Miguel Rodrigues

J'aime à penser que cette page sera la première, et cachera une forêt d'autres pages bien trop austères. Elle témoignera de la convivialité qui caractérise ce métier et reflétera, je l'espère, la profonde gratitude qui m'envahit au moment de ficeler cette généreuse tranche de vie.

Je souhaite en tout premier lieu rendre hommage à Frédéric et Nicolas qui ont encadré ma thèse. Mes premiers souvenirs à vos côtés sont ceux d'un voyage dans les méandres des espaces U q et V p , au bout du confinement, la certitude que l'analyse microlocale sera "toujours avec moi" -ce qui est vrai -et une magnifique épopée sur la sphère. Tout cela m'a bien occupé et passionné. La vôtre, de passion, dissipe toute forme de pression et d'angoisse. Vous m'avez accordé une confiance précieuse, une liberté totale et un soutien permanent qui m'accompagneront longtemps.

Je suis reconnaissant pour le travail qu'ont effectué Sebastian Herr et Gigliola Staffilani en rapportant consciencieusement ce long manuscrit. Vos commentaires enthousiastes m'ont touché. Je remercie Anne De Bouard, Patrick Gérard et Benoît Grébert qui composent le jury devant lequel j'ai l'honneur de présenter mes travaux.

Un grand merci à Nikolay Tzvetkov pour l'attention portée à chacune de mes contributions, son enthousiasme, ses idées merveilleuses et le souffle qu'il donne à notre communauté. Je remercie également Patrick Gérard qui me suit depuis un petit bout de temps déjà, qui est à l'origine de cette thèse et de mes goûts mathématiques.

Je salue chaleureusement "les Nantais". Benoît Grébert, qui a contribué à l'organisation du mémorable triptyque de l'Atlantique : Croisic, Vannes, Pornichet. Les démarches faites par Joackim Bernier pour mon postdoc m'ont particulièrement touché. Il me faut souligner la rare gentillesse et la minutie avec laquelle Joackim prépare ma venue. Il me tarde de travailler avec vous.

Je remercie les membres du très impressionnant laboratoire de mathématique d'Orsay. En premier lieu Stéphane Nonnenmacher pour son investissement total dans la réussite de chaque doctorant. Je pense à tous les membres de l'équipe, aux superbes journées de rentrée et aux séminaires, ainsi qu'aux enseignants pour qui j'ai eu l'honneur de dispenser les sessions d'exercices. Daniel Monclair et Jean-Benoit Bost pour les fascinantes discussions en amont des cours, Guy David et Matthieu Léautaud. Je salue Hans et Filipa pour leur gestion impeccable, malgré la tempête et les vagues successives, ainsi qu'à tous les personnels administratifs qui nous simplifient chaque jour la vie.

Les maths sont un sport collectif, une randonnée à plusieurs. Je remercie d'abord mes deux principaux compagnons de cordée. Chenmin, pour toutes ces craies réduites en poudre et ces heures passées à tourner autour de la sphère. Je pense à cet après-midi à Providence où Hartree nous a fait un peu mal. À cette soirée de défaite finalement illuminée par un water fire et un filet de pollock frit. Louise pour nos travaux que je redécouvre dans tes exposés. Cette analogie entre la convolution de l'ensemble pathologique et la couche de sucre qui enrobe le tanghuru. Pour ce vibrant hommage que tu as rendu à la musicalité des interactions "high low low ". Je remercie mes collaborateurs avec qui j'ai eu l'immense plaisir de découvrir la vraie recherche, celle que l'on fait à plusieurs. Mickaël, qui est venu me chercher très tôt dans ma thèse, Slim qui m'a sensibilisé aux équations faiblement dispersives, Andréa pour toute son énergie, Xueying, Haitian et enfin Ben. Je salue chaleureusement Changzhen qui m'a si bien épaulé dès mon master.

Merci à celles et ceux que j'ai eu le plaisir renouvelé de rencontrer dans de nombreuses conférences. En particulier Tristan Robert, sans qui une conf n'est plus vraiment une conf. Jonas Lührmann, dont la participation suffit à me convaincre d'y aller aussi. Je salue Bjoern Bringmann et sa générosité. Léonardo Toloméo et les passionnantes conversations dans la forêt noire, faisant fi de la menace des tiques. Je pense tout particulièrement à

