSD on E such that

P µ (X t ∈ • | t < τ ∂ ) -ν QSD T V ≤ C µ(ϕ) µ(η) e -γt , (3.8)
where 1 ≤ η ≤ ϕ is the right-eigenfunction of the infinitesimal generator of X with eigenvalue -λ 0 .

The above proposition appeared in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF] in a more general but weaker form, since µ(ϕ)/µ(η) was replaced by µ(ϕ)/µ(ϕ ), where ϕ : E → R + is a positive bounded function (while η is lower bounded away from 0 and may be unbounded). However the extension can be obtained in several ways: either as a consequence of the non-uniform exponential ergodic of the Q-process1 (using the fact that η is lower bounded) or extending [START_REF] Bansaye | A non-conservative Harris' ergodic theorem[END_REF]. Both methods use the existence of an eigenfunction (already established in [START_REF] Van Doorn | Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes[END_REF]), but the results may also be obtained directly by using the arguments of Chapter 7 and hence extended to more general processes, as those described in the next sections and chapters.

Finally, note that the subset of the domain of attraction provided by Proposition 3.6 is included (often strictly) in D of Theorem 3.5.

Multi-dimensional birth and death processes

We focus now on the extension of the above results to the case of multi-dimensional birth and death processes, as studied in [START_REF] Champagnat | Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes[END_REF][START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]. We focus first on the general case (see Subsection 3.4.1) and show that, in the case of Lotka-Volterra type parameters, we obtain uniform convergence to the quasi-stationary distribution with respect to the initial distribution (see Subsection 3.4.2).

General processes in discrete state space and continuous time

Let X be a non-explosive 2 Markov process in a countable state space E ∪{∂} absorbed in ∂, with infinitesimal generator L acting on nonnegative real functions f on E ∪{∂} such that y∈E ∪{∂} q x,y f (y) < ∞ for all x ∈ E as

L f (x) = y =x∈E ∪{∂} q x,y ( f (y) -f (x)), ∀x ∈ E , L f (∂) = 0, (3.9)
where q x,y is the jump rate of X from x to y = x and y∈E ∪{∂}\{x} q x,y < ∞ for all x ∈ E .

Theorem 3.7. Assume that there exists a finite subset D 0 of E such that P x (X 1 = y) > 0 for all x, y ∈ D 0 , so that the constant

λ 0 := inf λ > 0, s.t. lim inf t →+∞ e λt P x (X t = x) > 0 is finite and independent of x ∈ D 0 . If in addition there exist constants C > 0, λ 1 > λ 0 , a function ϕ : E ∪ {∂} → R + such that ϕ E ≥ 1, ϕ(∂) = 0, y∈E \{x} q x,y ϕ(y) < ∞ for all x ∈ E and such that L ϕ(x) ≤ -λ 1 ϕ(x) +C 1 x∈D 0 , ∀x ∈ E , (3.10)
then Assumption F is satisfied with L = D 0 , γ 1 = e -λ 1 , any γ 2 ∈ (e -λ 1 , e -λ 0 ) and ψ 1 = ϕ E . In addition, P ν QSD (t < τ ∂ ) = e -λ 0 t for all t ≥ 0, the function η of Proposition 2.3 satisfies P t η = e -λ 0 t η for all t ≥ 0 and y∈E \{x} q x,y η(y) < ∞ and L η(x) = -λ 0 η(x) for all x ∈ E .

Among earlier studies of the quasi-stationary behaviour of general continuous time Markov chains, we refer the reader to [START_REF] Nair | On the relationship between µ-invariant measures and quasistationary distributions for continuous-time markov chains[END_REF][START_REF] Pollett | The determination of quasistationary distributions directly from the transition rates of an absorbing markov chain[END_REF][START_REF] Pollett | Quasistationary distributions for continuous time Markov chains when absorption is not certain[END_REF][START_REF] Elmes | Further results on the relationship between µ-invariant measures and quasi-stationary distributions for absorbing continuous-time Markov chains[END_REF] and the survey [START_REF] Van Doorn | Quasi-stationary distributions for discrete-state models[END_REF] and references therein.

Example 3.1. We consider general multitype birth and death processes in continuous time, taking values in a connected (in the sense of the nearest neighbours structure of Z d ) subset E of Z d + for some d ≥ 1, with transition rates

q x,y =        b i (x) if y = x + e i , d i (x) if y = x -e i , 0
otherwise, with e i = (0, . . . , 0, 1, 0, . . . , 0) is the i th element of the canonical basis and with the convention that the process is sent instantaneously to ∂ when it jumps to a point y ∈ E according to the previous rates. To ensure irreducibility, it is sufficient (although not optimal) to assume that b i (x) > 0 and

d i (x) > 0 for all 1 ≤ i ≤ d and x ∈ E .
We show below that Theorem 3.7 applies under the assumption that .11) or that there exists δ > 1 such that

1 |x| d i =1 (d i (x) -b i (x)) ---------→ x∈E , |x|→+∞ +∞. ( 3
d i =1 (d i (x) -δ b i (x)) ---------→ x∈E , |x|→+∞ +∞. (3.12)
Let us first show that (3.11) implies that the assumptions of Theorem 3.7 are satisfied. In order to do so, we define ϕ(x) = |x| = x 1 + . . . + x d and ϕ(∂) = 0 and obtain

L ϕ(x) = d i =1 (b i (x) -d i (x)) = -ϕ(x) d i =1 (d i (x) -b i (x)) |x|
The proof is concluded by setting

D 0 = x ∈ E , s.t. d i =1 (d i (x)-b i (x)) |x| ≥ λ 0 + 1 .
Let us now show that (3.12) implies that the assumptions of Theorem 3.7 are satisfied. Setting ϕ(x) = exp〈a, x〉 for a given a ∈ (0, ∞) d and ϕ(∂) = 0, we obtain

L ϕ(x) ≤ -ϕ(x) d i =1 (1 -e -a i )d i (x) + (1 -e a i )b i (x) .
Choosing a = (ε, . . . , ε) with ε small enough, we have

lim inf x∈E , |x|→+∞ d i =1 (1 -e -a i )d i (x) + (1 -e a i )b i (x) = +∞. Taking D 0 = x ∈ E , s.t. d i =1 (1 -e -a i )d i (x) + (1 -e a i )b i (x) ≥ λ 0 + 1
allows us to conclude the proof.

Birth and death processes with Lotka-Volterra type parameters

The following result was obtained in [START_REF] Champagnat | Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes[END_REF] as part of a more general result based on Lyapunov type criterion. Since this general criterion is rather technical, we focus here on its application to Lotka-Volterra birth and death processes. A Lotka-Volterra birth and death process in dimension d ≥ 2 is a Markov process (X t ) t ∈R + on Z d + with transition rates q n,m from n = (n 1 , . . . ,

n d ) ∈ Z d + to m = n in Z d + given by q n,m =        n i (λ i + d j =1 γ i j n j ) if m = n + e i , for some i ∈ {1, . . . , d } n i (µ i + d j =1 c i j n j ) if m = n -e i , for some i ∈ {1, . . . , d } 0 otherwise.
We have q n,n-e i = 0 if n i = 0, so that the process remains in the state space Z d + . Since in addition q n,m = 0 for all n such that n i = 0 and m such that m i ≥ 1, the set ∂ = Z d + \ N d is absorbing for the process. We make the usual convention that q n,n := -q n := -

m =n q n,m .
From the biological point of view, the constant λ i > 0 is the birth rate per individual of type i ∈ {1, . . . , d }, the constant µ i > 0 is the death rate per individual of type i , c i j ≥ 0 is the rate of death of an individual of type i from competition with an individual of type j , and γ i j ≥ 0 is the rate of birth of an individual of type i from cooperation with (or predation of ) an individual of type j . In general, a Lotka-Volterra process could be explosive if some of the γ i j are positive, but the assumptions of the next theorem ensure that it is not the case and that the process is almost surely absorbed in finite time. Proposition 3.8. Consider a competitive Lotka-Volterra birth and death process (X t ) t ∈R + in Z d + as above. Assume that the matrix (c i j -γ i j ) 1≤i , j ≤d defines a positive operator on R d + in the sense that, for all (x 1 , . . . ,

x d ) ∈ R d + \ {0}, i j x i (c i j -γ i j )x j > 0.
Then the process has a unique quasi-stationary distribution ν QSD and there exist constants C , γ > 0 such that, for all probability measures µ on N d ,

P µ (X t ∈ • | t < τ ∂ ) -ν QSD T V ≤ C e -λt , ∀t ≥ 0.
Note that the existence of a quasi-stationary distribution for this kind of multi-dimensional birth and death processes can also be obtained using the theory of positive matrices, as exposed in [START_REF] Ferrari | R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata[END_REF], or using the result of the preceding section. However, neither approach provides the uniform convergence with respect to the initial distribution.

Chapter 4

One-dimensional diffusion processes

In this chapter, we consider general one-dimensional diffusion processes and expose necessary and sufficient conditions for the exponential convergence to a quasi-stationary distribution. After reminders on general one-dimensional diffusion processes in Section 4.1, we study their quasistationary distributions when there is no natural boundaries in Section 4.2. We consider the case of stochastic differential equations (SDEs) with possibly natural boundaries in Section 4. [START_REF] Alfonsi | Evolution of the Wasserstein distance between the marginals of two Markov processes[END_REF].

Convergence of conditioned one-dimensional diffusion processes has received a lot of attention in the past decades and general results have been obtained, using in general spectral theoretic arguments (self-adjoint operators, Sturm-Liouville theory, intrinsic ultra-contractivity), which proved to be extremely powerful (see for instance [START_REF] Steinsaltz | Quasistationary distributions for one-dimensional diffusions with killing[END_REF][START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF][START_REF] Littin | Uniqueness of quasistationary distributions and discrete spectra when ∞ is an entrance boundary and 0 is singular[END_REF][START_REF] Kolb | Quasilimiting behavior for one-dimensional diffusions with killing[END_REF][START_REF] Hening | Quasistationary distributions for one-dimensional diffusions with singular boundary points[END_REF][START_REF] Miura | Ultracontractivity for Markov semigroups and quasi-stationary distributions[END_REF]). Our main contribution to the theory of quasi-stationary distributions for one-dimensional diffusion processes, concerns the question of speed of convergence with respect to the initial distribution and the relaxation of the regularity of the coefficients. Moreover, our original approach can be easily adapted to other models, such as diffusion processes with jumps and time inhomogeneous diffusion processes (see Section 4.4 where several examples are provided).

Some reminders on general diffusion processes

In this section, we recall the definition and first properties of a general one-dimensional diffusion process

(Y t , t ≥ 0) on (a, b), -∞ ≤ a < b ≤ +∞, up to its exit time of (a, b) defined by τ ∂ = inf{t ≥ 0, lim sup s→t Y s = b or lim inf s→t Y s = a}. The process Y is sent to a cemetery point ∂ ∉ (a, b) for all t ≥ τ ∂ . Its distribution given Y 0 = x ∈ (a,
b) will be denoted P x . We refer the reader to [START_REF] Dynkin | Markov processes. Vols. I, II[END_REF][START_REF] Itô | Diffusion processes and their sample paths[END_REF][START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF][START_REF] Freedman | Brownian motion and diffusion[END_REF][START_REF] Cetin | An introduction to markov processes and their applications in mathematical economics[END_REF] for additional developments and proofs of the following properties.

A stochastic process (Y t , t ≥ 0) on (a, b) is called a diffusion process if it has a.s. continuous paths in (a, b) up to time τ ∂ , satisfies the strong Markov property and is regular. By regular, we mean that for all x, y ∈ (a, b), P x (T y < ∞) > 0, where T y is the first hitting time of y by the process Y . This notion is closely related to the concept of irreducibility of the set (a, b), since this open interval cannot be decomposed into strict subsets from which the process Y cannot evade.

To such a process, one can associate a continuous and strictly increasing function s on (a, b) such that, for all l < x < r ∈ (a, b),

P x (T l < T r ) = s(x) -s(l ) s(r ) -s(l ) .
The function s is called a scale function of Y and is unique up to affine transformation. If s(x) = x is a scale function for Y , then we say that Y is on natural scale. It can also be proved that, to any one dimensional diffusion process Y , one can associate a unique locally finite positive measure m Y (d x) on (0, +∞), called the speed measure of Y , which gives positive mass to any open subset of (0, +∞) and such that, for all l < x < r ∈ (a, b) and for all measurable functions f :

(a, b) → R + E x T l ∧T r 0 f (Y s ) d s = (l ,r ) G l ,r (x, y) f (y) m Y (d y), ( 4.1)
where G denotes the green function The meaning of the speed measure m Y is somewhat counter-intuitive given its name: the process slows down in subsets with higher speed measure. For instance, the sticky Brownian motion with parameter θ > 0, which is the general diffusion on natural scale on (-∞, +∞) and speed measure Λ + θδ 0 stays longer in 0 if θ is larger (where Λ is the Lebesgue measure on R). It is possible to give a precise meaning of this fact by looking as the construction of diffusion processes on natural scale as time changed Brownian motion (since this considerations are further away from the presentation at hand, we refer the reader to the original paper [START_REF] Champagnat | Uniform convergence of conditional distributions for absorbed one-dimensional diffusions[END_REF] and references therein).

G l ,r (x, y) = 2 (s(x) ∧ s(y) -s(l ))(s(r ) -s(x) ∨ s(y)) s(r ) -s(l ) .
It is also well known that one can classify the boundaries a and b of the state space as exit, regular, natural or entrance. Informally, they respectively mean (exit) that the process can reach the boundary and cannot come back to (a, b); (regular) that the process can reach the boundary and may be constructed so that it comes back to (a, b), depending on the boundary conditions; (natural) that the process cannot reach the boundary and that, when it starts near the boundary, it stays a long time near the boundary; (entrance) that the process cannot reach the boundary and that, when it starts near the boundary, it can reach any compact subsets of (a, b) in finite time. A boundary is said to be reachable if it is exit or regular, and non-reachable otherwise.

These definitions correspond to the following parameters (see for instance Section 5.11 of Itô's book [START_REF] Itō | Essentials of stochastic processes[END_REF]). Set, for some fixed c > 0,

I a = a<y<x<c m Y (d x) s(d y), I I a = a<y<x<c m Y (d y) s(d x), I b = c<x<y<b m Y (d x) s(d y), I I b = c<x<y<b m Y (d y) s(d x).
Then, for γ = a or b, we have

• γ is an exit boundary if I γ < ∞ and I I γ = ∞, • γ is a regular boundary if I γ < ∞ and I I γ < ∞, • γ is a natural boundary if I γ = ∞ and I I γ = ∞,
• γ is an entrance boundary if I γ = ∞ and I I γ < ∞.

GENERAL ONE-DIMENSIONAL DIFFUSION PROCESSES WITHOUT NATURAL BOUNDARIES35

Also one checks that, if γ ∈ {a, b} is a reachable boundary, then s(γ) ∈ (-∞, +∞) and that, if γ ∈ {a, b} is an entrance boundary, then s(γ) ∈ {-∞, +∞}.

Remark 4.1. The process X t = s(Y t ) is a diffusion on natural scale on (s(a), s(b)), whose speed measure is given by

m(d x) = m Y • s -1 (d x), ( 4.2)
where m Y •s -1 is the push-forward measure of m Y through the function s (this follows for instance from [232, Thm. VII.3.6]). Moreover, the boundaries s(a) and s(b) for X have respectively the same nature as a and b for Y . Since it is straightforward that the diffusion processes X and Y have the same quasi-stationary behaviour, we restricted most of the original article to diffusion processes X on natural scale.In the present manuscript, we present theses results in the general situation of diffusion processes that may not be on natural scale.

The case of solutions to stochastic differential equations.

One of the most widespread case of diffusion processes in the literature concerns the solutions to stochastic differential equations. Let Y be the solution to the general SDE on (a, b) with

-∞ < a < b ≤ +∞ d Y t = σ(Y t )d B t + β(Y t )d t , Y 0 = x ∈ (a, b), (4.3)
where

(1 + |β|)/σ 2 ∈ L 1 loc ((a, b)) (which

General one-dimensional diffusion processes without natural boundaries

We first consider the situation where none of the boundary is natural. This case is already well understood in the literature for solution to SDEs with regular coefficients. For instance, combining the works of [START_REF] Steinsaltz | Quasistationary distributions for one-dimensional diffusions with killing[END_REF][START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF][START_REF] Littin | Uniqueness of quasistationary distributions and discrete spectra when ∞ is an entrance boundary and 0 is singular[END_REF][START_REF] Kolb | Quasilimiting behavior for one-dimensional diffusions with killing[END_REF][START_REF] Hening | Quasistationary distributions for one-dimensional diffusions with singular boundary points[END_REF], one deduces that there exists a unique quasi-stationary distribution for solutions to SDE's on (0, +∞) with smooth coefficients, 0 reachable and ∞ entrance. These results also imply that the convergence toward a quasi-stationary distribution hold true pointwisely, for any initial distribution on (0, +∞) (some of the above cited works require the initial measures to be compactly supported, but the extension to non-compactly supported measures is not difficult). Of course, they can be extended without to much difficulty to the situation where (0, +∞) is replaced by an arbitrary non-empty open set (a, b). For general one-dimensional diffusion processes, Miura [START_REF] Miura | Ultracontractivity for Markov semigroups and quasi-stationary distributions[END_REF] used the abstract and powerful analytical results on the density of diffusion processes proved by Mastumoto in [START_REF] Matsumoto | Coalescing stochastic flows on the real line[END_REF] in order to prove that there exists a unique quasi-stationary distribution attracting all initial distribution. The conditions of Matsumoto are satisfied in most practical situations. Our two main aims are thus 1/ to extend the existing results by covering all the edge cases not covered by [START_REF] Matsumoto | Coalescing stochastic flows on the real line[END_REF] and 2/ to precise the convergence results, using the results of Chapters 1 and 2.

We begin with the following result, which answers to the first aim above. It can be obtained using the approach of Section 4 in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]. Proposition 4.1. Assume that a is reachable or entrance, and that b is reachable or entrance. Then Condition F of Chapter 2 is satisfied with ψ 1 = 1.

The main drawback of this result is that it does not imply uniform convergence with respect to the original distribution, while this is a very common situation as we explain now.

The following result was obtained in [START_REF] Champagnat | Uniform convergence of conditional distributions for absorbed one-dimensional diffusions[END_REF] for diffusion processes on natural scale evolving in (0, +∞) and absorbed at 0. Here, we translate this result to the general case of diffusion process that may not be on natural scale. The extension of this result to one-dimensional diffusions with killing is absorbed in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution for absorbed one-dimensional diffusions with killing[END_REF]. In the following result, t c > 0 is an arbitrarily fixed time.

Theorem 4.2. We have equivalence between (i) The boundary a is reachable and

P(t c < τ ∂ | Y 0 = y) ≤ As(y), ∀y ∈ (a, b),
or the boundary a is an entrance boundary. The boundary b is reachable and Of course, in order to apply this result in practical situations, one needs to be able to check whether a given general diffusion process satisfies condition (i) above. This is the subject of Propositions 4.3. Its proof, detailed in [START_REF] Champagnat | Uniform convergence of conditional distributions for absorbed one-dimensional diffusions[END_REF] only uses elementary tools. In the next result, c is an arbitrary fixed point in (a, b). for some constants C > 0 and ρ > 0. Then, for all t > 0, there exists A t < ∞ such that

P(t c < τ ∂ | Y 0 = y) ≤ A(s(b) -s(y)), ∀y ∈ (a, b),
P x (t < τ ∂ ) ≤ A t (s(x) -s(a)), ∀x > 0.
We immediately deduce the next corollary.

Corollary 4.4. Assume that a is regular or entrance, and that b is regular or entrance. Then the conditions of Theorem 4.2 are satisfied.

Proposition 4.3 also covers many situations where the boundary is exit, although it does not cover all cases. To understand the generality of this criterion, let us consider the case of stochastic diffusion processes without drift terms. One easily checks that a diffusion process X evolving in (0, +∞) and solution to the SDE

d X t = σ(X t )d B t
satisfies (4.5) as soon as σ(x) ≥ C x 1-ε in the neighbourhood of 0 for some constants ε > 0 and C > 0. We recall that, if σ(x) ≤ C x in a neighbourhood of 0, the boundary 0 is not reachable. As a consequence, this simple criterion covers most practical situations. However, it does not cover all cases: for instance, if σ(x) = x log x 1/2-ε , then 0 is a reachable (exit) boundary, but (4.5) is not satisfied. This case is covered by a more involved criterion described in the original article [START_REF] Champagnat | Uniform convergence of conditional distributions for absorbed one-dimensional diffusions[END_REF] (which, however, does not cover all reachable boundary cases either). This leads to the following natural open question (which admits a positive answer for birth and death processes).

Open Question. Is it true that any diffusion process on (a, b) with a and b either entrance or reachable satisfies Condition A of Chapter 1? This open question can also be translated into an open problem from strict martingale theory, as we detail in the original article [START_REF] Champagnat | Uniform convergence of conditional distributions for absorbed one-dimensional diffusions[END_REF].

Construction of a diffusion process with prescribed quasi-stationary distribution

Our goal here is to give a sufficient condition on a given positive measure α Y ensuring that it is the unique quasi-stationary distribution of a diffusion process on (a, b) with identified speed measure and scale function. The problem of finding diffusion processes with prescribed quasistationary distributions have been independently studied by other authors since the publication of the original article. New advances with original applications to Monte Carlo methods can be found for instance in [START_REF] Pollock | The Scalable Langevin Exact Algorithm: Bayesian Inference for Big Data[END_REF][START_REF] Wang | Theoretical Properties of Quasistationary Monte Carlo Methods[END_REF]. Proposition 4.5. Fix -∞ ≤ a < b ≤ +∞, λ 0 > 0, a continuous strictly increasing function s on (a, b) and a probability measure α on (a, b). Assume that s and 1 t(x)∧1 α(d x) are respectively the scale function and the speed measure of a regular diffusion process X on (0, +∞) satisfying the conditions of Theorem 4.2 with a or b reachable, where

t(x) =        s(x) -s(a)
if a is reachable and b is entrance for X , s(b)s(x) if a is entrance and b is reachable for X , (s(x)s(a))(s(b)s(x)) if both a and b are reachable for X .

Then the diffusion process Y with scale function s and speed measure

m Y (d x) = α(d x) λ 0 ∞ 0 G a,b (x, y)α(d y) , ∀x ∈ (0, ∞)
satisfies the conditions of Theorem 4.2, its unique quasi-stationary distribution is α and it satisfies

P α (t < τ ∂ ) = e -λ 0 t , ∀t ≥ 0.

One dimensional SDEs with possibly natural boundaries

In this section, we consider the case of one-dimensional diffusion processes solution to a SDE and admitting a natural boundary. The case of general one-dimensional diffusion processes can be handled using a similar approach, but the construction of Lyapunov functions compatible with their infinitesimal generator is more tricky [START_REF] Itô | Diffusion processes and their sample paths[END_REF]. The question of existence and convergence to a quasi-stationary distribution for such diffusion processes is largely solved when the coefficients are regular (see in particular [START_REF] Collet | Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption[END_REF][START_REF] Martínez | Rates of decay and h-processes for one dimensional diffusions conditioned on non-absorption[END_REF][START_REF] Martínez | Classification of killed one-dimensional diffusions[END_REF][START_REF] Steinsaltz | Quasistationary distributions for one-dimensional diffusions with killing[END_REF][START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF][START_REF] Kolb | Quasilimiting behavior for one-dimensional diffusions with killing[END_REF][START_REF] Hening | Quasistationary distributions for one-dimensional diffusions with singular boundary points[END_REF]). However, the question of finding a non-trivial subset of the domain of attraction of the quasi-stationary distribution is less understood (although some partial answers can be found in [START_REF] Collet | Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption[END_REF]). Our main goal is thus to extend those result to less regular coefficients and to give a criterion ensuring that Assumption F of Chapter 2 holds true, providing precisions on the domain of attraction and the rate of convergence. Note that we cannot recover the sub-exponential rate of convergence of [START_REF] Collet | Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption[END_REF].

Let X be the solution in (a, b), where -∞ ≤ a < b ≤ +∞, to the SDE

d X t = σ(X t ) d B t + β(X t ) d t , X 0 ∈ (a, b),
where σ : (a, b) → (0, +∞) and β : (a, b) → R are measurable functions such that (1 + |β|)/σ 2 is locally integrable on (a, b). We assume that the process is sent to a cemetery point ∂ when it exits the set (a, b) and that it is subject to an additional killing rate κ : (a, b) → R + which is measurable and locally integrable w.r.t. Lebesgue's measure. This assumption implies that the killed process is regular.

We define λ 0 as for some arbitrary c ∈ (a, b). We recall that s is the scale function of X (unique up to an affine transformation).

λ 0 := inf λ > 0, s.t. lim inf t →+∞ e λt P x (X t ∈ [c, d ]) > 0 (4.6) for some x ∈ [c, d ] ⊂ (a
Theorem 4.6. Assume that one among the following conditions (i), (ii) or (iii) holds true:

(i) a and b are reachable boundaries and ϕ = 1;

(ii) a is reachable and there exist

λ 1 > λ 0 , a C 2 ((a, b)) function ϕ : (a, b) → [1, +∞) and x 1 ∈ (a, b) such that, for all x ∈ [x 1 , b), σ(x) 2 2 ϕ (x) + β(x)ϕ (x) -κ(x)ϕ(x) ≤ -λ 1 ϕ(x); (4.7) (iii) there exist λ 1 > λ 0 , a C 2 ((a, b)) function ϕ : (a, b) → [1, +∞) and x 0 < x 1 ∈ (a, b) such that (4.7) holds true for all x ∈ (a, x 0 ) ∪ (x 1 , b).
Then X admits a quasi-stationary distribution ν QSD which satisfies ν QSD (ϕ 1/p ) < +∞ for all p > 1.

Moreover, for all p ∈ (1, λ 1 /λ 0 ), there exist a constant α p ∈ (0, 1), a constant C p and a positive function ϕ 2,p : (a, b) → (0, +∞) uniformly bounded away from 0 on compact subsets of (a, b) such that, for all

µ ∈ M (ϕ 1 /p 1 ), P µ (X t ∈ • | t < τ ∂ ) -ν QSD T V ≤ C p α t p µ(ϕ 1/p ) µ(ϕ 2,p ) , ∀t ∈ [0, +∞).
In particular, ν QSD is the only quasi-stationary distribution of X which satisfies ν QSD (ϕ 1/p ) < +∞ for at least one value of p ∈ (1, λ 1 /λ 0 ).

In order to apply this result in practice, one needs to find computable estimates for λ 0 and candidates for ϕ. One may for instance use the sharp bounds for the first eigenvalue of the (Dirichlet) infinitesimal generator of (X t ) t ∈R + obtained in a L 2 (symmetric) setting in [START_REF] Pinsky | Explicit and almost explicit spectral calculations for diffusion operators[END_REF][START_REF] Wang | First eigenvalue of one-dimensional diffusion processes[END_REF][START_REF] Wang | Sharp bounds for the first eigenvalue of symmetric markov processes and their applications[END_REF], as observed in [START_REF] Kolb | Quasilimiting behavior for one-dimensional diffusions with killing[END_REF]. We propose also in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF] two different upper bounds for λ 0 which follow from the characterisation (4.6) of the eigenvalue λ 0 and Dynkin's formula.

Examples

Example 4.1. In the settings of the last section, assume that (a, b) = (0, +∞), κ is locally bounded and that X is solution to the SDE in (a, b)

d X t = X t d B t -X t d t .
Then 0 is reachable for X and since

σ(x) 2 δ(x) 2 8s(x) 2 -----→ x→+∞ +∞,
we deduce from [58, Proposition 4.7] and Theorem 4.6 that X admits a quasi-stationary distribution ν QSD and, for all p ≥ 1, there exist positive constants C p , γ p and a positive function ϕ 2,p on (0, +∞) such that

P µ (X t ∈ • | t < τ ∂ ) -ν QSD T V ≤ C p (0,+∞) exp(x/p) µ(d x) µ(ϕ 2,p ) e -γ p t ,
for all probability measures µ on (0, +∞). In particular, one deduces that the domain of attraction ν QSD contains any initial distribution µ admitting a finite exponential moment. Note that, in the case where κ ≡ 0, the process X is a continuous state branching process (Feller diffusion), for which quasi-stationarity was already studied (see [START_REF] Lambert | Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct[END_REF] and the references therein).

Example 4.2. Assume that (a, b) = R, that β ≡ 0 and σ is bounded measurable on R. Assume also that the absorption of X is due to the killing rate κ(x) = κ 0 1 -1 1+|x| for some constant κ 0 > 0. We deduce from [58, Proposition 4.7] that

λ 0 ≤ π 2 σ 2 ∞ 8b 2 + κ 0 1 - 1 1 + b ≤ κ 0 1 - 1 1 + 2b
for b large enough. Moreover, choosing ϕ = 1 and x 0 = -3b, x 1 = 3b, one deduces that, for all

x ∈ [-x 1 , x 1 ], σ(x) 2 2 ϕ (x) -κ(x)ϕ(x) ≤ -κ 0 1 - 1 1 + 3b ϕ(x).
Hence Theorem 4.6 implies that there exists a unique quasi-stationary distribution ν QSD for X and that it attracts all probability measures µ on R. for some constant κ 0 > π 2 + 3. This corresponds to a SDE d X t = d B t + ∇U (X t )d t where the potential U (x) = sin xx cos x has infinitely many wells with arbitrarily large depths, meaning that the process X without killing has a tendency to be "trapped" away from zero for large initial conditions. Nevertheless, thanks to the killing, we are able to prove convergence to a unique quasi-stationary distribution. Indeed, using [58, Proposition 4.7], we obtain

λ 0 ≤ sup x∈(0,1) π 2 2 + sin x + x cos x + x 2 sin 2 x 2 + κ 0 1 - 1 1 + x ≤ π 2 2 + 3 2 + κ 0 /2.
Moreover, 0 is a reachable boundary for X and, taking ϕ = 1, one has, for all x 1 > 0 and all x > x 1 ,

σ(x) 2 2 ϕ (x) + b(x)ϕ (x) -κ(x)ϕ(x) ≤ -κ 0 1 - 1 1 + x 1 ϕ(x)
Hence, since we assumed that κ 0 > π 2 + 3, one deduces that there exists a unique quasi-stationary distribution ν QSD for X and that it attracts all probability measures µ on (0, +∞).

Example 4.4 (Sticky Brownian motion absorbed at -1 and +1). We recall that a diffusion process on R with speed measure Λ+δ 0 is called a sticky Brownian motion [START_REF] Itô | Diffusion processes and their sample paths[END_REF][START_REF] Amir | Sticky brownian motion as the strong limit of a sequence of random walks[END_REF], where Λ is the Lebesgue measure on R. It is called "sticky" because it slows down at 0, giving the impression that the trajectory of the process is glued to 0. We consider here a diffusion process X evolving as a sticky Brownian motion in (-1, 1) and absorbed at -1 and 1. This means that X is a diffusion on natural scale with speed measure m(d x) = Λ(d x) + δ 0 (d x) on (-1, 1), absorbed at -1 and 1.

In this case, the conditions of Theorem 4.2 are satisfied since both boundaries -1 and 1 are regular for X (see Corollary 4.4). Moreover, since the unique quasi-stationary distribution ν QSD of X satisfies

d ν QSD d m (x) = λ 0 1 -1 (x ∧ y + 1)(1 -x ∨ y) ν QSD (d y), ∀x ∈ (-1, 1), careful computations show that ν QSD (d x) = γ * 2 sin γ * (1 + x) ∧ (1 -x) m(d x),
where γ * is the unique solution in (0, π] of cotan γ = γ/2.

Example 4.5 (Diffusion process with multiple sticking points). We consider here a diffusion process X on (0, ∞) on natural scale and with absorption at 0, where ∞ is an entrance boundary and which "sticks" at the points a 1 , a 2 , . . ., where (a i ) i ≥1 is decreasing, converges to 0 and a 1 < 1. Denoting by m the speed measure of this diffusion, this means that ∞ 1 y m(d y) < ∞ and

m (0,1) = Λ (0,1) + i ≥1 δ a i ,
where Λ is the Lebesgue measure on R.

Assuming that there exist constants C , ρ > 0 such that for all j ≥ 1,

i ≥ j a i ≤ C a ρ j , (4.8)
then for all x ∈ (0, 1), defining i 0 := inf{ j ≥ 1 : a j < x},

(0,x) y m(d y) = x 2 2 + i ≥i 0 a i ≤ x 2 2 +C a ρ i 0 ≤ x 2 2 +C x ρ ,
and we can apply Proposition 4.3. For example, the choice a i = i -1

1-ρ , for all i ≥ 1, satisfies (4.8).

Example 4.6 (A simple model with jumps). The following example uses a simple extension of the results presented above. We consider a diffusion process (X t ) t ∈R + on (0, ∞) with speed measure m, on natural scale, and satisfying the conditions of Theorem 4.2. Let us denote by L the infinitesimal generator of X . We consider the Markov process ( X t ) t ∈R + with infinitesimal generator

L f (x) = L f (x) + ( f (x + 1) -f (x))1 x≥1 ,
for all f in the domain of L . In other words, we consider a càdlàg process following a diffusion process on natural scale with speed measure m between jump times, which occur at the jump times of an independent Poisson process (N t ) t ∈R + of rate 1, with jump size +1 if the process is above 1, and 0 otherwise. We denote by τ ∂ its first hitting time of 0.

In this situation, we show in [START_REF] Champagnat | Uniform convergence of conditional distributions for absorbed one-dimensional diffusions[END_REF] that X admits a unique quasi-stationary distribution ν QSD on (0, ∞) and that there exist two constants C , γ > 0 such that, for all initial distribution µ on (0, ∞),

P µ ( X t ∈ • | t < τ ∂ ) -ν QSD (•) T V ≤ C e -γt , ∀t ≥ 0. (4.9)
Example 4.7 (One-dimensional diffusion processes with time-inhomogeneous coefficients). We consider a time inhomogeneous diffusion process X on (0, +∞) stopped when it exits (0, +∞) at time T X 0 = inf{t ≥ 0, X t -= 0}, which is assumed finite almost surely, and solution, for all s ≥ 0, on [s, T X 0 ) to

d X t = σ(t , X t )d B t , X s ∈ (0, +∞), (4.10)
where B is a standard one-dimensional Brownian motion and σ is a measurable function on (0, +∞)× (0, +∞) to (0, +∞). We assume that

σ * (x) ≤ σ(t , x) ≤ σ * (x), (4.11)
for some measurable functions σ * and σ * from (0, +∞) to (0, +∞) satisfying (0,+∞)

x d x σ * (x) 2 < ∞ and (a,b) d x σ * (x) 2 > 0, ∀0 < a < b < ∞.
We also assume that σ * (x) ≥ C x log 1+ε 2 1

x for some constants C > 0 and ε > 0 in a neighbourhood of the boundary 0.

Under the above assumptions, for all probability measures µ 1 and µ 2 on E , and for all t ≥ 0, we show in [START_REF] Champagnat | Uniform convergence of penalized timeinhomogeneous markov processes[END_REF] using coupling with the diffusion studied in the previous sections, that

P µ 1 (X t ∈ • | t < τ ∂ ) -P µ 2 (X t ∈ • | t < τ ∂ ) ≤ C e -γt
for some positive constants C , γ > 0.

A more complete result is provided in [START_REF] Champagnat | Uniform convergence of penalized timeinhomogeneous markov processes[END_REF], where we also state a general criterion for timeinhomogeneous semi-groups and provide applications to birth and death processes in random environments. Note that the classical (without absorption) version of these results (e.g. when both 0 and ∞ are an entrance boundary for σ * and σ * ), can be obtained using existing timeinhomogeneous Doblin type criteria, known to apply since decades [START_REF] Cohn | On a paper by Doeblin on nonhomogeneous Markov chains[END_REF].

Chapter 5

Multi-dimensional diffusion processes

In this chapter, we consider the application of the results of Chapters 1 and 2 to diffusion processes absorbed at the boundary of a domain. We give a general criterion in Section 5.1 and apply it to uniformly elliptic diffusions in Section 5.2 and to an example with vanishing diffusion coefficient at the boundary of the domain in Section 5.3. Of course the study of the quasi-stationary behaviour of multi-dimensional diffusion processes is not new and has been largely understood in many situation, see for instance [START_REF] Pinsky | On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes[END_REF][START_REF] Deblassie | The lifetime of conditioned brownian motion in certain lipschitz domains[END_REF][START_REF] Gong | Killed diffusions and their conditioning[END_REF][START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF][START_REF] Del Moral | Exponential mixing properties for time inhomogeneous diffusion processes with killing[END_REF][START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF]. Our main contribution is to strongly reduce regularity requirements of these works (both on the boundary of the domain and on the coefficients), to prove exponential convergence in total variation norm and to provide a non-trivial subset of the domain of attraction.

A general criterion

The results of this section and of the following one first appeared in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF].

We consider a diffusion process X on a connected, open domain D ⊂ R d for some d ≥ 1, solution to the SDE

d X t = b(X t )d t + σ(X t )d B t , (5.1)
where B is a standard, r -dimensional Brownian motion and b : D → R d and σ : D → R d ×r are locally Hölder functions, such that σ is locally uniformly elliptic in D, i.e.

∀K ⊂ D compact, inf

x∈K inf s∈R d \{0} s * σ(x)σ * (x)s |s| 2 > 0,
where | • | is the standard Euclidean norm on R d . We assume that the process is immediately absorbed 1 at some cemetery point ∂ ∈ D at its first exit time of D, denoted τ ∂ . The existence and basic properties of this process are detailed in Subsection 12.1 of the original article [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]. We can observe that, for all k ≥ 1, defining the compact set 1 The study of diffusion processes with additional soft killing can also be derived from the same lines, see for instance

K k = x ∈ D : |x| ≤ k and d (x, D c ) ≥ 1/k ,
τ ∂ = sup k≥1 τ K c k . (5.2)
We introduce the differential operator associated to the SDE (5.1), related to the infinitesimal generator of the process X : for all f ∈ C 2 (D), we define for all x ∈ D

L f (x) := d i =1 b i (x) ∂ f ∂x i (x) + 1 2 d i , j =1 r k=1 σ i k (x)σ j k (x) ∂ 2 f ∂x i ∂x j (x). (5.3)
We also define the constant

λ 0 := inf λ > 0, s.t. lim inf t →+∞ e λt P x (X t ∈ B ) > 0 (5.4)
for some x ∈ D and some open ball B such that B ⊂ D. It is standard to prove using Harnack inequalities that, under the previous assumptions, λ 0 < +∞ and its value is independent of the choice of x ∈ D and of the non-empty, open ball B such that B ⊂ D.