Dans une première partie, on établit l'existence de deux types de données initiales : les données statistiques, qui donnent lieu à des solutions fortes, et les données pathologiques qui sont instantanément instables. Ces deux types de données initiales sont génériques, respectivement au sens de la me-sure et de la topologie. Dans une seconde partie, des méthodes d'analyse sont combinées avec l'approche probabiliste pour décrire le comportement en temps long des données statistiques, ainsi que la stabilité de dynamiques non-linéaires sous l'effet de perturbations aléatoires. Enfin, une troisième partie est consacrée au développement de schémas itératifs quasi-linéaires dans un contexte probabiliste. Ces schémas exploitent l'information statistique des données initiales, et permettent l'analyse d'équations faiblement dispersives dans des régimes singuliers. La troisième partie se conclut par une prémisse au problème de l'invariance de la mesure de Gibbs pour l'équation de Schrödinger cubique sur la sphère. Une interaction singulière est isolée, ce qui permet la mise en place future d'un schéma quasi-linéaire adapté à ce problème. Abstract : The purpose of this thesis is to study nonlinear Schrödinger equations with random initial data, in singular regimes. Nonlinear Schrödinger equations are involved in many wave models to describe the envelope of a rapidly oscillating wave packet. The statistical study of these nonlinear waves in singular regimes is motivated by the presence of instabilities that compromise the analysis of individual trajectories.

In a first part, we establish the existence of two types of initial data : statistical data, which give rise to strong solutions, and pathological data, which are instantaneously unstable. These two types of initial data are generic, respectively in the sense of measure theory and topology. In a se-cond part, analytical methods are combined with the probabilistic approach to describe the longtime behavior of the statistical data, as well as the stability of nonlinear dynamics under random perturbations. Finally, a third part is devoted to the development of quasilinear iterative schemes in a probabilistic context. These schemes exploit the statistical information of the initial data and allow the analysis of weakly dispersive equations in singular regimes. The third part concludes with a premise for the problem of invariance of the Gibbs measure for the cubic Schrödinger equation on the sphere. A singular interaction is isolated, which allows the future implementation of a quasilinear scheme adapted to this problem.

Introduction

Generalities and main questions

In classical physics, dispersion is a linear wave phenomenon that occurs in media where waves of different frequencies propagate differently. Dispersion is mathematically captured by a class of linear equations called dispersive equations. Specifically, these equations describe vibratory phenomena in dispersive media, without friction or dissipation. A typical feature shared by dispersive equations is the presence of conserved quantities, such as the mass or the energy of the waves. On the other hand, due to the dispersion, the amplitude of the solutions decreases, at least when the domain is unbounded so that the waves have room to spread.

The propagation of waves in weakly nonlinear dispersive media is described by nonlinear dispersive equations that are typically of the form

i∂ t u = Lu + f (u) ,
where the solution u = u(t, x) has values in R d or C d , L is a self-adjoint operator usually given by a Fourier multiplier, and f (u) is the nonlinear term, which may contain derivatives of u. The theory of nonlinear dispersive equations has been greatly developed over the last 30 years, and we refer to the books [Bou99; Caz03; Dod19a; Tao06], to the note [START_REF] Tao | Why are solitons stable?[END_REF], and to the references therein for general presentation and discussion on many aspects of the rich mathematical study of these equations. Illustrious examples of nonlinear dispersive models are the Korteweg-de Vries equation (KdV), the nonlinear Schrödinger equation (NLS) or the wave equation (NLW), the Klein-Gordon equation (KG), the Benjamin-Ono equation (BO), and the Hartree equation.

In this dissertation, we are interested in the study of nonlinear dispersive equation in singular regimes where the solutions are highly oscillating, in the presence of rough and stochastic perturbations. Before discussing some motivations, we emphasize that unlike solutions to parabolic equations, such as the heat equation, the solutions to dispersive equations generally do not experience immediate smoothing effects. Thus, when working at a prescribed regularity, one has to isolate the different scales of oscillations and to finely exploit the dispersive mechanisms that govern wave propagation. In addition, at the non-linear level, it is necessary to understand how the different waves interact with each other. Dispersive mechanisms are expressed by means of functional norms linked together by a bunch of powerful inequalities. Some of these inequalities, state them in the following paragraph, are widely used in the work that constitutes this manuscript.

The motivations to study nonlinear dispersive equations in the presence of randomness are of very different natures. First, randomness can be imposed by the model to describe microscopic fluctuations and collective behavior of large number of particles. In this situation, and when the equation is Hamiltonian, the relevant objects that provide a statistical description of the system at equilibrium are the Gibbs measures. Second, nonlinear dispersive equations display instabilities or blow-up solutions in certain regimes that compromise the full description of the evolution for individual initial data. Nevertheless, by endowing the set of initial data with a relevant probability measure, we can inject randomness in the equation to grab a statistical ensemble of initial data leading to strong solutions that are limits of smooth regularized solutions. To motivate the study of dispersive equations with rough data, let us mention that models involving small parameters, in a semiclassical regime for instance, can be studied by considering a fixed equation with a sequence of oscillatory initial data, possibly with an unbounded energy.