We obtain the following result. Assume also that there exists a time s 1 > 0 such that sup

x∈D 0 P x (s 1 < τ K k ∧ τ ∂ ) ----→ k→∞ 0. (5.6)
Then X admits a quasi-stationary distribution ν QSD which satisfies ν QSD (ϕ 1/p ) < +∞ for all p > 1. Moreover, for all p ∈ (1, λ 1 /λ 0 ), there exist a constant α p ∈ (0, 1), a constant C p and a positive function ϕ 2,p : D → (0, +∞) uniformly bounded away from 0 on compact subsets of D such that, for all probability measures µ on E satisfying µ(ϕ 1/p ) < ∞,

P µ (X t ∈ • | t < τ ∂ ) -ν QSD T V ≤ C p α t p µ(ϕ 1/p ) µ(ϕ 2,p ) , ∀t ∈ [0, +∞).
In particular, ν QSD is the only quasi-stationary distribution of X which satisfies ν QSD (ϕ 1/p ) < +∞ for at least one value of p ∈ (1, λ 1 /λ 0 ).

Remark 5.1. The assumptions of Theorem 5.1 do not ensure the non-explosion of the Markov process X . In the event of an explosion, the absorption time τ ∂ is equal to the explosion time.

The last result has other consequences of interest, gathered in the next corollary.

Corollary 5.2. Under the assumptions of Theorem 5.1, the infimum defining the constant λ 0 in (5.4) is actually a minimum and it satisfies P ν QSD (t < τ ∂ ) = e -λ 0 t for all t ≥ 0. In addition, the function η of Theorem 2.4 satisfies P t η = e -λ 0 t η for all t ≥ 0. In particular, η belongs to the domain of the infinitesimal generator of the semigroup of the process X defined as acting on the Banach space L ∞ (ϕ 1 ), and it is an eigenfunction for the eigenvalue -λ 0 . In addition, η ∈ C 2 (D) and L η(x) = -λ 0 η(x) for all x ∈ D.

It is natural to ask if ϕ 2,p may be replaced by η in the conclusion of Theorem 5.1. It is not too difficult to see that this is the case if η is bounded (this comes down to the fact that one can take ϕ 2 = η in Assumption E when η is bounded). Actually, it is also the case if η is not bounded, in which case one needs to apply the above criteria to the ϕ 1,p -transform of the diffusion process X , following the same approach as in Chapter 7.

Application to uniformly elliptic diffusion processes

We consider the case where σ can be extended as a locally uniformly elliptic matrix to R d . We emphasise that, contrary to previous results on the existence of quasi-stationary distributions for diffusions in a domain (see [START_REF] Pinsky | On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes[END_REF][START_REF] Deblassie | The lifetime of conditioned brownian motion in certain lipschitz domains[END_REF][START_REF] Gong | Killed diffusions and their conditioning[END_REF][START_REF] Knobloch | Uniform conditional ergodicity and intrinsic ultracontractivity[END_REF][START_REF] Del Moral | Exponential mixing properties for time inhomogeneous diffusion processes with killing[END_REF][START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF]), no regularity on the boundary of D is required.

Corollary 5.3. Let D be an open connected subset

of R d , d ≥ 1. Let X be solution to the SDE d X t = b(X t )d t + σ(X t )d B t , t < τ ∂ ,
> 0, λ 1 > λ 0 , a C 2 (D) function ϕ : D → [1, +∞) and a bounded subset D 0 ⊂ D closed in D such that L ϕ(x) ≤ -λ 1 ϕ(x) +C 1 x∈D 0 , ∀x ∈ D.
(5.7)

Then the process X absorbed at the boundary of D satisfies the assumptions of Theorem 5.1.

Again, we do not assume that ϕ is a norm-like function, hence the process X may be explosive (see Remark 5.1). We give now three examples of application.

Example 5.1. Assume that D is bounded. Then, one can choose D 0 = D and ϕ 1 = 1 in Corollary 5.3. Hence the process X has a unique quasi-stationary distribution ν QSD whose domain of attraction is the whole set of probability measures on D. Since η is bounded in this case, one can actually prove that

P µ (X t ∈ • | t < τ ∂ ) -ν QSD T V ≤ C µ(η) α t , ∀t ∈ [0, +∞),
This completely solve the question of existence, uniqueness and convergence to a quasi-stationary distribution for such diffusion processes. However, finding the correct speed of convergence remains an open problem, since the factor 1/µ(η) is not optimal. Indeed, as we will see in Section 5.4 under regularity assumptions on the boundary of D, the convergence can be proved uniform in µ. where 〈•, •〉 is the standard Euclidean product in R d and λ 0 is defined in (5.4). Then the process X absorbed at the boundary of D satisfies the assumptions of Theorem 5.1.

Example 5.2. Assume that D ⊂ R d + is open connected and that d X t = b(X t )d t + σ(X t )d B t in D,
To apply this criterion, one can use a priori bounds on λ 0 . If the best available bound is λ 0 < +∞, one may still apply the above criterion using that (5.

Feller diffusion with competition

We study now the case of a diffusion matrix σ that cannot be extended out of D as a locally uniformly elliptic matrix. This example deals with Feller diffusions with competition and is motivated by models of population dynamics with d species in interaction, where absorption corresponds to the extinction of one of the populations [START_REF] Cattiaux | Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned to non-extinction[END_REF]. Assume that D = (0, ∞) d and

d X i t = γ i X i t d B i t + X i t b i (X t ) d t ,
where γ i > 0 for all 1 ≤ i ≤ d , B 1 , . . . , B d are independent standard Brownian motions and b i are locally Hölder in (0, ∞) d and locally bounded in R d + .

Proposition 5.4. Assume that there exist constants c 0 , c 1 > 0 such that

d i =1 x i b i (x) γ i ≤ c 0 -c 1 |x|, ∀x ∈ (0, ∞) d .
Then the process X absorbed at the boundary of D satisfies the assumptions of Theorem 5.1.

In order to prove Proposition 5.4, one shows that the assumptions of Theorem 5.1 hold true with ϕ(x) = exp(c(x 1 /γ 1 + . . . + x n /γ n )), where c = c 1 min i γ i / d . See Section 4.3 in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF] for details.

Compared to the existing literature on multi-dimensional Feller diffusions (see [START_REF] Cattiaux | Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned to non-extinction[END_REF]), this result covers cases where the process does not come down from infinity, e.g. b i (x) = r i -d j =1 c i j x j 1+x j , for some positive constants r i and c i j such that r i < c i i for all 1 ≤ i ≤ d . Also, the case considered in [START_REF] Cattiaux | Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned to non-extinction[END_REF] is restricted to (transformations of) Kolmogorov diffusions where the drift derives from a potential (b = ∇V ), which allows the authors to use a spectral theoretic approach as in the one-dimensional case [START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF]. In the case of logistic Feller diffusions, where b i (x) = r i -d j =1 c i j x j , this requires the additional assumption that the matrix (c i j γ j ) 1≤i , j ≤d is symmetric. While our results on existence and convergence to quasi-stationary distributions are more general than those of [START_REF] Cattiaux | Competitive or weak cooperative stochastic Lotka-Volterra systems conditioned to non-extinction[END_REF], we do not recover the fine results they obtain on the spectrum of the process, such as its discreteness.

Multidimensional Feller diffusions absorbed when one of the coordinates hits 0. A competitive Lotka-Volterra Feller diffusion process in dimension

d ≥ 2 is a Markov process (X t ) t ∈R + on R d + , where X t = (X 1 t , . . . , X d t )
, is solution to the stochastic differential equation

d X i t = γ i X i t d B i t + X i t r i - d j =1 c i j X j t d t , ∀i ∈ {1, . . . , d }, ( 5.9)
where B 1 , . . . , B d are independent standard Brownian motions. The Brownian terms and the linear drift terms correspond to classical Feller diffusions, and the quadratic drift terms correspond to Lotka-Volterra interactions between coordinates of the process. The variances per individual γ i is a positive number, and the growth rates per individual r i can be any real number, for all 1 ≤ i ≤ d . The competition parameters c i j are assumed nonnegative for all 1 ≤ i , j ≤ d , which corresponds to competitive Lotka-Volterra interaction. While this model enters the settings of Proposition 5.4, Theorem 5.1 cannot provide uniform convergence toward a quasi-stationary distribution. However, this can be done for competitive Lotka-Volterra diffusion processes using the criterion of Chapter 1, through the specialised drift condition of [START_REF] Champagnat | Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes[END_REF], where we obtain the following Proposition 5.5. Consider a competitive Lotka-Volterra Feller diffusion (X t ) t ∈R + in R d + as above. Assume that c i i > 0 for all i ∈ {1, . . . , d }. Then the process has a unique quasi-stationary distribution ν QSD and there exist constants C , γ > 0 such that, for all probability measures µ on (0, ∞) d ,

P µ (X t ∈ • | t < τ ∂ ) -ν QSD T V ≤ C e -λt , ∀t ≥ 0.
(5.10)

Uniform convergence using gradient estimates

The results of this section first appeared in [START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF].

We state here that gradient estimates on the semi-group of the an absorbed Markov process (X t ) t ∈R + can imply the exponential convergence in the situation described in Example 5.1.

Let X be a diffusion process2 in a compact manifold with boundary M absorbed at the boundary ∂M . We assume that one of the two following assumptions S1 or S2 is satisfied: S1. M is a bounded, connected and closed C 2 Riemannian manifold with C 2 boundary ∂M and the infinitesimal generator of the diffusion process X is given by L = 1 2 ∆ + Z , where ∆ is the Laplace-Beltrami operator and Z is a C 1 vector field. S2. M is a compact subset of R d with non-empty, connected interior and C 2 boundary ∂M and for some function g such that 1 0 g (r )d r < ∞. For instance, (5.11) is satisfied as soon as σ is uniformly α-Hölder on M for some α > 0.

X is solution to the SDE d X t = σ(X t )d B t + b(X t )d t ,
In both situations, one can use the gradient estimates obtained by Wang in [START_REF] Wang | Gradient estimates of Dirichlet heat semigroups and application to isoperimetric inequalities[END_REF] and Priola and Wang in [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF]: there exists t 1 > 0 such that the process satisfies a gradient estimate of the form

∇P t 1 f ∞ ≤ C f ∞ , ∀ f ∈ B b (M ),
(5.12)

where P t f (x) = E x ( f (X t )1 t <τ ∂ ) denotes the Dirichlet semi-group of X and the notation ∇P t 1 f ∞ has to be understood as

∇P t 1 f ∞ := sup x,y∈E ∪{∂} |P t 1 f (x) -P t 1 f (y)| ρ(x, y) .
Theorem 5.6. Assume that the diffusion process (X t ) t ∈R + satisfies Assumption S1 or Assumption S2.

Then Condition A of Chapter 1 and hence (1.5) are satisfied. Moreover, there exist two constants B, γ > 0 such that, for any initial distributions µ 1 and µ 2 on E ,

P µ 1 (X t ∈ • | t < τ ∂ ) -P µ 2 (X t ∈ • | t < τ ∂ ) T V ≤ B e -γt µ 1 (ρ ∂ ) ∨ µ 2 (ρ ∂ ) µ 1 -µ 2 T V , (5.13)
where ρ ∂ is the Euclidean distance to the boundary.

Remark 5.2. The gradient estimates of [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF] are proved for diffusion processes with space-dependent killing rate V : M → [0, ∞). More precisely, they consider infinitesimal generators of the form

L = 1 2 d i , j =1 [σσ * ] i j ∂ i ∂ j + d i =1 b i ∂ i -V
with V bounded measurable. The proof proposed in [START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF] can be adapted to this setting.

Time inhomogeneous diffusion processes

Let D be a bounded open subset of R d (d ≥ 1) whose boundary ∂D is of class C 2 and consider the stochastic differential equation

d Z t = σ(t , Z t )d B t + b(t , Z t )d t , Z 0 ∈ D, (5.14)
where B is a standard d dimensional Brownian motion. We assume that the functions

σ : [0, +∞[×R d → R d × R d (t , x) → σ(t , x) and b : [0, +∞[×R d → R d (t , x) → b(t , x)
are continuous on [0, +∞[×R d . Moreover, we assume that they are time-periodic and Lipschitz in x ∈ D uniformly in t ∈ [0, +∞[. This means that there exist two constants Π > 0 and C 0 > 0 such that, for all x, y ∈ D and t ≥ 0,

σ(t + Π, x) = σ(t , x) and b(t + Π, x) = b(t , x), σ(t , x) -σ(t , y) + |b(t , x) -b(t , y)| ≤ k 0 |x -y|. (5.15)
We also assume that the coefficients are elliptic, meaning here that there exists a constant c 0 > 0 such that

c 0 |y| ≤ |σ(t , x)y|, ∀(t , x, y) ∈ [0, +∞[×D × R d .
For all s > 0 and any probability distribution µ on D, we denote by (Z µ s,t ) t ≥s the unique solution to this stochastic differential equation starting at time s > 0 with distribution µ, killed when it hits the boundary and killed with a rate κ(t , Z µ s,t ) ≥ 0, where κ : [0, +∞[×D → R + is a uniformly bounded non-negative measurable function which is also time-periodic (with period Π).

This model was studied in [START_REF] Del Moral | Exponential mixing properties for time inhomogeneous diffusion processes with killing[END_REF], where mixing properties for the conditioned process were proved, using, as in the preceding section, the gradient estimates proved in [START_REF] Priola | Gradient estimates for diffusion semigroups with singular coefficients[END_REF] and adapted to the time inhomogeneous setting, but where we assumed an additional involved differentiability assumption in order to make use of an involved tightness result elaborated in [START_REF] Villemonais | Uniform tightness for time-inhomogeneous particle systems and for conditional distributions of time-inhomogeneous diffusion processes[END_REF]. This last assumption proved to be useless in [START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF] by controlling the distance to the boundary with a coupling argument involving one-dimensional drifted Brownian motions and the fact that the distance to the boundary is C 2 in a vicinity of the boundary (see e.g. [START_REF] Delfour | Shapes and geometries, volume 4 of Advances in Design and Control[END_REF]Chapter 7,Section 8]). This coupling argument can be repeated here (the details are left to the reader), allowing us to state the following result, which is thus a direct improvement over [START_REF] Del Moral | Exponential mixing properties for time inhomogeneous diffusion processes with killing[END_REF].

Theorem 5.7. Under the assumptions of this section, there exist two constants C > 0 and γ > 0 such that, for all 0 ≤ s ≤ t , and all probability measures µ 1 and µ 2 on D,

sup µ 1 ,µ 2 ∈M 1 (D) P(Z µ 1 s,t ∈ • | t < τ ∂ ) -P(Z µ 2 s,t ∈ • | t < τ ∂ ) T V ≤ C e -γ(t -s) .
Additional general results on time-inhomogeneous processes conditioned not to be absorbed are also presented in [START_REF] Champagnat | Uniform convergence of penalized timeinhomogeneous markov processes[END_REF]. They make use of the simple fact that the coupling methods used in the proof of Theorem 1.1 in Chapter 1 can be effortlessly adapted to the time inhomogeneous setting. We also refer the reader to [START_REF] Oçafrain | Quasi-stationarity and quasi-ergodicity for discrete-time markov chains with absorbing boundaries moving periodically[END_REF][START_REF] Oçafrain | Q-process and asymptotic properties of Markov processes conditioned not to hit moving boundaries[END_REF] for processes in time inhomogeneous environments.

Part III

Application of the criteria of Part I to the study of other problems

In fact, the convergence (6.1) holds in total variation norm, uniformly over the initial distribution under Assumption A. (A weak converse result is also provided in the original paper.) We even obtain an exponential speed of convergence, which is useful to deduce quasi-ergodic properties of the process. In the following result, η is the right-eigenfunction whose existence has been established under Assumption A, E or F in Chapters 1 and 2. The result was proved in [START_REF] Champagnat | Uniform convergence to the Q-process[END_REF] under Assumption A and can be generalised as follows (the proof of the generalisation is omitted here, but it follows the same lines as the original one). Proposition 6.2. Assume that Assumption A or Assumption E or Assumption F holds true. Then there exist two constants C , γ > 0 such that, for all 0 ≤ t ≤ T and for all Γ ∈ F t ,

Q x (Γ) -P x (Γ | T < τ ∂ ) T V ≤ C µ f η e -γ(T -t ) , (6.2)
where f = η under Assumption A, f = ϕ 1 under Assumption E and f = ψ 1 under Assumption F.

One of the main feature of the Q-process as described above is that it is a η-transform of the original absorbed process. Note that the η transform includes a source term e λ 0 t , so that it is conservative. Without this term, it would be a sub-Markov process with killing rate λ 0 . In the case of diffusion processes, this is a general long time feature of h-transforms, as proved in [START_REF] Deblassie | Doob's conditioned diffusions and their lifetimes[END_REF][START_REF] Pinsky | The lifetimes of conditioned diffusion processes[END_REF]. Proposition 6.3. Under the assumptions of the above proposition, for all measurable set A ⊂ E and t ≥ 0,

Q x (X t ∈ •) = e λ 0 t η(x) E x η(X t )1 A (X t ) , (6.3)
where λ 0 ≥ 0 is such that P ν QSD (t < τ ∂ ) = e -λ 0 t for all t ≥ 0.

This feature is particularly interesting and has been used on several occasions. For instance, the notion of Q-process can be used to derive results on stochastic representations of the eigenvectors of sub-Markovian semi-groups. This is the subject of Proposition 2.4 in [START_REF] Martínez | Some properties of quasi-stationary distributions for finite Markov chains[END_REF] (although the cited study is restricted to the finite dimensional case for simplicity, most of the results and proofs of the authors extend directly to models with infinite state space). Interestingly, the Q-process approach of this paper can also be used to derive very simply the main result of [START_REF] Cerf | A Markov chain representation of the normalized Perron-Frobenius eigenvector[END_REF], where the authors obtain a "Markov chain representation of the normalised Perron-Frobenius eigenvector".

The above representation of the Q-process naturally suggests the following use of h-transforms to deduce quasi-stationary properties. Assume that there exists a positive eigenfunction h for the semi-group (P t ) t ≥0 of an absorbed2 process X , associated to some eigenvalue e -λ 0 t ∈ (0, 1]. Then one can define the h-transform of P as follows:

Q t f (x) = e λ 0 t h(x) P t ( f (x)h(x)).
This defines the semi-group of a Markov process (without absorption) to which classical results for convergence to a stationary distribution can be used. Then one recovers the asymptotic behaviour of (P t ) t ≥0 through the formula e λ 0 t P t f (x)

= Q t ( f /h)(x).
The difficulties are two-fold: first the existence of h must be pre-established, second the limiting behaviour of Q t ( f /h)(x) must be obtained, although 1/h is typically unbounded. This general method has been used successfully for instance in [START_REF] Gong | Killed diffusions and their conditioning[END_REF][START_REF] Miura | Ultracontractivity for Markov semigroups and quasi-stationary distributions[END_REF][START_REF] Takeda | Existence and uniqueness of quasi-stationary distributions for symmetric markov processes with tightness property[END_REF][START_REF] Ferré | More on the long time stability of Feynman-Kac semigroups[END_REF] and may be used in several other cases, in particular when explicit formulas or at least good estimates are available for the eigenfunction h.

Exponential ergodicity

We are interested now in the ergodic behaviour of the Q-process. These results were proved under Assumption A and under Assumption E in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF] and [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF] respectively, and the extension to Assumption F is straightforward.

Proposition 6.4. Assumption A, E or F implies that the probability measure β on E defined by

β(d x) = η(x)ν QSD (d x) is the unique invariant distribution of the Markov process X under (Q x ) x∈E .
Moreover, there exist constants C > 0 and α ∈ (0, 1) such that, for all initial distributions µ on E such that µ( f /η) < ∞,

Q µ (X t ∈ •) -β(•) M ( f /η) ≤ C α t µ f /η , ∀t ≥ 0, (6.4)
where f = η under Assumption A, f = ϕ 1 under Assumption E and f = ψ 1 under Assumption F.

Note that one can reformulate the last result as follows: for all g ∈ L ∞ ( f ),

e -λ 0 t P t g (•) -η(•)ν QSD (g ) L ∞ ( f ) ≤ C α t f (x) g L ∞ ( f ) , ∀x ∈ E , ∀t ≥ 0.

Quasi-ergodic properties

Quasi-ergodic theorems go back at least to [START_REF] Breyer | A quasi-ergodic theorem for evanescent processes[END_REF]. We refer the interested reader to [START_REF] Chen | Some limit theorems of killed Brownian motion[END_REF][START_REF] Zhang | Quasi-stationarity and quasi-ergodicity of general Markov processes[END_REF][START_REF] He | On quasi-ergodic distribution for one-dimensional diffusions[END_REF][START_REF] Chen | Quasi-ergodicity for absorbing Markov processes via deviation inequality[END_REF][START_REF] Chen | A deviation inequality and quasi-ergodicity for absorbing Markov processes[END_REF][START_REF] He | A note on the quasi-ergodic distribution of one-dimensional diffusions[END_REF] for further developments. The proof of the following result is developed in [START_REF] Champagnat | Uniform convergence to the Q-process[END_REF] under Assumption A. The proof of its extension to Assumptions E or F follows the same lines and is omitted here.

Corollary 6.5. Assume that Assumption A or Assumption E or Assumption F holds true. Then there exists a positive constant C such that, for all T > 0 and all bounded measurable functions g :

E → R, E x 1 T T 0 g (X t ) d t | T < τ ∂ - E g d β ≤ C g ∞ f (x) T η(x) , ∀x ∈ E , (6.5)
where f = η under Assumption A, f = ϕ 1 under Assumption E and f = ψ 1 under Assumption F.

One month before the release of the first preprint version of [START_REF] Champagnat | Uniform convergence to the Q-process[END_REF], where the above result has been announced, the quasi-ergodic result (6.5) has been obtained by He, Zhang and Zu [143, Thm. 2.1] under Assumption A, without the convergence estimate in 1/T .

Recently, double quasi-ergodic properties have been developed by Zhang, Li and Song in [START_REF] Zhang | Quasi-stationarity and quasi-ergodicity of general Markov processes[END_REF]Theorem 3.2]. They can also be obtained under Assumption A, E or F, following very similar proofs, and are stated as follows. Proposition 6.6. Assume that Assumption A or Assumption E or Assumption F holds true. Then there exists a constant C > 0 such that, for all bounded measurable functions g 1 , g 2 : E → R and constants 0 < p < q < 1, we have

E x 1 T T 0 g 1 (X pt )g 2 (X q t ) d t | T < τ ∂ -β(g 1 )β(g 2 ) ≤ C g 1 ∞ g 2 ∞ f (x) η(x)T , ∀x ∈ E ,
where f = η under Assumption A, f = ϕ 1 under Assumption E and f = ψ 1 under Assumption F.

In [START_REF] Zhang | Quasi-stationarity and quasi-ergodicity of general Markov processes[END_REF]Theorem 3.6], the authors prove a nice consequence to the above double quasi-ergodic result, whose proof holds true here without modification. We thus obtain, using their approach combined with the above results, the following Corollary 6.7. Assume that Assumption A or Assumption E or Assumption F holds true. Then, for all ε > 0 and all bounded measurable functions g : E → R,

P x 1 T T 0 g (X t ) d t -β(g ) ≥ ε | T < τ ∂ -----→ T →+∞ 0, ∀x ∈ E .
Chapter 7

R-positive recurrence of unbounded semi-groups

The aim of this chapter is to show how the results of Chapter 2 can be used to deduce effortlessly general criteria for the geometric convergence of normalised unbounded semigroups. This natural extension provides practical criteria for the R-positive recurrence of unbounded semigroups as developed in [213, Section 6.2] and [START_REF] Niemi | On nonsingular renewal kernels with an application to a semigroup of transition kernels[END_REF]. It has applications to penalized Markov processes [START_REF] Moral | Feynman-Kac formulae. Probability and its Applications[END_REF][START_REF] Moral | Mean field simulation for Monte Carlo integration[END_REF], to the study of the long time behaviour of Markov branching processes (see for instance [START_REF] Ikeda | Branching Markov processes[END_REF][START_REF] Ikeda | Branching Markov processes[END_REF][START_REF] Ikeda | Branching Markov processes[END_REF][START_REF] Biggins | Measure change in multitype branching[END_REF][START_REF] Jagers | General branching processes as Markov fields[END_REF][START_REF] Cloez | Limit theorems for some branching measure-valued processes[END_REF][START_REF] Bertoin | A probabilistic approach to spectral analysis of growthfragmentation equations[END_REF][START_REF] Bertoin | On a Feynman-Kac approach to growth-fragmentation semigroups and their asymptotic behaviors[END_REF][START_REF] Bertoin | The strong Malthusian behavior of growth-fragmentation processes[END_REF]), of non-conservative PDEs (see e.g. [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF][START_REF] Bansaye | A non-conservative Harris' ergodic theorem[END_REF] and references therein).

Let E be a measurable space and (P n ) n∈Z + be a positive semigroup on the set of bounded measurable functions on E . We shall consider cases where there exists a measurable (possibly unbounded) function ψ 1 : E → (0, +∞) such that P 1 ψ 1 ≤ cψ 1 for some constant c, so that P n f is naturally defined for all measurable f ∈ L ∞ (ψ 1 ) and all positive measure µ such that µ(ψ 1 ) < +∞ (this corresponds to the settings described in [START_REF] Nummelin | General irreducible Markov chains and nonnegative operators[END_REF]Section 6.2]). In this settings, the recent article [START_REF] Bansaye | A non-conservative Harris' ergodic theorem[END_REF] makes use of the methods developed in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF][START_REF] Champagnat | Uniform convergence to the Q-process[END_REF] (which correspond to Chapters 1 and 2) to give a necessary and sufficient condition for the existence of a positive eigenfunction η of P 1 with eigenvalue θ 0 and the geometric convergence of θ -n 0 µP n f . We show below that this result can be strengthened as an immediate corollary of the results of Chapter 2 applied to the sub-Markov semigroup

P n (•ψ 1 )
c n ψ 1 for the sufficient condition, and standard results on ergodicity of Markov processes applied to a well-chosen h-transform of P n for the necessary condition. Section 7.1 is devoted to the general statement of this result. We then explain in Section 7.2 how large classes of semigroups satisfying our hypotheses can be deduced from those studied in [START_REF] Champagnat | Uniform convergence to the Q-process[END_REF]. We focus on two applications: penalized semigroups associated to perturbed (discrete-time) dynamical systems in Subsection 7.2.1 and diffusion processes in Subsection 7.2.2.

Main result

We first introduce the assumptions on which our results are based. We state them following the same structure as Assumption E in Chapter 2 to emphasise their similarity.

Condition G. There exist positive real constants θ 1 , θ 2 , c 1 , c 2 , c 3 , an integer n 1 ≥ 1, two functions ψ 1 : E → (0, +∞), ψ 2 : E → R + and a probability measure ν on a measurable subset K of E such that G1 (Local A1-A2). ∀x ∈ K and all measurable A ⊂ K ,

P n 1 (ψ 1 1 A )(x) ≥ c 1 ν(A)ψ 1 (x) and sup n∈Z + sup y∈K P n ψ 1 (y)/ψ 1 (y) inf y∈K P n ψ 1 (y)/ψ 1 (y) ≤ c 2 .
G2 (Global Lyapunov criterion). We have

θ 1 < θ 2 and inf x∈K ψ 2 (x)/ψ 1 (x) > 0, sup x∈E ψ 2 (x)/ψ 1 (x) ≤ 1, P 1 ψ 1 (x) ≤ θ 1 ψ 1 (x) + c 3 1 K (x)ψ 1 (x), ∀x ∈ E P 1 ψ 2 (x) ≥ θ 2 ψ 2 (x), ∀x ∈ E .
G3 (Aperiodicity). For all x ∈ K , there exists n 4 (x) such that for all n ≥ n 4 (x),

P n (1 K ψ 1 ) > 0.
Theorem 7.1. Assume that Condition G holds true. Then there exists a positive measure ν P on E such that ν P (ψ 1 ) = 1 and ν P (ψ 2 ) > 0, and two constants C < +∞ and α ∈ (0, 1) such that, for all f ∈ L ∞ (ψ 1 ) and all positive measure µ on E such that µ(ψ 1 ) < +∞ and µ(ψ 2 ) > 0,

µP n f µP n ψ 1 -ν P ( f ) ≤ C α n µ(ψ 1 ) µ(ψ 2 ) , ∀n ∈ Z + . (7.1)
In addition, there exists θ 0 > 0 such that ν P P n = θ n 0 ν P and a function η : E → R + such that θ -n 0 P n ψ 1 converges uniformly and geometrically toward η in L ∞ (ψ 1 ) and such that P 1 η = θ 0 η and ν P (η) = ν P (ψ 1 ) = 1. Moreover, there exist two constants C > 0 and β ∈ (0, 1) such that, for all f ∈ L ∞ (ψ 1 ) and all positive measures µ on E such that µ(ψ 1 ) < +∞,

θ -n 0 µP n f -µ(η)ν P ( f ) ≤ C β n µ(ψ 1 ). (7.2)
Remark 7.1. One can check that replacing ψ 1 by ψ 2 in G1 and/or G3 give equivalent versions of Condition G. In [START_REF] Bansaye | A non-conservative Harris' ergodic theorem[END_REF], a similar result is obtained, but with the additional assumptions that ψ 2 > 0 on E and n 1 = 1. In this restricted case, one can easily check that their assumptions on the discretetime semigroup are equivalent to ours. The fact that ψ 2 can vanish in Assumption G allows to deal with reducible processes (as in Section 2.6).

The proof of the above theorem is straightforward, since the semi-group (Q n ) n∈N defined by

Q n ( f ) = P n ( f ψ 1 ) (θ 1 + c 2 ) n ψ 1 , ∀n ≥ 0, f ∞ ≤ 1 satisfies Condition E in Chapter 2 with ϕ 1 = 1 and ϕ 2 = ψ 2 /ψ 1 , using θ 1 /(θ 1 + c 2 ) in place of θ 1 , θ 2 /(θ 1 + c 2 ) in place of θ 2 and c 1 /(θ 1 + c 2 ) n 1 in place of c 1 .
See the original paper [START_REF] Champagnat | Practical criteria for R-positive recurrence of unbounded semigroups[END_REF] for the details.

Remark 7.2. The elementary method consisting in studying the sub-Markov semi-group (Q n ) instead of (P n ) is neither new nor specific to our approach. It can also be used to derive immediately sufficient criteria for the convergence of unbounded semi-groups from the abundant theory of sub-Markovian semi-groups, as developed for instance in [START_REF] Collet | Quasi-stationary distributions[END_REF][START_REF] Collet | Quasi-stationary distributions for structured birth and death processes with mutations[END_REF][START_REF] Velleret | Unique Quasi-Stationary Distribution, with a possibly stabilizing extinction[END_REF][START_REF] Ferré | More on the long time stability of Feynman-Kac semigroups[END_REF][START_REF] Kolb | Quasilimiting behavior for one-dimensional diffusions with killing[END_REF][START_REF] Hening | Quasistationary distributions for one-dimensional diffusions with singular boundary points[END_REF]. Note that a similar approach has been used in [START_REF] Bertoin | A probabilistic approach to spectral analysis of growthfragmentation equations[END_REF] to describe the asymptotic behaviour of the growthfragmentation equation using Krein-Rutman theorem and other criteria for R-positivity.

Whether Assumption G is necessary for (7.1) is still an open problem up to our knowledge. However, if one assumes that there exists a positive eigenfunction η such that (7.2) holds true, then one recovers easily Assumption G by applying the classical counterpart of Forster-Lyapunov criteria for conservative semigroups. Here, the conservative semigroup is the one associated to the η-transform of P n defined by R n f := θ -n 0 η P n (η f ) (which corresponds to the Q-process in the sub-Markovian case, cf. Chapter 6). The only difficulty in the proof of the following theorem is that η may vanish on some subset of E . Proposition 7.2. Assume that there exist a positive function ψ : E → (0, +∞) and a non-negative eigenfunction η ∈ L ∞ (ψ) of P 1 for the eigenvalue θ 0 > 0, such that

θ -n 0 P n f (x) -η(x)ν P ( f ) ≤ ζ n ψ(x), ∀x ∈ E , f ∈ L ∞ (ψ 1 ), (7.3)
where (ζ n ) n≥0 is some positive sequence converging to 0. Then Assumption G is satisfied with ψ 2 = η and with some function

ψ 1 ∈ L ∞ (ψ) such that ψ ∈ L ∞ (ψ 1 ).
Remark 7.3. A similar partial counterpart to Proposition 7.2 was proven in [START_REF] Bansaye | A non-conservative Harris' ergodic theorem[END_REF], where the authors assume that ζ n is geometrically decreasing, that η is positive and use the approach of [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF] to conclude.

For continuous time semigroups (P t ) t ∈[0,+∞) , the conclusions of Theorem 7.1 can be easily deduced from properties on the discrete skeleton (P nt 0 ) n∈N (similar properties where already observed in Theorem 5 of [START_REF] Tuominen | Exponential decay and ergodicity of general Markov processes and their discrete skeletons[END_REF] and in [START_REF] Champagnat | Uniform convergence to the Q-process[END_REF]). In the following result, the function η and the positive measure ν P are the one of Theorem 2.1 applied to the discrete skeleton (P nt 0 ) n∈N . Corollary 7.3. Let (P t ) t ∈[0,+∞) be a continuous time semigroup. Assume that there exists t 0 > 0 such that (P nt 0 ) n∈N satisfies Assumption G,

P t ψ 1 ψ 1 t ∈[0,t 0 ]
is upper bounded by a constant c > 0 and

P t ψ 2 ψ 2 t ∈[0,t 0 ]
is lower bounded by a constant c > 0. Then there exist some constants C > 0 and γ > 0 such that, for all f ∈ L ∞ (ψ 1 ) and all positive measure µ on E such that µ(ψ 1 ) < +∞ and µ(ψ 2 ) > 0,

µP t f µP t ψ 1 -ν P ( f ) ≤ C e -γt µ(ψ 1 ) µ(ψ 2 ) , ∀t ∈ [0, +∞), (7.4)
In addition, there exists λ 0 ∈ R such that ν P P t = e λ 0 t ν P for all t ≥ 0, and e -λ 0 t P t ψ 1 converges uniformly and exponentially toward η in L ∞ (ψ 1 ) when t → +∞. Moreover, there exist some constants C > 0 and γ > 0 such that, for all f ∈ L ∞ (ψ 1 ) and all positive measures µ on E such that µ(ψ 1 ) < +∞,

e -λ 0 t µP t f -µ(η)ν P ( f ) ≤ C e -γ t µ(ψ 1 ), ∀t ∈ [0, +∞). (7.5)
Remark 7.4. These results can be seen as an extension to bounded non-conservative semigroups of criteria of convergence for semigroups associated to Markov processes (in particular, Harris theorem and all its extensions based on Doblin's conditions and Foster-Lyapunov criteria, see e.g. [START_REF] Meyn | Markov chains and stochastic stability[END_REF][START_REF] Douc | Markov chains[END_REF]) and as a practical alternative to R-recurrent Markov chains theory [START_REF] Tuominen | Exponential decay and ergodicity of general Markov processes and their discrete skeletons[END_REF][START_REF] Nummelin | General irreducible Markov chains and nonnegative operators[END_REF][START_REF] Niemi | On nonsingular renewal kernels with an application to a semigroup of transition kernels[END_REF].

In particular, it provides an alternative to spectral theoretic results dealing with existence of eigenfunctions and convergence to them (e.g. Krein-Rutman theorem, spectral theory of symmetric operators, or the theorem of convergence of normalised semigroups of Birkhoff [START_REF] Birkhoff | Extensions of Jentzsch's theorem[END_REF] and its extensions).

Some applications

Given a positive semigroup P acting on measurable functions on E , one can try to directly check Assumption (G) by finding appropriate functions ψ 1 and ψ 2 . Another natural and equivalent strategy is to find a function ψ such that the semigroup defined by

Q n f = P n (ψ f )
c n ψ is sub-Markovian and check that it satisfies Assumption E of Chapter 2. The main advantage of this last approach is that Q has a probabilistic interpretation as the semigroup of a sub-Markov process. As such, one can apply all the criteria developed in the first part of this manuscript and, more generally, use the intuitions and toolboxes of the theory of stochastic processes. Since both approaches are equivalent, this is rather a question of taste.

Perturbed dynamical systems

Let f : R d → R d be a locally bounded measurable function and consider the perturbed dynamical system X n+1 = f (X n ) + ξ n with (ξ i ) i ∈Z + i.i.d. non-degenerate Gaussian random variables. We are interested in the asymptotic behaviour of the associated Feynman-Kac semigroup

P n f (x) = E x n k=1 G(X k )1 X k ∈E f (X n ) ,
where E is a measurable subset of R d with positive Lebesgue measure and G : E → (0, +∞) is a measurable function.

Proposition 7.4. Assume that 1/G is locally bounded, that G(x) ≤ C exp(|x|) for all x ∈ E and some constant C > 0, and there exists p > 1 such that |x| -p| f (x)| → +∞ when |x| → +∞, then the semigroup (P n ) n∈N satisfies Assumption G.

Diffusion processes

Let (X t ) t ∈R + be solution to the SDE

d X t = d B t + b(X t ) d t , X 0 ∈ (0, +∞) d , (7.6)
where B is a standard d -dimensional Brownian motion and b : R d → R d is locally Hölder. Let r : (0, +∞) d → R be locally bounded and consider the semigroup (P t ) t ∈R + defined by

P t f (x) = E x e t 0 r (X u ) d u f (X t ) 1 X s ∈(0,+∞) d , ∀s∈[0,t ] . (7.7)
The term 1 X s ∈(0,+∞) d , ∀s∈[0,t ] above corresponds to a killing at the boundary of the domain (0, +∞) d . Note that the solution to (7.6) may explode in finite time if b does not satisfy the linear growth condition. However, we assume by convention that X t ∈ (0, +∞) d after the explosion time, so that (7.7) makes sense. We refer to [61, Sections 4.1 and 12.1] for the precise construction of the process.

One motivation for the study of this semigroup comes from the Feynman-Kac formula. Indeed, when the coefficients are smooth enough (see for instance [START_REF] Pham | Continuous-time stochastic control and optimization with financial applications[END_REF]Section 1.3.3]), this semigroup is solution to the Cauchy linear parabolic partial differential equation

r v - ∂v ∂t + L v = 0, on [0, +∞) × (0, +∞) d v(0, •) = f , on (0, +∞) d ,
where L is the differential operator of second order

L ϕ(x) = 1 2 ∆ϕ(x) + b(x) • ∇ϕ(x), ∀ϕ ∈ C 2 (R d ),
with Dirichlet boundary conditions.

Proposition 7.5. Assume that r (x)

+ d i =1 b i (x) ------------→ |x|→∞, x∈(0,∞) d -∞. (7.8)
Then the semigroup (P t ) t ∈[0,+∞) satisfies the assumptions of Corollary 7.3.

Chapter 8

Measure-valued Pólya urn processes

Measure-valued Pólya processes (MVPPs) are a generalisation of Pólya urns to the infinitely-manycolour case. Pólya urns date back to Pólya & Eggenberger [START_REF] Eggenberger | Über die statistik verketetter vorgäge[END_REF], and have been thoroughly studied since then; highlights include, e.g., the seminal works of Athreya & Karlin [START_REF] Athreya | Embedding of urn schemes into continuous time Markov branching processes and related limit theorems[END_REF] and Janson [START_REF] Janson | Functional limit theorems for multitype branching processes and generalized Pólya urns[END_REF].

Although the question of generalising Pólya urns to infinitely-many colours was posed in 2004 in [START_REF] Janson | Functional limit theorems for multitype branching processes and generalized Pólya urns[END_REF], MVPPs were only introduced recently by Bandyopadhyay & Thacker [START_REF] Bandyopadhyay | A New Approach to Pólya Urn Schemes and Its Infinite Color Generalization[END_REF] and Mailler & Marckert [START_REF] Mailler | Measure-valued pólya urn processes[END_REF].

In this chapter, we focus on almost-sure convergence of a large class of MVPPs using stochasticapproximation methods (in the spirit of Duflo [START_REF] Duflo | Random iterative models[END_REF] and Benaïm [START_REF] Benaïm | Dynamics of stochastic approximation algorithms[END_REF]). The main difficulty comes from the fact that the stochastic-approximation algorithm that we consider is defined on the space of measures on a non-compact space. All the results below originate from a collaboration with Cécile Mailler [START_REF] Mailler | Stochastic approximation on non-compact measure spaces and application to measure-valued P\'olya processes[END_REF].

The stochastic-approximation approach is a classical method for the study of Pólya urn processes when the colour-set is finite, see for instance [START_REF] Benaïm | Dynamics of stochastic approximation algorithms[END_REF], [START_REF] Laruelle | Randomized urn models revisited using stochastic approximation[END_REF], [START_REF] Laruelle | Nonlinear randomized urn models: a stochastic approximation viewpoint[END_REF], Zhang [START_REF] Zhang | Central limit theorems of a recursive stochastic algorithm with applications to adaptive designs[END_REF] and the survey paper [START_REF] Pemantle | A survey of random processes with reinforcement[END_REF]. Our main contribution from the stochastic-approximation point of view is to prove convergence of a stochastic-approximation algorithm defined on a non-compact space, namely the set of probability measures on the colour-space (being an arbitrary Polish space). To our knowledge, very little is known for measure valued stochastic-approximation algorithm on non-compact spaces, with some exceptions such as [START_REF] Janson | Asymptotic degree distribution in random recursive trees[END_REF] and [START_REF] Maillard | Choices and intervals[END_REF].

Our main contribution to the theory of MVPPs is to prove almost-sure convergence for a large class of MVPPs (instead of the convergence in probability shown in [START_REF] Bandyopadhyay | A New Approach to Pólya Urn Schemes and Its Infinite Color Generalization[END_REF][START_REF] Mailler | Measure-valued pólya urn processes[END_REF]). Furthermore, we generalise the definition of measure-valued Pólya processes to allow different colours to have different "weights", and to allow the so-called "replacement rule" to be random (two features that are classical in the context of Pólya urns). We are also able to treat the "non-balanced" case, which was not treated at all by Bandyopadhyay & Thacker [START_REF] Bandyopadhyay | A New Approach to Pólya Urn Schemes and Its Infinite Color Generalization[END_REF] or Mailler & Marckert [START_REF] Mailler | Measure-valued pólya urn processes[END_REF].

The link between Pólya urns and quasi-stationary distributions already exists in the literature; for example, Aldous, Flannery and Palacios [START_REF] Aldous | Two applications of urn processes the fringe analysis of search trees and the simulation of quasi-stationary distributions of markov chains[END_REF] apply the convergence results of Athreya and Karlin [START_REF] Athreya | Embedding of urn schemes into continuous time Markov branching processes and related limit theorems[END_REF] to approximating quasi-stationary distributions on a finite state space. Our main result generalises this work to the case of measure-valued Pólya processes.

Definition of the model and main result

Let E be a Polish space endowed with its Borel sigma-field. A measure-valued Pólya process (MVPP) is a Markov chain (m n ) n≥0 taking values in M (E ). It depends on three parameters: its initial composition m 0 a non-zero non-negative measure on E , a sequence of i.i.d. replacement kernels 1 (R (n) ) n≥1 on E , and a non-negative weight kernel P on E . We assume that T >0 . almost surely, for all x ∈ E , R (n) x is a non-negative measure.

Given m n , we define m n+1 as follows: pick a random element Y n+1 of E according to the probability distribution proportional to m n P , i.e., for all Borel set A of E ,

P(Y n+1 ∈ A | m n ) = E P x (A) d m n (x) E P x (E ) d m n (x) , (8.1)
and then set

m n+1 = m n + R (n+1) Y n+1
.

Measure-valued Pólya processes were originally introduced by [START_REF] Bandyopadhyay | A New Approach to Pólya Urn Schemes and Its Infinite Color Generalization[END_REF] and [START_REF] Mailler | Measure-valued pólya urn processes[END_REF], as a generalisation of d -colour Pólya urns, although they did not consider "weighted" MVPPs (they always had P x = δ x for all x ∈ E ). Several examples are developed in Section 8.2. For now, let us recall the definition of a Pólya urn and show why MVPPs generalise this model. A d -colour Pólya urn is a Markov process (U (n)) n≥0 on N d that depends on three parameters: the initial composition vector U (0), the replacement matrix M , and weights w 1 , . . . , w d ∈ (0, ∞). The vector U (n) represents the content of an urn that contains balls of d different colours; balls of colour i all have weight w i . Given U (n), one defines U (n + 1) by picking a ball at random in the urn with probability proportional to its weight, denoting the colour of this random ball ξ n+1 , and setting U (n + 1) = U (n) + M ξ n , where M 1 , . . . , M d are the lines of M .

If we let E = {1, . . . , d } and m n = d i =1 U i (n)δ i for all n ≥ 0, then m n is a measure-valued Pólya process with replacement kernel

R (n) x = d i =1 M x,i δ i ( almost surely for all n ≥ 0, 1 ≤ x ≤ d ),
and weight kernel P x = w x δ x for all 1 ≤ x ≤ d . Therefore, the MVPP process (m n ) n≥0 can be thought of as a composition measure on a set E of colours, and the random variable Y n+1 can be seen as the colour of the "ball" drawn at time n + 1. The main advantage of this wider model is that one can consider Pólya urns defined on an infinite, and even uncountable, set.

Our main result is to prove almost-sure convergence of the sequence ( m n/m n (E )) n≥0 to a deterministic measure under the following assumptions. We denote by R the common expectation of the R (n) 's and set Q (n) = R (n) P for all n ≥ 1, and Q = RP , meaning that, for all x ∈ E and all Borel set A ⊆ E ,

Q (n) x (A) = E P y (A) d R (n) x (y) and Q x (A) = E P y (A) d R x (y).
We assume that 1 A kernel (resp. a non-negative kernel) on E is, by definition, a function from E into the set of measures (resp. nonnegative measures) on E . In particular, for all x ∈ E , R (n) x is a measure on E almost surely.

C1. for all x ∈ E , we have Q x (E ) ≤ 1, and there exists a probability measure µ on R with positive mean such that, for all x ∈ E , the law of Q (i ) x (E ) stochastically dominates µ. In particular, setting

c 1 = ∞ 0 x dµ(x), 0 <c 1 ≤ inf x∈E Q x (E ) ≤ sup x∈E Q x (E ) ≤ 1;
C2. there exists a locally bounded function V : E → [1, +∞) such that, (i) for all N ≥ 0, the set {x ∈ E : V (x) ≤ N } is relatively compact;

(ii) there exist two constants θ ∈ (0, c 1 ) and K ≥ 0 such that

Q x • V ≤ θV (x) + K , ∀x ∈ E ,
(iii) and that there exist three constants r > 1, p > ln θ ln( θ /c 1 ) ∨ 2, A > 0 such that E R (1) x (E ) r ∨ E Q (1) x (E ) p ≤ AV (x), ∀x ∈ E .

Under Assumption C1, Q is a non-negative kernel such that sup x Q x (E ) ≤ 1, so that Q -I is the jump kernel (or infinitesimal generator) of a unique sub-Markovian transition kernel (P t ) t ≥0 on E . We consider the continuous-time pure-jump Markov process (X t ) t ≥0 on E ∪ {∂}, where ∂ ∉ E is an absorbing state, with Markovian transition kernel P t + (1 -P t (E ))δ ∂ . C3. the continuous-time pure jump Markov process X with sub-Markovian jump kernel Q -I admits a quasi-stationary distribution ν QSD ∈ M 1 (E ). We further assume that the convergence of P α (X t ∈ • | X t = ∂) holds uniformly with respect to the total variation norm on {α ∈ M 1 (E ) | α • V 1/q ≤ C }, for each C > 0, where q = p/(p -1).

Finally, we need the following technical assumption:

C4. for all bounded continuous functions f :

E → R, x ∈ E → R x f and x ∈ E → Q x f are continu- ous.
Under these assumptions, we are able to prove almost-sure convergence of the normalised MVPP mn := m n /m n (E ). This result is applied to concrete models in Section 8.2 Theorem 8.1. Under Assumptions T >0 and C1-4, if m 0 •V < ∞ and m 0 P •V < ∞, then the sequence of random measures (m n /n) n≥0 converges almost surely to νR with respect to the topology of weak convergence. Moreover, sup n {m n P • V 1 /q /n} < +∞ almost surely, where q = p/(p -1).

Furthermore, if νR(E ) > 0, then ( mn ) n∈N converges almost surely to νR/νR(E ) with respect to the topology of weak convergence. Remark 8.1. Several refinements and precisions are provided on this result in the original article [START_REF] Mailler | Stochastic approximation on non-compact measure spaces and application to measure-valued P\'olya processes[END_REF]. In particular, our more general result includes the possibility to remove ball from the urns, which is useful in several applications. This leads to several additional technicalities, both in the proof of our results and in the exposition of our assumptions. Remark 8.2. If R = Q, then the quasi-stationary distribution ν is a left eigenfunction for R, with associated eigenvalue θ 0 ∈ (0, 1]. In particular, Theorem 8.1 implies that the average mass of m n , i.e. m n (E )/n, converges almost surely to θ 0 .

Examples

The finite state space case

To illustrate how this theorem applies, let us first consider the simple case of a classical d -colour Pólya urn of random replacement matrix M (n) with no weights, where (M (n) ) n is a sequence of i.i.d. random matrices with non-negative entries and mean M . We assume that d i =1 M x,i > 0 for all 1 ≤ x ≤ d and that M is irreducible. Let S = max d x=1 d i =1 M x,i , and let m n = 1 S d i =1 U i (n)δ i , where U i (n) is the number of balls of colour i in the urn at time n. One can check that (m n ) n≥0 is an MVPP with replacement kernel R (n) x = 1

S d i =1 M (n)
x,i δ i on E = {1, . . . , d }, for all n ≥ 0 and 1

≤ x ≤ d , such that R = M /S.
Note that, since we have no weights, R = Q. Let µ be the distribution of min x∈{1,...,d } Y x , where Y 1 , . . . , Y d are independent random variables respectively distributed as Q (1) 1 (E ), . . . ,Q (1) d (E ). Assumption C1 is satisfied since µ has positive mean c 1 ≤ Q x (E ) ≤ 1 for all 1 ≤ x ≤ d . Assumption C2 is automatically satisfied since the colour space E is compact. Consider the process X on E ∪ {∂} absorbed at ∂ and whose jump matrix restricted to E is given by M /S -I . Then, since M /S is irreducible, the process X conditioned on not hitting ∂ has a unique quasi-stationary distribution ν QSD = n i =1 v i δ i , which is given by the unique non-negative left eigenvector v of M /S -I and hence of M . It is also known (see e.g. Darroch & Seneta [START_REF] Darroch | On quasi-stationary distributions in absorbing continuous-time finite Markov chains[END_REF]) that there exists C , δ > 0 such that

P α (X t ∈ •|X t = ∂)-ν QSD T V ≤ C e -δt
for all α ∈ M 1 (E ), which thus implies C3. Finally, Assumption C4 is trivially satisfied since E is discrete.

Thus, Theorem 8.1 applies, and we get that, almost surely when n tends to infinity, mn → ν QSD R/ν QSD R(E ) = ν QSD (with respect to the topology of weak convergence), and thus, U (n)/n → v, a result that dates back to Athreya & Karlin's work on generalised Pólya urns [START_REF] Athreya | Embedding of urn schemes into continuous time Markov branching processes and related limit theorems[END_REF].

Ergodic Markov chains

In [START_REF] Mailler | Measure-valued pólya urn processes[END_REF], the following example is treated: take E = Z + , fix 0 < λ < µ, and set

R x = λ xµ + λ δ x+1 + xµ xµ + λ δ x-1 ,
for all x = 0, and R 0 = δ 1 . This example is not weighted, meaning that P x = δ x for all x ∈ E , and balanced since R x (E ) = 1 for all x ∈ E . Note that the Markov chain of transition kernel R is the M /M /∞ queue. The authors proved that that this MVPP satisfies

n -1 m n → γ in probability,
where γ is the stationary measure of the M /M /∞ queue, i.e.

γ(x) = λ µ

x e -λ/µ

x! ∀x ∈ Z + .

Our result also applies to this situation and thus implies the stronger almost-sure convergence of this MVPP. Since this example is simple, let us detail how one show that our result applies. First note that, in this example, the R (i ) are deterministic and equal to R, P x = δ x ; therefore,

Q (i ) = Q = R (∀i ≥ 1).
Since R x (E ) = 1 for all x ∈ N, then C1 is satisfied (we can take µ = δ 1 , and thus, c 1 = 1). Assumption C2 also holds: one can take V (x) = e x , implying that

R x • V = λe x+1 + µxe x-1 λ + µx = λe 2 + µx λ + µx e x-1 = λe 2 + µx e(λ + µx) V (x).
Note that λe 2 + µx

e(λ + µx) < 2 e ⇔ x > λ(e 2 -2) µ , therefore, R x • V ≤ θ V (x) + K , ∀x ∈ Z +
where θ = 2 e ∈ (0, c 1 ) and K = sup x≤λ(e 2 -2)/µ R x • V . Also note that, for all r, p > 1, we have ER (1) x (E ) r ∨ EQ (1) x (E )

p = R x (E ) r ∨ R x (E ) p = 1,
implying that C2-(iii) holds, while the rest of C2 is clear. Since the queue M /M /∞ is ergodic with stationary distribution ν SD , we can infer that the continuous-time Markov process of generator R -I is also ergodic. Moreover, one can show that, for any q > 1, Q x •V 1 /q ≤ θ 1/q V (x)+K 1 /q , where θ 1 /q < 1. This and the Foster-Lyapunov type criteria of [START_REF] Meyn | Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes[END_REF] provide the uniform convergence to ν SD required in Assumption C3. Finally, since Z + is discrete, C4 is trivially satisfied. Thus, Theorem 8.1 applies and we can conclude that if k≥0 e k m 0 (k) is finite, then n -1 m n → ν SD almost surely when n → ∞.

Quasi-ergodic Markov chains

Let us now consider the more general case where E = Z + and, for all x ∈ E ,

R x = λ x δ x+1 + µ x δ x-1 ,
where (λ x ) x and (µ x ) x are families of positive numbers such that µ 0 = 0, λ 0 > 0, inf x≥1 µ x > 0, sup x µ x < ∞ and λ x = o(µ x ) when x → +∞. In this situation, the MVPP is not weighted, so that P x = δ x and Q x = R x for all x ∈ E , and it is not balanced (hence the results of [START_REF] Bandyopadhyay | A New Approach to Pólya Urn Schemes and Its Infinite Color Generalization[END_REF] and [START_REF] Mailler | Measure-valued pólya urn processes[END_REF] do not apply). We assume, without loss of generality, that sup x (λ x + µ x ) = 1, so that Q x (E ) ≤ 1 for all x ∈ E . The situation is reminiscent of the simple example of Section 2.2 and the calculus are similar. In particular, we deduce that, for the irreducible process X with infinitesimal generator Q -I , there exist a quasi-stationary distribution ν QSD for X and two positive constants Cst, δ > 0 such that, for all probability measures α ∈ E , satisfying α•V 1 /q < +∞, where V = exp(ax) with a > 0 large enough, -----→

P α (X t ∈ • | t < τ ∂ ) -ν QSD T V ≤ Cst α • V 1 /q e -
n→+∞ ν QSD R ν QSD R(E ) = ν QSD .
with respect to the topology of weak convergence, as soon as m 0 (V ) < +∞.

Random trees

As discussed in Janson [START_REF] Janson | Functional limit theorems for multitype branching processes and generalized Pólya urns[END_REF]Examples 7.5 and 7.6], infinitely-many-colour urns are particularly useful for the study of some functionals of random trees. We give below two examples where our main result applies, and gives stronger convergence results.

Definition 8.1 (Outdegree profiles). We define the out-degree profile of a rooted tree τ as

Out(τ) = ν∈τ δ outdeg(ν) ,
where for all nodes ν in τ, outdeg(ν) is the out-degree of ν (i.e. its number of children).

Out-degree profile in the random recursive tree. The random recursive tree (RRT n ) n≥1 is a sequence of random rooted trees defined recursively as follows:

• RRT 1 has one node (the root);

• we build RRT n+1 from RRT n by choosing a node of RRT n uniformly at random, and adding a child to this node.

It is straightforward to see that the sequence (Out(RRT n )) n≥1 of the out-degree profile of the random recursive tree is a MVPP on Z + of initial composition m 1 = δ 0 , and replacement kernel

R x = -δ x + δ 0 + δ x+1 , ∀x ∈ Z + .
Note that the replacement measures R x are not positive, but the process satisfies the additional assumptions detailed in Section 1.4 of [START_REF] Mailler | Stochastic approximation on non-compact measure spaces and application to measure-valued P\'olya processes[END_REF] for unbalanced MVPPs. In this case, P x = δ x , and R (i ) = R = Q almost surely for all i ≥ 1. Note that Q x (Z + ) = 1 for all x ∈ Z + , and, therefore, Assumption C1 holds with µ = δ 1 and c 1 = 1.

Choosing ε ∈ (0, 1 /2) and setting V (x) = (2 -ε) x for all x ≥ 0, we show in [START_REF] Mailler | Stochastic approximation on non-compact measure spaces and application to measure-valued P\'olya processes[END_REF] that Theorem 8.1 applies and that

n -1 Out(RRT n ) → ν weakly, almost surely when n → ∞, (8.2)
where ν x = 2 -x-1 , for all x ∈ Z + . Different versions of this result can be found in the literature: Bergeron, Flajolet & Salvy [30, Corollary 4] prove it using generating functions, Mahmoud & Smythe [START_REF] Mahmoud | Asymptotic joint normality of outdegrees of nodes in random recursive trees[END_REF] prove a joint central limit theorem for the number of nodes of out-degree 0, 1 and 2, Janson [START_REF] Janson | Functional limit theorems for multitype branching processes and generalized Pólya urns[END_REF]Example 7.5] extends this result by considering out-degrees 0, 1, . . . , M for all M ≥ 0, which implies (8.2). The approach of [START_REF] Mahmoud | Asymptotic joint normality of outdegrees of nodes in random recursive trees[END_REF] and [START_REF] Janson | Functional limit theorems for multitype branching processes and generalized Pólya urns[END_REF] relies on the remarkable fact that, in that particular example, one can reduce the problem to finitely many types.

Our main contribution for this example is to prove the convergence in a stronger sense, and thus answer a question of Janson (see Remark 1.2 [START_REF] Janson | Asymptotic degree distribution in random recursive trees[END_REF]). Indeed, Theorem 8.1 also gives that, for all q ∈ (1, 2),

sup n Out(RRT n ) n • V 1 /q < +∞,
since P x = δ x for all x, in this example. This leads to the following proposition.

Proposition 8.2. For all ε ∈ (0, 1 /2), for all q ∈ (1, 2), for all functions f :

Z + → R such that f (x) = o (2 -ε)
x /q when x → ∞, we have

1 n f d Out(RRT n ) → ∞ x=0 2 -x-1 f (x)
, almost surely when n → ∞.

Our approach also has the advantage of providing a framework that can be easily generalised, as, for example, in the next application to which Janson's finitely-many-types approach wouldn't apply.

Out-degree profile in a random recursive forest with multiple children. Let us now consider the following generalisation of the random recursive tree studied above. The random recursive forest (RRF n ) n≥1 with multiple children is defined as a sequence of random rooted forests defined recursively as follows: consider a probability measure α on {-1} ∪ {1, 2, . . .} (with 0 < α -1 < 1) and a probability measure β on {1, 2, . . .};

• RRF 1 has one node (the root);

• we build RRF n+1 from RRF n by choosing a node of RRF n uniformly at random, and, if this node has at least one child, -with probability α -1 , remove the edge between the node and one of his children (hence forming an other tree in the forest),

with probability α k (k ≥ 1), add k children to this node, while, if this node has 0 child, with probability β k (k ≥ 1), add k children to this node.

We define Out(RRF n ) as the sum of the out-degree profiles (see Definition 8.1) of the trees composing the forest RRF n and obtain the following result.

Proposition 8.3. Assume that α and β both admit an exponential moment of order λ, for some fixed λ > 0. Then there exists a probability distribution ν QSD such that, for all q ∈ (1, 2), for all a > 0 satisfying

+∞ k=1 α k e ak < 2 ∞ k=1 α k ,
and for all function f : Z + → R such that f (x) = o(e ax/q ) when x → ∞, we have

f d Out(RRF n ) Out(RRF n )(E ) → f d ν QSD , almost surely when n → ∞. (8.3)
Protected nodes in the random recursive tree. A node ν of a tree τ is 2-protected if the closest leaf is at distance at least 2 from ν; in a social network, 2-protected nodes can be users who used to invite new users to the network but have not done so recently. The proportion of such nodes in different models of random trees have been studied in the literature: Motzkin trees in Cheon & Shapiro [START_REF] Cheon | Protected points in ordered trees[END_REF], random binary search tree in Bóna [START_REF] Bóna | k-protected vertices in binary search trees[END_REF], and more recently in the m-ary search tree in Holmgren, Janson & Šileikis [START_REF] Holmgren | Multivariate normal limit laws for the numbers of fringe subtrees in m-ary search trees and preferential attachment trees[END_REF]. Devroye & Janson [START_REF] Devroye | Protected nodes and fringe subtrees in some random trees[END_REF] show how results of Aldous [START_REF] Aldous | Asymptotic fringe distributions for general families of random trees[END_REF] about fringe trees can be used to study this question with a unified approach for different models of random trees, including simply generating trees and the random recursive tree. We show here how our main result allows to get information about protected nodes in random trees. For all n ≥ 1 and x ≥ 0, let us denote by X n,x the number of internal nodes in RRT n having exactly x leaf-children. The random measure

m n = x∈N X n,x δ x is a MVPP E = Z + of initial composition m 0 = δ 1 .
The replacement kernel of (m n ) n≥0 is (for all i ≥ 1 and x ≥ 1)

R (i ) 0 = -δ 0 + δ 1 and R (i ) x = B (i ) 1 /x+1 δ x+1 + 1 -B (i ) 1 /x+1 (δ x-1 + δ 1 ) -δ x , where B (i )
1 /x+1 is a sequence of i.i.d. random Bernoulli-distributed variables of parameters 1 /x+1 for all x ≥ 1. The weight kernel of (m n ) n≥0 is P x = (x + 1)δ x , for all x ∈ Z + . We therefore have

R 0 = -δ 0 + δ 1 and R x = 1 x + 1 δ x+1 + x x + 1 (δ x-1 + δ 1 ) -δ x ,
and

Q x = x + 2 x + 1 δ x+1 + x x + 1 (xδ x-1 + 2δ 1 ) -(x + 1)δ x , for all x ≥ 0. Note that Q x (Z + ) = 1 for all x ∈ Z + .
We prove in [START_REF] Mailler | Stochastic approximation on non-compact measure spaces and application to measure-valued P\'olya processes[END_REF] that the assumptions of Theorem 8.1 holds true and obtain the following Proposition 8.4. For all x ≥ 1, the proportion p n,x of internal nodes having exactly x leaf-children in the n-node random recursive tree converges almost surely to

2 e i ≥x+1 1 i ! .
The proportion p n,0 of protected internal nodes converges almost surely to 1 -2 /e. Moreover, for all q ∈ (1, 2) and all function f : Z + → R such that f (x) = o x i =2 (i -ε) 1 /q for some ε > 0 when x → ∞,

we have i ≥0 p n,i f (i ) → (1 -2 /e) f (0) + 2 e i ≥1 f (i ) j ≥i +1 1 j ! almost surely when n → ∞.
Using this result, one can show for instance that the proportion of protected internal nodes converges almost surely to 1 /2-1 /e, improving on the convergence in probability already established by Ward & Mahmoud [START_REF] Mahmoud | Asymptotic properties of protected nodes in random recursive trees[END_REF].

Chapter 9

Reinforced processes

In this chapter, we study a random process with reinforcement, which evolves following the dynamics of a given absorbed Markov process and is resampled according to its occupation measure when it reaches the absorption point. We show in different situations that its occupation measure converges to the minimal quasi-stationary distribution of the absorbed Markov process.

Let X be a time homogeneous Markov process with state space E ∪ {∂}, where ∂ ∈ E is an absorbing state for the process. We assume that P x (τ ∂ < ∞) = 1 and P x (t < τ ∂ ) > 0 for all t ≥ 0 and ∀x ∈ E .

We consider a random process (Y t ) t ≥0 with reinforcement, which evolves following the dynamic of X when it lies in E and which is resampled according to its occupation measure when it reaches ∂. More precisely, given a probability measure µ on E , we set

Y t = ∞ k=1 1 t ∈[θ k-1 ,θ k ) X (k) t -θ k-1
, ∀t ≥ 0, where θ 0 = 0,

• (X (1) t ) t ∈R + is a realisation of the process (X t ) t ∈R + with X (1) 0 ∼ µ (i.e. under P µ ) and the stopping time θ 1 is defined as θ 1 = τ (1) ∂ the first hitting time of ∂ by X (1) ,

• given X (1) , (X (2) t ) t ∈R + is a realisation of the process (X t , t ≥ 0) with X (2) 0 ∼ µ θ 1 , where

µ θ 1 = 1 θ 1 θ 1 0 δ Y s d s and θ 2 -θ 1 = τ (2)
∂ the first hitting time of ∂ by X (2) ,

• for all k ≥ 1, given X (1) , X (2) , . . . , X (k) ,(X (k+1) t ) t ∈R + is a realisation of the process (X t ) t ∈R + with X (k+1) 0 ∼ µ θ k , where

µ θ k = 1 θ k θ k 0 δ Y s d s and θ k+1 -θ k = τ (k+1)
∂ the first hitting time of ∂ by X (k+1) .

We also define, for all t ∈ R + ,

µ t = 1 t t 0 δ Y s d s, i.e. µ t ( f ) = 1 t t 0 f (Y s ) d s, ∀ f ∈ B b (E ).
This process has been studied in several situations, with the main goal of proving an almost sure convergence result for the occupation measure µ t when t → +∞. In the finite state space case and in a discrete time setting, Aldous, Flannery and Palacios [START_REF] Aldous | Two applications of urn processes the fringe analysis of search trees and the simulation of quasi-stationary distributions of markov chains[END_REF] solved this problem by showing that the proportion of colours in a Pólya urn type process converges almost surely to the left eigenfunction of the replacement matrix, which was also identified as the quasi-stationary distribution of a corresponding Markov chain. Under a similar setting but using stochastic approximation techniques, Benaïm and Cloez [START_REF] Benaïm | A stochastic approximation approach to quasi-stationary distributions on finite spaces[END_REF] and Blanchet, Glynn and Zheng [START_REF] Blanchet | Analysis of a stochastic approximation algorithm for computing quasi-stationary distributions[END_REF] independently proved the almost sure convergence of the occupation measure µ t toward the quasi-stationary distribution of X . These works have since been generalised to the compact state space case by Benaïm, Cloez and Panloup [START_REF] Benaim | Stochastic approximation of quasi-stationary distributions on compact spaces and applications[END_REF] under general criteria for the existence of a quasi-stationary distribution for X . The case of continuous time diffusion processes with smooth bounded killing rate on compact Riemannian manifolds has been recently solved by Wang, Roberts and Steinsaltz [START_REF] Wang | An Approximation Scheme for Quasistationary Distributions of Killed Diffusions[END_REF], who show that a similar algorithm with weights also converges toward the quasi-stationary distribution of the underlying diffusion process.

In Section 9.1, we solve the question of convergence of the occupation measure toward the quasi-stationary distribution of X when this process is a uniformly elliptic diffusion process evolving in an open bounded connected open set D with C 2 boundary ∂D, with hard killing when the process hits the boundary. This answers positively the open problem stated in Section 8 of [START_REF] Benaim | Stochastic approximation of quasi-stationary distributions on compact spaces and applications[END_REF].

In section 9.2, we state such a convergence result for processes with smooth and bounded killing rate evolving in unbounded spaces using a measure-valued Pólya process representation of this reinforced algorithm. This result strongly relies on the convergence of Measure-valued Pólya processes as stated in Chapter 8.

Stochastic approximation of a quasi-stationary distributions for diffusion processes in a bounded domain

Let (X t ) t ∈R + be a diffusion process in a connected bounded open set D of R d , d ≥ 2 with C 2 boundary ∂D and absorbed at ∂D. We assume that X is solution to the SDE

d X t = σ(X t )d B t + b(X t )d t , (9.1)
where B is a r -dimensional Brownian motion, b : D → R d is bounded and continuous and σ : D → R d ×r is continuous, σσ * is uniformly elliptic and for all ρ > 0, sup

x,y∈D, |x-y|=ρ |σ(x) -σ(y)| 2 ρ ≤ g (ρ) (9.2)
for some function g such that 1 0 g (r )d r < ∞. Note that, in this case, the process (Y t , µ t ) t ≥0 described in the introduction is well-defined since one can prove that θ k → +∞ a.s. [START_REF] Benaim | Stochastic approximation of quasi-stationary distributions on compact spaces and applications[END_REF]Lemma 8.1].

Recall that, the results of Chapter 5 entail that, under the above regularity assumptions, the killed diffusion process X satisfies Assumption A of Chapter 1 and hence that it admits a unique quasi-stationary distribution α. We denote by λ 0 the positive constant such that P α (t < τ ∂ ) = exp(-λ 0 t ) for all t ≥ 0. Remark 9.1. All the results of this chapter, and in particular the next one, can be extended to the one-dimensional diffusion processes studied in Chapter 4 and to the diffusion processes on compact manifolds studied in Chapter 5.

STOCHASTIC APPROXIMATION OF A QUASI-STATIONARY DISTRIBUTIONS FOR DIFFUSION PROCESSES WITH

We obtain the following Theorem 9.1. For all bounded measurable function f : D → R, one has

µ t f ----→ t →+∞ α f a.s.
Moreover, θ n /n → 1/λ 0 almost surely when n → +∞.

The proof of this result, detailed in [START_REF] Benaïm | Stochastic approximation of quasistationary distributions for diffusion processes in a bounded domain[END_REF], relies, among other methods, on the properties of the Green operator A on B b (E ) for X , defined as

A f (x) = E x τ ∂ 0 f (X s ) d s = ∞ 0 P s f (x) d s. (9.3)
Assuming that X satisfies Conditions A of Chapter 1 (and hence in the present situation), one easily checks that this operator is bounded on (B b (E ), • ∞ ). For all µ ∈ M 1 (E ), we also define the notation µA f = E A f (x) µ(d x). We obtain the following Proposition 9.2. Assume that Condition A of Chapter 1 is satisfied. Then, for all µ ∈ M 1 (E ), all f ∈ B b (E ) and all n ≥ 1, we have

µA n f - α( f )µ(η) λ n 0 ≤ f ∞ C (λ 0 + γ) n , ( 9.4)
for some positive constant C , γ > 0. We also have for some constant B > 0

µA n µA n 1 -α T V ≤ B µ(η) λ 0 λ 0 + γ n (9.5)
and, for all t ≥ 0,

µe t A µe t A 1 -α T V ≤ B µ(η) e -t
γ λ 0 (λ 0 +γ) . (9.6)

Stochastic approximation of a quasi-stationary distributions for diffusion processes with soft killing

Let (X t ) t ∈R + be the solution in E = R d to the stochastic differential equation

dX t = dB t + b(X t )dt ,
where B is a standard d -dimensional Brownian motion and b : R d → R d is locally Hölder continuous in R d . We assume that X is subject to an additional soft killing κ : x → [0, +∞), which is continuous, uniformly bounded and such that κ ≥ 1. Note that the quasi-stationary distribution of X with killing rate κ is the same as the quasi-stationary distribution of X with a killing rate κ -1. We also assume that lim sup

|x|→+∞ 〈b(x), x〉 |x| < - 3 2 κ 1 /2 ∞ ,
so that the process X admits a unique quasi-stationary distribution ν QSD such that ν QSD •V < +∞, where

V : x ∈ R d → exp( κ 1 /2
∞ |x|) (this is an application of the results of Chapter 5). We consider the self-interacting process (Y t ) t ≥0 evolving with the same dynamic of X but, at rate κ, instead of being killed, it jumps to a new position chosen accordingly to its empirical occupation measure, as described in the beginning of this chapter.

Chapter 10

Coarse Ricci curvature

Let (E , d ) be a Polish space. Fix N ≥ 1 and consider a continuous time pure jump particle system of N particles ( Xt ) t ≥0 = (X 1 t , . . . , X N t ) t ≥0 evolving in E N . We assume that the process is non-explosive and that its infinitesimal generator L is given, for all x = (x 1 , . . . , x N ) ∈ E N and any bounded measurable function f :

E N → R, by L f ( x) = N i =1 E f (x 1 , . . . , x i -1 , y, x i +1 , . . . , x N ) -f (x 1 , . . . , x n ) F i x i , x, d y ,
where the terms F i (x i , x, •) are finite non-negative measures on E , measurable with respect to x i and x and such that, for some (and hence for all) x ∈ E N , d (x i , y)F i (x i , x, d y) < ∞. Our aim is to provide, using coupling methods, a lower bound for the coarse Ricci curvature of X evolving in E N endowed with the metric

d ( x, ȳ) = 1 N N i =1 d (x i , y i ), ∀ x = (x 1 , . . . , x N ), ȳ = (y 1 , . . . , y N ) ∈ E N .
We recall that the coarse Ricci curvature of the continuous-time Markov process X is the largest constant σ satisfying, for all t ≥ 0,

W d P( Xt ∈ • | X0 = x), P( Xt ∈ • | X0 = ȳ) ≤ e -σt d ( x, ȳ), ∀ x, ȳ ∈ E N ,
where W d denotes the Wasserstein distance. All the results of this chapter originally appeared in [START_REF] Villemonais | Lower bound for the coarse ricci curvature of continuous-time pure-jump processes[END_REF].

A lower bound on σ provides a measure of the instantaneous convergence rate to a unique stationary distribution (see for instance [START_REF] Chen | Optimal Markovian couplings and applications[END_REF]). This concept is closely related to the optimal coupling theory developed by Chen (see for instance [START_REF] Chen | Optimal Markovian couplings and applications[END_REF][START_REF] Chen | Eigenvalues, inequalities, and ergodic theory[END_REF]). It also entails spectral gap inequalities and concentration inequalities (see [START_REF] Ollivier | Ricci curvature of Markov chains on metric spaces[END_REF][START_REF] Joulin | Curvature, concentration and error estimates for Markov chain Monte Carlo[END_REF][START_REF] Joulin | Poisson-type deviation inequalities for curved continuous-time Markov chains[END_REF][START_REF] Joulin | A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature[END_REF][START_REF] Veysseire | Coarse Ricci curvature for continuous-time Markov processes[END_REF][START_REF] Veysseire | Coarse Ricci curvature of Markov processes[END_REF][START_REF] Djellout | Transportation cost-information inequalities and applications to random dynamical systems and diffusions[END_REF]). We refer the reader to [START_REF] Alfonsi | Evolution of the Wasserstein distance between the marginals of two Markov processes[END_REF] for a different approach, based on Kantorovich potentials. We also refer the reader to [START_REF] Chafaï | Intertwining and commutation relations for birth-death processes[END_REF]Section 3.2] for a link between coarse Ricci curvature and functional inequalities. For general state space processes and for diffusion processes, we refer the reader to the works of Veysseire, where a systematic study of the coarse Ricci curvature has been conducted (see [START_REF] Veysseire | Coarse Ricci curvature for continuous-time Markov processes[END_REF][START_REF] Veysseire | Coarse Ricci curvature of Markov processes[END_REF]) with nice implications on concentration inequalities and spectral gap estimates. Let us also mention that estimates on the coarse Ricci curvature of a continuous time process immediately provide estimates for the curvature of its discrete time included Markov chain, which also implies several interesting properties (see the works of Ollivier [START_REF] Ollivier | Ricci curvature of Markov chains on metric spaces[END_REF][START_REF] Ollivier | A survey of Ricci curvature for metric spaces and Markov chains[END_REF] and references therein).

Reconstituting transport distance bounds on Markov chains on product spaces from the behaviour of marginals via suitable couplings was already used by Talagrand and Marton, see for instance [START_REF] Marton | A measure concentration inequality for contracting Markov chains[END_REF][START_REF] Marton | Logarithmic Sobolev inequalities in discrete product spaces: a proof by a transportation cost distance[END_REF] and references therein. These methods also apply to Markov processes that are not of pure jump types and to cost functions d that are not distance functions. For diffusion processes, we refer the reader to [START_REF] Chen | Coupling methods for multidimensional diffusion processes[END_REF] and to [START_REF] Von Renesse | Transport inequalities, gradient estimates, entropy, and Ricci curvature[END_REF]Corollary 1.4] for necessary and sufficient conditions in the case where the drift derives from a potential. We also refer the reader to [START_REF] Eberle | Reflection coupling and Wasserstein contractivity without convexity[END_REF][START_REF] Eberle | Reflection couplings and contraction rates for diffusions[END_REF] with an introduction to parallel coupling and the construction of ad hoc distances on the state space. Computation of the coarse Ricci curvature for diffusion processes on manifold has also been studied by Veysseire [START_REF] Veysseire | Coarse Ricci curvature of Markov processes[END_REF]. For piecewise deterministic processes, we refer the reader to [75, Lemma 5.2] and [START_REF] Chafaï | On the long time behavior of the TCP window size process[END_REF]Theorem 2.3]. Original coupling approaches are also provided in [START_REF] Majka | Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling[END_REF][START_REF] Majka | Coupling and exponential ergodicity for stochastic differential equations driven by L\'{e}vy processes[END_REF][START_REF] Cattiaux | Semi log-concave Markov diffusions[END_REF].

Below, we state our main result in Section 10.1, provide applications to the case N = 1 in Section 10.2, and conclude with an application to a simple model of agents in interaction in Section 10.3. We refer the reader to the original paper [START_REF] Villemonais | Lower bound for the coarse ricci curvature of continuous-time pure-jump processes[END_REF] for the proofs, additional details and references. Therein, we also compute a lower bound for the coarse Ricci curvature of other interacting particle systems, including zero range dynamics, Fleming-Viot type systems and some of their natural extensions, birth and death processes in mean-field type interaction, and finally systems of particles whose jump measures admit a density with respect to the Lebesgue measure or the counting measure.

Main result

One of the difficulties of the continuous time setting, compared to the discrete time setting [START_REF] Ollivier | Ricci curvature of Markov chains on metric spaces[END_REF], is that the jump measures do not, in general, share the same mass. In order to overcome this difficulty, we introduce the family of functions (J

x,y d ) x,y∈E from M d (E ) 2 to R, defined for all m 1 , m 2 ∈ M d (E ) by J x,y d (m 1 , m 2 ) = W d (m 1 + m 2 (E )δ x , m 2 + m 1 (E )δ y ) -(m 1 (E ) + m 2 (E ))d (x, y),
where δ x denotes the Dirac measure at point x and m 2 (E )δ x is the product of the scalar m 2 (E ) by δ x . Note that the finite measures m 1 and m 2 can have different masses. Proper generalisations of the Wasserstein distance between measures with different masses already exist in the literature (such as the flat metric [START_REF] Dudley | Real analysis and probability[END_REF] and the generalised W 1,1 1 Wasserstein distance [START_REF] Piccoli | On properties of the generalized Wasserstein distance[END_REF], see also the recent developments in [START_REF] Chizat | Unbalanced optimal transport: dynamic and Kantorovich formulations[END_REF][START_REF] Kondratyev | A new optimal transport distance on the space of finite Radon measures[END_REF]182] with applications to convergence of measure valued dynamical systems), but are not directly relevant in our context. Theorem 10.1. Consider the Markov process X with generator L . Then there exists a coupling operator L c of L such that, for all x, ȳ ∈ E N ,

L c d ( x, ȳ) = 1 N N i =1 J x i ,y i d (F i (x i , x, •), F i (y i , ȳ, •)).
In particular, the coarse Ricci curvature σ of the process ( Xt ) t ≥0 satisfies

σ ≥ -sup x, ȳ∈E N 1 N N i =1 J x i ,y i d (F i (x i , x, •), F i (y i , ȳ, •)) d ( x, ȳ) .
One main feature of this result is that one does not need to build an explicit coupling between processes to conclude. This is particularly useful for involved jump matrices.

Fix x, y ∈ E . We now provide some interesting properties of the functional J

(m 1 + n 1 , m 2 + n 2 ) ≤ J x,y d (m 1 , m 2 ) + J x,y d (n 1 , n 2 ). (10.2)
The following inequality is in general a crude estimate, but it is in some cases useful and sharp (such as for one dimensional birth and death processes, see Example 10.2).

Proposition 10.3. We have, for all m

1 , m 2 ∈ M d (E ), J x,y d (m 1 , m 2 ) ≤ E [d (u, y) -d (x, y)] m 1 (d u) + E [d (x, v) -d (x, y)] m 2 (d v).
The following property implies in particular that, if m 1 and m 2 are two probability measures, then J where a, b are taken in the set of real numbers such that m 1 + aδ x and m 2 + bδ y are non-negative measures on E with equal mass, i.e. such that m 1 (E )+a ≥ 0, m 2 (E )+b ≥ 0 and m 1 (E )+a = m 2 (E )+b. In addition, the minimum is attained for all a ≥ m 2 (E ) (or equivalently b ≥ m 1 (E )).

x,y d (m 1 , m 2 ) is smaller than W d (m 1 , m 2 )-d (x, y). It also implies that, for measures m 1 and m 2 on E such that m 1 (E ) ≥ m 2 (E ), then J x,y d (m 1 , m 2 ) ≤ W d (m 1 , m 2 +(m 1 (E )-m 2 (E ))δ y )-m 1 (E )d (x, y).

First applications in the particular case N = 1

In this section, we state our result in the simpler case N = 1. The following corollary is an immediate consequence of Theorem 10.1.

Corollary 10.5. Let L be the infinitesimal generator of a pure jump non-explosive Markov process on E defined, for any bounded measurable function f

: E → R, by L f (x) = E ( f (u) -f (x)) q(x, d u), ∀x ∈ E ,
where (q(x, d u)) x∈E is a jump kernel of finite non-negative measures. Then the coarse Ricci curvature σ of the Markov process generated by L satisfies

σ ≥ -sup x,y∈E J x,y d q(x, •), q(y, •) d (x, y) .
For continuous time birth and death processes, Mielke [START_REF] Mielke | Geodesic convexity of the relative entropy in reversible Markov chains[END_REF] recently computed a lower bound for an other notion of discrete Ricci curvature, related to the fact that the evolution of the law of a continuous time birth and death process can be described through a gradient flow system. To relate both definitions is still an open problem, but the lower bound obtained in Mielke's work has a similar expression (see Section 5 in [START_REF] Mielke | Geodesic convexity of the relative entropy in reversible Markov chains[END_REF] and Example 10.2 below) and may be a good starting point to compare both approaches. This example has also been considered by Fathi and Maas in [START_REF] Fathi | Entropic Ricci curvature bounds for discrete interacting systems[END_REF]Theorem 4.1] in the setting of Entropic Ricci curvature.

Example 10.1. In this example, d is the trivial distance on E (so that W d is the total variation distance). Assume that there exist a non-negative measure ζ on E and a measurable function

α : E × E → R + such that q(x, d z) = α(x, z) ζ(d z), ∀x ∈ E .
Then one carefully checks that

J x,y d (q(x, •), q(y, •)) = - E α(x, z) ∧ α(y, z) ζ(d z) -α(y, x)ζ({x}) -α(x, y)ζ({y}).
In particular, the coarse Ricci curvature σ of the process satisfies

σ ≥ inf x =y E α(x, z) ∧ α(y, z) ζ(d z) + α(y, x)ζ({x}) + α(x, y)ζ({y}) .
Example 10.2. Consider the particular case where E = Z + and L is the infinitesimal generator of a birth and death process with birth rates (b x ) x∈Z + and death rates (d x ) x∈Z + , all positive but d 0 = 0. In this case, for all x, y ∈ Z + ,

q(x, y) =        b x if y = x + 1 d x if x ≥ 1 and y = x -1 0 otherwise
We also assume that the distance d is given by d (x, y) = x-1 k=0 u k -y-1 k=0 u k , where (u k ) k≥0 is a sequence of positive numbers. After careful computations, we deduce from Proposition 10.3 and Corollary 10.5 that the coarse Ricci curvature σ of the process satisfies

σ ≥ inf x∈Z + b x + d x+1 -d x u x-1 u x -b x+1 u x+1 u x .
In [START_REF] Chen | Optimal Markovian couplings and applications[END_REF], [START_REF] Joulin | A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature[END_REF] and [START_REF] Chafaï | Intertwining and commutation relations for birth-death processes[END_REF], it is shown that there is equality in the above equation. This implies that, at least in some cases, Corollary 10.5 and hence Theorem 10.1 are sharp. Note that, in this case, Proposition 10.3 provides an explicit expression for the quantity J

x,y d (q(x, •), q(y, •)).

Example 10.3. The choice of the classical coupling (i.e. the use of Proposition 10.3) in the previous example was judicious because the measures involved for a birth and death process are stochastically ordered. This is not the case in the present example, where we assume that

q(x, y) =        b x if y = x + 2 d x if y = x -1, 0 otherwise.
In this case, using a slight extension of [START_REF] Vallender | Calculations of the Vasseršteȋn distance between probability distributions on the line[END_REF], one obtain that the coarse Ricci curvature of the process satisfies

σ ≥ inf x∈Z + b x + d x+1 -d x u x-1 u x -|b x+1 -b x | u x+1 u x -b x+1 u x+2 u x .

A model of interacting agents

We study now a simple model of interacting agents whose individual behaviour is influenced in a non-linear way by the behaviour of the other agents: each agent wanders randomly in a complete graph and also changes its position to a new one, depending on a function of the number of agents in this position. This dynamic is modelled by a system of N particles evolving in the complete finite graph E of size #E ≥ 2: we assume that there exist T > 0 and a function f : [0, 1] → R + such that any agent jumps from state x to y ∈ E with the following rate

x → y with rate

T #E + f Number of agents in y N . ( 10.3)
In this model, T is the temperature of the system and f is a preference function. For instance, with an increasing function f with high convexity, the agents will give higher preferences to positions that are already favoured by many other agents; with a larger temperature T , the agents act more independently. Our aim is to determine characteristics of f and values of T for which a herd behaviour occurs or not in this model. By a herd behaviour, we mean a meta-stable state of the whole particle system where a majority of the agents share the same position for a long time. Note that this model can be written in the settings of the present paper, by setting, for all x, y ∈ E and x ∈ E N ,

F i (x, x, {y}) = T #E + f N i =1 1 x i =y N , ∀y ∈ E .
This process is exponentially ergodic and the marginal of its empirical stationary distribution is the uniform probability measure on E (this is an immediate consequence of the symmetry of the state space and of the dynamic of the particles). The existence of the phase without herd behaviour is obtained using the results of Section ??, while the existence of the phase with herd behaviour is proved using large deviation results obtained in [START_REF] Dupuis | Large deviations for Markov processes with discontinuous statistics. I. General upper bounds[END_REF][START_REF] Dupuis | Large Deviation Principle For Finite-State Mean Field Interacting Particle Systems[END_REF]. Since the publication of the original article, Erbar, Fathi and Schlichting [START_REF] Erbar | Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces[END_REF] have studied this model (among others) in the settings of Entropic Ricci curvature.

In this first proposition, we assume that f is Lipschitz and provide a coarse Ricci curvature's lower bound independent of N . In the next proposition, we assume that f is a non-decreasing strictly convex function and show that, for small values of T , the process exhibits a meta-stable state, so that the agents have a herd behaviour for large values of N : if all the agents start with the same choice x ∈ E , then, during a time of order exp(cN ), where c > 0, x is favoured by the majority of the agents. This happens despite the fact that, during this interval of time, most agents have changed their choices at multiple times. If the temperature is sufficiently small, namely if 0 ≤ T < m * #E z * #E -1 , then there exists a positive constant δ > 0 such that, for all x ∈ E , In order to check that m * > 0 in the above result, one simply uses the fact that f is strictly convex with f (0) = 0, so that, for all z ∈ (1/2, 1), f (1z)/(1z) < f (z)/z. Hence we deduce from Proposition 10.6 and Proposition 10.7 that

• if T > 2 -1/#E , then the Wasserstein curvature of the particle system is positive (bounded from below by T -2 + 1/#E ) and the system of agents does not exhibits a herd behaviour;

• if 0 ≤ T < #E /((3 + 3 3)#E -6 3), then the system of agents exhibits a herd behaviour.

Chapter 11

Maintenance of biodiversity and perpetual integrals

Often, demogenetics model are obtained from a specific scaling of the parameters in the individualbased model, leading to a stochastic differential equation with a diffusion term proportional to the square root of the size of the population (Feller type diffusion processes). Other scaling will lead to different coefficients and we refer the reader to [START_REF] Coron | Stochastic modeling and eco-evolution of a diploid population[END_REF][START_REF] Bansaye | of Mathematical Biosciences Institute Lecture Series[END_REF][START_REF] Coron | Slow-fast stochastic diffusion dynamics and quasi-stationarity for diploid populations with varying size[END_REF] for an in-depth discussion of such models.

Our aim here is to emphasise the importance of this diffusion term in fixation problems. Our main question is whether, in a given demogenetic model, one allele gets fixed almost surely before the population goes extinct. In a collaboration with Camille Coron and Sylvie Méléard [START_REF] Coron | Impact of demography on extinction/fixation events[END_REF], we prove that this is the case almost surely for Feller type diffusion coefficients. We also show that, in fact, it depends on the behaviour of the diffusion coefficient near extinction in the equation satisfied by the population size, as detailed below. The next theorem notably highlights the major effect of the demography on the maintenance of genetic diversity by giving a necessary and sufficient criterion ensuring almost sure fixation before extinction. The main tool of the proof has its own interest, since it derives from finiteness criteria for perpetual integrals, which we detail at the end of this chapter.

Demography and maintenance of biodiversity

Let us consider the process (N t , X t ) t ∈R + solution to the system of stochastic differential equations

d N t = σ(N t ) d B t + N t (ρ -αN t )d t , N 0 > 0, α > 0 d X t = X t (1-X t ) f (N t ) dW t , t < T N 0+ , (11.1)
where B,W are independent one-dimensional Brownian motions, σ : (0, +∞) → (0, +∞) is locally Lipschitz and f : (0, +∞) → (0, +∞) is locally bounded away from 0 and where

T N 0+ := lim n→+∞ T N 1/n
denotes the extinction time of the population. The system admits a pathwise unique strong solution up to the extinction time and we denote by , with r = -1 and c = 0.1. We plot the number of simulations for which fixation does not occur before extinction.

T f := inf{t ≥ 0, X t ∈ {0, 1}} 0
(0, +∞), ∀y ∈ (0, +∞), P z (T y < +∞) > 0). This implies that for any a < b ∈ (0, +∞) and a ≤ z ≤ b, E z (T a ∧T b ) < +∞ and we can associate with Z a scale function s and a locally finite speed measure m on [0, +∞) (see Chapter 4). We moreover assume that for all z ∈ (0, +∞),

P z (T 0 = T 0 ∧ T e < +∞) = 1, (11.3)
where T e is the explosion time (this is equivalent to s(+∞) = +∞, s(0) > -∞ and 0+ (s(y)-s(0)) m(d y) < +∞). Since the function s is defined up to a constant, we choose by convention s(0) = 0 as soon as s(0) > -∞.

Theorem 11.2. Let (Z t ) t ∈R + be a regular diffusion process on [0, +∞) with scale function s and speed measure m on (0, +∞) satisfying (11.3). Let also f be a non-negative locally integrable function on (0, +∞). Then, for all z > 0 and all n ≥ 1, (11.4)

In the particular case α = 1, the authors of [START_REF] Foucart | Stable continuous-state branching processes with immigration and Beta-Fleming-Viot processes with immigration[END_REF] propose an other approach based on selfsimilarity properties.

Chapter 12

The individual's signature of telomere length distribution.

In a recent collaboration with Éliane Albuisson (CHRU of Nancy and IECL), Athanase Benetos (CHRU of Nancy), Anne Gégout-Petit (IECL), Daphné Germain (former student at École des Mines de Nancy) and Simon Toupance (CHRU of Nancy), we studied the evolution of telomere length distribution over time in adults. This statistical study was published in [START_REF] Toupance | The individual's signature of telomere length distribution[END_REF].

A short introduction to telomere lengths

Telomeres are specialised non-coding double-stranded repetitive DNA-protein complexes that form protective caps on the ends of chromosomes. They safeguard their extremity and maintain genomic integrity by allowing cells to distinguish telomeres from sites of DNA damage [START_REF] Griffith | Mammalian telomeres end in a large duplex loop[END_REF][START_REF] De Lange | How telomeres solve the end-protection problem[END_REF]. Telomere length displays progressive shortening in replicating somatic cells with age [START_REF] Lindsey | In vivo loss of telomeric repeats with age in humans[END_REF][START_REF] Harley | Telomeres shorten during ageing of human fibroblasts[END_REF]. Eventually cells will acquire critically short and dysfunctional telomeres that, consequently, activate a DNA damage response and growth arrest known as replicative senescence [START_REF] Shay | Defining the molecular mechanisms of human cell immortalization[END_REF][START_REF] Di Fagagna | A dna damage checkpoint response in telomere-initiated senescence[END_REF]. Therefore, all somatic cells have limited cell proliferation capacity called the Hayflick limit [START_REF] Hayflick | The serial cultivation of human diploid cell strains[END_REF][START_REF] Olovnikov | A theory of marginotomy: the incomplete copying of template margin in enzymic synthesis of polynucleotides and biological significance of the phenomenon[END_REF]. Short leukocyte telomere length is associated with many degenerative diseases linked to ageing and with higher mortality risk. Epidemiological studies use leukocyte telomere length to examine the potential role of telomere length in health and disease. It is known that leukocyte telomere length decreases with age and thus is considered as a biomarker of chronological ageing [START_REF] Rizvi | Telomere length variations in aging and age-related diseases[END_REF][START_REF] Blackburn | Human telomere biology: a contributory and interactive factor in aging, disease risks, and protection[END_REF][START_REF] Baird | The extent and significance of telomere loss with age[END_REF]. In humans, a long leukocyte telomere length is associated with better survival in the elderly [START_REF] Bakaysa | Telomere length predicts survival independent of genetic influences[END_REF][START_REF] Kimura | Telomere length and mortality: a study of leukocytes in elderly danish twins[END_REF][START_REF] Fitzpatrick | Leukocyte telomere length and mortality in the cardiovascular health study[END_REF][START_REF] Deelen | Leukocyte telomere length associates with prospective mortality independent of immune-related parameters and known genetic markers[END_REF] and a recent meta-analysis has indicated a strong relation between short telomeres and mortality risk, particularly at younger ages [START_REF] Boonekamp | Telomere length behaves as biomarker of somatic redundancy rather than biological age[END_REF]. A shorter leukocyte telomere length is associated with many degenerative diseases linked to ageing such as cardiovascular disease [START_REF] Haycock | Leucocyte telomere length and risk of cardiovascular disease: systematic review and metaanalysis[END_REF][START_REF] D´mello | Association between shortened leukocyte telomere length and cardiometabolic outcomes: systematic review and meta-analysis[END_REF], neurodegenerative disease [START_REF] Cai | Telomere shortening and alzheimer's disease[END_REF][START_REF] Forero | Telomere length in parkinson's disease: a meta-analysis[END_REF] and metabolic diseases [START_REF] D´mello | Association between shortened leukocyte telomere length and cardiometabolic outcomes: systematic review and meta-analysis[END_REF][START_REF] Balasubramanyam | Telomere shortening & metabolic/vascular diseases[END_REF].

Several methods have been developed to measure the length of telomere repeats from cells or extracted DNA [START_REF] Aubert | Telomere length measurement-caveats and a critical assessment of the available technologies and tools[END_REF]. The most used are: Southern blot analysis of the terminal restriction fragments length (TRF); quantitative PCR (qPCR) amplification of telomeric DNA, expressed as the ratio of telomere repeats relative to a single copy gene; single telomere length analysis (STELA), a PCR and Southern blot combining method that measures telomere lengths from individual chromosomes; and fluorescent in situ hybridisation (FISH) techniques, quantitative FISH (qFISH) based on microscopy and flow FISH using flow cytometry. There are ongoing debates as to which method is the most suitable to measure telomere's length in clinical studies, particularly between TRF and qPCR [START_REF] Elbers | Comparison between southern blots and qpcr analysis of leukocyte telomere length in the health abc study[END_REF][START_REF] Martin-Ruiz | Reproducibility of telomere length assessment: an international collaborative study[END_REF][START_REF] Verhulst | Commentary: The reliability of telomere length measurements[END_REF]. The advantages of the qPCR method are the high throughput and low cost but TRF is considered the "gold standard" since it displays less variability [START_REF] Aviv | Impartial comparative analysis of measurement of leukocyte telomere length/dna content by southern blots and qpcr[END_REF], gives absolute values of telomere length and gives access to telomere length distribution [START_REF] Baird | New developments in telomere length analysis[END_REF]. However, the vast majority of studies using TRF measurements only use mean telomere length and not distribution as a result. Assessing telomere length distribution with TRF measurement can give access to new information concerning telomere length dynamics since a cell's shortest telomeres and the "load" (amount) of short telomeres appear to play a role independently of mean telomere length [START_REF] Bendix | The load of short telomeres, estimated by a new method, universal stela, correlates with number of senescent cells[END_REF][START_REF] Hemann | The shortest telomere, not average telomere length, is critical for cell viability and chromosome stability[END_REF]. Mean telomere attrition rates do not capture changes in telomere distribution that may play a role in pathology development (see Figure 12.2). The aim of the study described in this chapter was to analyse the distribution of telomere lengths and its evolution in time, using data generated by TRF in a longitudinal study in which two sequential measurements of telomeres were performed at the beginning and the end of the study corresponding to a mean time distance of 8 years.

Telomere length signature

A quick look at the datas at the beginning and at the end of the study, suggests that the telomere length distribution conserves the same shape at base line and at follow up. To study the conservation of the shape, the leukocyte telomere length distribution have then been translated in order to have the same median, and then drowned (see Figure 12.3). For most patients (although not systematically), the translated distributions at time 1 and at time 2 coincide very well.

In order to measure numerically the distance between distributions, we used the Kolmogorov distance between distributions, defined as the infinite norm distance between their cumulative functions.

To show how the shape of the distributions of one Subject is well conserved between the two times, we have computed the Kolmogorov distances between successive distributions for each of the 72 Subjects. The distribution of these 72 distances is given in the normalised histogram in red of the Figure 12.4. We have also computed the 72×71 Kolmogorov distances between the translated telomere length distribution at time 1 for one Subject and time 2 for another one. The normalised histogram is given in blue at Figure 12.4. We see clearly that intra-subject distances seem to be lower than the inter-subjects distances.

We have performed a T -test to confirm the tendency: it strongly rejects the equality between the observed mean of the intra-Subject distances (mean 1 = 0.0258) and the mean of the inter-Subject distances that we consider like a theoretical expectancy (mean 2 = 0.0639). We can say that the shapes of the leukocyte telomere length distributions of one Subject are significantly closer than two shapes of two different Subjects.

Our conclusion is that leukocyte telomere length distribution characterises an individual and

Discussion

The contribution of this study is to show a strong stability over time not only of ranking but also of the telomere length distribution. The telomere length signature could become a new criterion to describe patients since we have seen that two individuals can have the same mean leukocyte telomere length but different distributions. We also acknowledge limitations of this study. First, the sample size is modest with less than hundred Subjects. Second, our cohort comprised participants who were all above 60 years of age at the beginning of the study. Third, the follow-up duration was only 8 years and we can't conclude on variation on longer period. However, the clear results obtained on only 72 patients are strong enough to overcome difficulties and sources of errors linked to this type of studies: blood samples have been taken 8 years apart by different nurses and DNA extracted 8 years apart by different researchers. In the future, in clinical studies, maybe that telomere length signature could capture new associations of telomere dynamics with clinical parameters or disease markers or help to better clusterise patients.

In an ongoing collaboration, we are currently working on a larger cohort (around 500 individuals of all ages), which allowed us to confirm the finding of this study and to provide new insights on the evolution of telomere length distributions over time. Our next step will be to relate, when possible, the telomere length signature of the patients to their medical records.

  • D. Villemonais. Lower bound for the coarse ricci curvature of continuous-time pure-jump processes. J. Theoret. Probab. Probability, May 2019. • C. Coron, S. Méléard, and D. Villemonais. Impact of demography on extinction/fixation events. J. Math. Biol., 78(3):549-577, Feb 2019. • S. Toupance, D. Villemonais, D. Germain, A. Gegout-Petit, E. Albuisson, and A. Benetos. The individual's signature of telomere length distribution. Sci. Rep., 9(1):685, 2019.

  and D. Villemonais. Convergence of the Fleming-Viot process toward the minimal quasi-stationary distribution. ArXiv e-prints, Oct. 2018. xiii • E. Horton, A. E. Kyprianou, and D. Villemonais. Stochastic Methods for the Neutron Transport Equation I: Linear Semigroup asymptotics. ArXiv e-prints, Oct. 2018. • A. M. G. Cox, E. L. Horton, A. E. Kyprianou, and D. Villemonais. Stochastic Methods for Neutron Transport Equation III: Generational many-to-one and k e f f . ArXiv e-prints, Sept. 2019.

Proposition 1 . 3 .

 13 Suppose that Assumption A holds true. If f ∈ B b (E ∪ {∂}) is a right eigenfunction for L for an eigenvalue λ, then either 1. λ = 0 and f is constant, 2. or λ = -λ 0 and f

16CHAPTER 2 .

 2 NON-UNIFORM CONVERGENCE TOWARD A QUASI-STATIONARY DISTRIBUTIONLet us now recall the statement of[START_REF] Ferrari | R-positivity, quasi-stationary distributions and ratio limit theorems for a class of probabilistic automata[END_REF] Theorem 1]. Here R is defined as 1

Figure 2 . 1 :

 21 Figure 2.1: Transition graph displaying the relation between the sets D 1 , D 2 , D 3 and ∂.

  This formula can be extended to l = a or r = b by letting l and r tend to a and b respectively in the above expressions. For instance, if s(a) = 0 and s(b) = +∞, equation (4.1) remains valid with l = a, r = b, and G a,b (x, y) = 2 s(x) ∧ s(y).

  ensures the existence of a weak solution[START_REF] Kallenberg | Foundations of modern probability. Probability and its Applications[END_REF] Ch. 23] up to the exit time of (a, b)). In this case, one obtains the following semi-explicit expressions for the scale function and the speed measure (see for instance [232, Chapter VII, Section 3]) s(x) = (c,x) exp -(c,y) 2β(z)σ -2 (z) d z d y and m(d x) = 2d x s (x) σ 2 (x) (4.4) for any arbitrarily fixed point c ∈ (a, b).

  or the boundary b is an entrance boundary.(ii) Assumption A of Chapter 1 is satisfied.In this case, if either a or b is reachable, then there exists λ 0 > 0 such that P ν QSD (t < τ ∂ ) = e -λ 0 t and the unique quasi-stationary distribution ν QSD of the process is absolutely continuous with respect to m Y with Radon-Nikodym densityd ν QSD d m Y (x) = 2b a G a,b (x, y)ν QSD (d y), where G a,b is the Green function defined in the previous section. In addition, b a s(y)ν QSD (d y) < ∞, where s(y) = s(y)s(a) if a is reachable and b entrance, s(y) = s(b)s(y) if a is entrance and b is reachable and s(y) = (s(y)s(a))(s(b)s(y)) if both a and b are entrance. Remark 4.2. The Green function G a,b is well defined only if s(a) or s(b) is finite, which is the case when a or b is reachable. When both a and b are entrance boundaries, the classical theory of conservative Markov processes applies and one checks that α Y = m Y /m Y (a, b) and λ 0 = 0.

4. 2 .Proposition 4 . 3 .

 243 GENERAL ONE-DIMENSIONAL DIFFUSION PROCESSES WITHOUT NATURAL BOUNDARIES37 Assume that b is reachable or entrance, and that, for all x ∈ (a, c), I (x) := (a,x) (s(y)s(a)) m Y (d y) ≤ C (s(x)s(a)) ρ (4.5)

  , b). The fact that λ 0 does not depend on x nor [c, d ] is a consequence of the regularity of the process. This defines the decay parameter, analogously to (3.5) for birth and death processes. Let δ : (a, b) → R + and s : (a, b) → R be defined by δ(x) = exp -2

Example 4 . 3 .

 43 We consider the case (a, b) = (0, +∞), σ(x) = 1, β(x) = x sin x, and κ(x) = κ 0 1 -1 1+x

[ 58 , 44 CHAPTER 5 .

 58445 Section 4.4]. MULTI-DIMENSIONAL DIFFUSION PROCESSES a weak solution to (5.1) can be constructed up to the first exit time τ K c k of K k . The proper definition of the absorption time τ ∂ is then

Theorem 5 . 1 .

 51 Assume that there exist some constants C > 0, λ 1 > λ 0 , a C 2 (D) function ϕ : D → [1, +∞) and a subset D 0 ⊂ D closed for the relative topology on D such that sup x∈D 0 ϕ(x) < +∞ andL ϕ(x) ≤ -λ 1 ϕ(x) +C 1 x∈D 0 , ∀x ∈ D.(5.5)

  where b : R d → R d and σ : R d → R d ×r are locally Hölder continuous in R d and σ is locally uniformly elliptic on R d . Recall the definition (5.4) of λ 0 and assume that there exist constants C

1 . 5 . 3 .

 153 where b : R d → R d and σ : R d → R d ×r are locally Hölder continuous in R d , σ is locally uniformly elliptic on R d and 〈b(x), 1〉 〈x, 1〉 ------→ |x|→+∞ -∞, where 〈•, •〉 is the standard Euclidean product in R d and | • | is the associated norm. Then (5.7) is satisfied for ϕ(x) = 1+ x 1 +. . .+ x d and hence the process X absorbed at the boundary of D satisfies the assumptions of Theorem 5.Example Assume that D ⊂ R d is open connected and that d X t = b(X t )d t + d B t in D, where b : R d → R d is locally Hölder continuous in R d

  8) is implied by lim |x|→+∞ 〈b(x), x〉 |x| = -∞.

2 r

 2 where B is a r -dimensional Brownian motion, b : M → R d is bounded and continuous and σ : M → R d ×r is continuous, σσ * is uniformly elliptic and for all r > 0, sup x,y∈M , |x-y|=r |σ(x) -σ(y)|

1 . 10 . 2 .

 1102 are also useful to derive upper bounds and hence to apply Theorem 10.Proposition For all m 1 , n 1 , m 2 , n 2 ∈ M d (E ) and all α > 0, we have J x,y d (αm 1 , αm 2 ) = αJ

Proposition 10 . 4 .

 104 We have, for all m 1 , m 2 ∈ M d (E ), J x,y d (m 1 , m 2 ) = min a,b W d (m 1 + aδ x , m 2 + bδ y ) -(m 1 (E ) + a) d (x, y),

Proposition 10 . 6 .

 106 Assume that f is a Lipschitz function and define the Lipschitz constant of f as fLi p = sup u =v∈[0,1] | f (u)f (v)|/|u -v|.Then the coarse Ricci curvature σ of the particle system described above satisfiesσ ≥ T -2 f Li p + inf µ x∈E f (µ(x)),where the infimum is taken over the probability measures µ on E . Moreover, if f is monotone, thenσ ≥ Tf Li p + inf µ x∈E f (µ(x)).

Proposition 10 . 7 .

 107 Assume that f is a strictly convex function such that f (0) = 0, let z * ∈ (1/2, 1) such that z * = argmax z∈[1/2,1] f (z)z( f (z) + f (1z))and setm * = f (z * )z * ( f (z * ) + f (1z * )) > 0.

1 N 1 NNi

 11 log P ∃s ∈ [0, t ], µ N s (x) ≤ z * = N →+∞ O min δ(µ N 0 (x)z * ) 2 + , δε 2 -log t N ,uniformly in t ≥ 0 and where µ N s = =1 δ X i s .

Example 10 . 4 .

 104 Assume that f is an affine function : f (x) = ax + b for some a ∈ R and b ≥ 0 such that a + b ≥ 0. Then f is Lipschitz with f Li p = |a| and x∈E f (µ(x)) = a + b #E for any probability measure µ on E . Hence Proposition 10.6 implies that the Wasserstein curvature of the process is bounded from below by T + b #E + a -|a|. In particular, it is positive sinceT + b #E + a -|a|= T + b#E > 0 if a ≥ 0, T + b(#E -2) + 2(b + a) > 0 if a < 0,and hence the system of agents does not exhibit a herd behaviour. Example 10.5. Assume that f (x) = x 2 . Then f Li p = 2 and inf argmax z∈[1/2,1] z 2z(z 2 + (1z)

  s ) d s < +∞ P zalmost surely 0 + s(y) f (y) m(d y) = +∞ ⇐⇒ T 0 0 f (Z s ) d s = +∞ P zalmost surely. Let us give two examples for population size processes. Example 11.1 (Branching process with immigration). Let us consider the solution of the stochastic differential equation d N t = σ N t d B t + βd t , β > 0. Computing s and m as in (4.4) of Chapter 4, we easily obtain that (11.3) ⇐⇒ β/σ 2 < 1/2. Hence T 0 0 1 (N s ) α d s = +∞ a.s. ⇐⇒ α ≥ 1 ;

  s ) α d s < +∞ a.s. ⇐⇒ α < 1.

Example 11 . 2 .

 112 Logistic diffusion process. Let us consider the processd N t = N t d B t + N t (bc N t ) d t ; N 0 > 0,where c > 0. Then s(y) = y 0 e c z 2 -2bz d z and m(d y) = 2e -c y 2 +2b y y d y and 0 + s(y)m(d y) < +∞, since s(y) s (y) y → y→0 1.(Note that if c = 0, the condition s(+∞) = +∞ is not satisfied). It is immediate to check that (11.4) also holds.

Figure 12 . 1 :

 121 Figure 12.1: Human chromosomes (grey) capped by telomeres (black). Wikipedia.

Figure 12 . 2 :

 122 Figure 12.2: Telomere length distribution of two patients. The mean of the two distributions is the same, although the distributions differs.

Figure 12 . 3 :

 123 Figure 12.3: Telomere length distribution of two patients and their translation.

Figure 12 . 4 :

 124 Figure 12.4: Histogram of Kolmogorov distances

  and D. Villemonais. Uniform convergence of penalized time-inhomogeneous Markov processes. ESAIM Probab. Stat., 22:129-162, 2018.

• N.

Champagnat

, K. A. Coulibaly-Pasquier, and D. Villemonais. Criteria for Exponential Convergence to Quasi-Stationary Distributions and Applications to Multi-Dimensional Diffusions. Séminaire de Probabilités, XLIX:165-182, 2018. xii

  Figure 11.2: For different values of ε, we simulate 10000 trajectories of the 2-dimensional diffusion process(N , X ) such that d N t = N (1-ε) t d B 1 t + N t (rc N t )d t and d X t = X t (1-X t )

	Nombre d'extinctions avant fixation	200 400 600 800 1000
		0
		.1	0.2	0.3	0.4	0.5
			epsilon
				N t

All my publications can be accessed from my web-page https://www.normalesup.org/~villemonais/

This is proved in[START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF] with c

(µ 1 ) ∧ c 2 (µ 2 ) instead of c 2 (µ 1 ) ∨ c 2 (µ 2 ), however the proof of the result with the latter stronger estimate is almost identical.

The presentation is slightly different, although equivalent, to the one in the original article[START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF]. Our aim here is to make appear clearly the fact that Condition A from Chapter 1 is assumed to hold locally.

In the original article[START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF], we also consider situations where the random variables ξ n does not admits a bounded density with respect to Lebesgue's measure. The same arguments would also work, at the expense of additional technical difficulties, if X n+1 = f (X n ) + ξ n (X n ), where the sequence of random maps (x → ξ n (x)) n≥0 are i.i.d.

See Chapter 6 for the definition and properties of Q-processes

One could actually consider the case of explosive Markov processes, but then τ ∂ shall be defined as the infimum between the first hitting time of ∂ and the explosion time.

General processes with gradient estimates are studied in the original article[START_REF] Champagnat | Criteria for exponential convergence to quasi-stationary distributions and applications to multi-dimensional diffusions[END_REF].

E is the set defined in

under Assumption E (and similarly under Assumption F). We define it as equal to the state space E under Assumption A.

actually, this development can also be used to derive the asymptotic stability of unbounded semi-groups, in which case λ 0 may be negative.

Remerciements

Chapter 6

The Q-process and quasi-ergodic properties

Definition and existence

The Q-process is obtained as a Markov process conditioned to never be absorbed. The event "the process is never absorbed" has probability 0 in most practical cases, where we have P x (τ ∂ < +∞) = 1. Hence conditioning on this event is ill defined and the Q-process is instead obtained here as the limit, when T → +∞, of the process conditioned not be absorbed before time T . We show in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF][START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF] that it is well defined under Assumption A of Chapter 1 and under Assumption E of Chapter 2. It is also well defined under Assumption F. Earlier studies of Q-processes go back at least to [START_REF] Darroch | On quasi-stationary distributions in absorbing discrete-time finite Markov chains[END_REF][START_REF] Darroch | On quasi-stationary distributions in absorbing continuous-time finite Markov chains[END_REF], see also [START_REF] Pinsky | On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes[END_REF][START_REF] Breyer | A quasi-ergodic theorem for evanescent processes[END_REF][START_REF] Cattiaux | Quasi-stationary distributions and diffusion models in population dynamics[END_REF] and [START_REF] Jacka | Examples of convergence and non-convergence of Markov chains conditioned not to die[END_REF], where unusual behaviour of processes conditioned to survive are exhibited. These processes belong to the more general class of penalized Markov processes, as studied in [START_REF] Roynette | Some penalisations of the Wiener measure[END_REF], used in order to derive the following result (stated here both for discrete and continuous time models). Proposition 6.1. Assume that either Assumption A, or Assumption E or Assumption F is satisfied. Then there exists a family (Q x ) x∈E of probability measures on Ω defined by

for all F s -measurable set A. It defines an E -valued homogeneous Markov process 1 . Moreover, if X is a strong Markov process under P, then so is X under Q.

Remark 6.1. There are other ways of conditioning a Markov process to never be absorbed. For instance, for a non-explosive one dimensional diffusion process evolving in (0, +∞) and absorbed when it reaches 0, one may define a process as the limit, when A → +∞, of the process conditioned to reach A before reaching 0 (see [START_REF] Perkowski | Conditioned martingales[END_REF] for a detailed investigation of this situation); the resulting conservative process is then different from the Q-process. Other interesting limiting processes may be studied when the quasi-stationary behaviour of the process is known, such as the two-side taboo limit introduced and studied in [START_REF] Glynn | Two-sided taboo limits for Markov processes and associated perfect simulation[END_REF].

Proposition 9.3. The empirical occupation measure 1 t t 0 δ Y s ds converges almost-surely when t → +∞, with respect to the topology of weak convergence, to the unique quasi-stationary distribution ν QSD of X such that ν QSD (V ) < ∞.

The proof of this result uses the theory of Measure-Valued Pólya processes exposed in Chapter 8. More precisely, it derives from a larger class of models, called sample paths Pólya-Urns, whose study is developed in [START_REF] Mailler | Stochastic approximation on non-compact measure spaces and application to measure-valued P\'olya processes[END_REF] for continuous and discrete-time models.

Part IV

Some other works

the fixation time of the process.

Theorem 11.1. Fixation occurs before extinction with probability one if and only if

Consider the particular case where f is the identity function. Whereas for the usual demographic term σ(N ) = N , fixation occurs almost surely before extinction, a small perturbation of this diffusion term, taking for example σ(N ) = N (1-ε)/2 , ε > 0, leads to extinction before fixation with positive probability. An example of trajectory for which fixation does not occur before extinction is given in Figure 11.1, and the effect of ε on the probability of extinction before fixation is numerically studied in Figure 11.2.

Integrability properties for diffusion processes

We state a result implying that, depending on the behaviour of the diffusion and drift coefficients near absorption, the integral of the paths of diffusion processes are either almost surely finite or almost surely infinite. This 0-1 law criterion has already been proved by various methods, using a combination of the local time formula and Ray-Knight theorem [START_REF] Engelbert | Integral functionals of strong Markov continuous local martingales[END_REF][START_REF] Mijatović | Convergence of integral functionals of one-dimensional diffusions[END_REF][START_REF] Khoshnevisan | A note on a.s. finiteness of perpetual integral functionals of diffusions[END_REF] (see also [START_REF] Engelbert | On functionals of a Wiener process with drift and exponential local martingales[END_REF][START_REF] Foucart | Stable continuous-state branching processes with immigration and Beta-Fleming-Viot processes with immigration[END_REF] for proofs in particular settings). In [START_REF] Coron | Impact of demography on extinction/fixation events[END_REF], we give a simpler proof of this criterion, which also provides explicit bounds for the moments of perpetual integrals and can be easily extended to more general one dimensional Markov processes.

General diffusion processes on [0, +∞)

Let us consider a general one-dimensional diffusion process (Z t ) t ∈R + (see Chapter 4) with values in (0, +∞). Let us denote by P z the law of Z starting from z. We assume that Z is regular (∀z ∈