We end this paragraph by pointing out that, generally speaking, probabilistic methods are used to exhibit mathematical objects with prescribed properties. In this perspective, the use of the probabilistic method is an old idea in harmonic analysis where the study of random Fourier series has been a subject of intense research (see for instance the book [START_REF] Kahane | Some Random Series of Functions[END_REF] and the references therein). In the context of probabilistic Cauchy theory, however, the use of randomness is slightly different since, up to a certain extent, the relevant probability measures are prescribed. Indeed, we want the statistical ensemble to be representative of the functions in the regime we are interested in. On the top of that, the Gibbs measures, or other relevant invariant measures that are usually supported at very low regularities, are imposed by the equation and its Hamiltonian structure.

The model We will mostly discuss the nonlinear Schrödinger equation in the three-dimensional Euclidean space R 3 , but also on the periodic case T d and the sphere S 2 . Our motivation is to investigate the Cauchy theory and the long-time dynamics of NLS at low regularity under stochastic perturbations that may be rough. The general form of the nonlinear Schrödinger equation posed on a Riemannian manifold (M, g) is

i∂ t u + ∆ g u = µ|u| p-1 u , (t, x) ∈ R × M . u |t=0 = u 0 , (NLS)
Here, ∆ g is the Laplace-Beltrami operator on (M, g) and p > 1 measures the strength of the semilinear perturbation of power type, seen as a self-interacting potential. The sign of the parameter µ ∈ R dictates the focusing or defocusing nature of the equation, leading to drastically different dynamics. Roughly speaking, in the defocusing case (µ > 0), the potential is repulsive and we expect the waves to propagate linearly. In the focusing case (µ < 0), however, the potential is attractive and nonlinear phenomena are expected. We recall that equation NLS has the conserved Hamiltonian

E[u](t) := 1 2 M |∇u(t, x)| 2 dx + µ p + 1 M |u(t, x)| p+1 dx .
In addition, the L 2 -mass is also conserved along the evolution

M [u](t) := 1 2 M |u(t, x)| 2 dx .
The nonlinear Schrödinger equation serves as a theoretical model that appears in a number of dispersive models to describes the amplitude of the envelope of a fast oscillating wave packet. We refer to [START_REF] Sulem | The nonlinear Schrödinger equation: Self-focusing and wave collapse[END_REF] for physical perspectives on the nonlinear Schrödinger equation. In light of the properties of the linear flow e it∆ t∈R and of the conservation laws, the natural phase spaces generally associated with dispersive equations are the Sobolev spaces H s (M) with s ∈ R, endowed with the norm

∥u∥ H s (M) := M |∆ s 2 g u(x)| 2 dx 1 2 .
The parameter s measures the regularity and the oscillations of the function. Even if the mass of the solution to NLS is conserved, its H s (M)-norm can, on the other hand, inflate for some s > 0 as time evolves. In such a case, the mass distributes on the high frequencies as small-scale oscillations appear. Such a low-to-high frequency cascade is referred as weak turbulence.

In this dissertation, we only consider equations that are not known to be integrable and we follow a perturbative approach materialized by the Duhamel's integral formula u(t, x) = e it∆g u 0 (x) -iµ t 0 e i(t-τ )∆g |u(τ, x)| p-1 u(τ, x)dτ .

Specifically, the nonlinear evolution is seen as a perturbation of the linear evolution. The Duhamel's integral formula is used to prove local existence and uniqueness by a contraction mapping argument or, alternatively, a Picard's iteration scheme in suitable space-time functional space. Notice that this strategy is more likely to be successful when the data and the time of existence are small. In order to understand the long-time behavior, however, we need to iterate this local argument thanks to conserved quantities that control the growth of Sobolev norms, possibly combined with other uniform space-time bounds obtained from monotonicity formulas.

In some supercritical regimes where the initial data oscillates at high frequencies, however, instabilities rule out the Picard's iteration scheme. We face two types of instabilities. On the one hand, we have the scaling-supercriticality that leads to the norm-inflation mechanism discussed in Chapter 1. Roughly speaking, because of some scaling-symmetries, studying small data in small-time amounts to study large data. On the other hand, some instabilities are due to the presence of particular stationary solutions that concentrate their mass, for instance when the dispersion is too weak. In these regimes, illustrated in Chapters 4 and 5, the system is quasilinear in the sense that the linear evolution is not a good approximation of the nonlinear evolution.

Finally, we emphasize that when studying a nonlinear equation, one needs to control some L p -norms to ensure that the nonlinearity makes sense. Such L p -bounds measure the concentration of the solutions. Specifically, if a function in L 2 has a huge L p -norm for some p > 2, then the function has its mass concentrated at a finite distance from the origin. Guided by physical intuition, one can imagine that dispersive effects induce bounds on the L p -norms of the linear evolution, at least when averaging in time. In this perspective, we recall some fundamental estimates that capture dispersive features shared by solutions to the linear Schrödinger equation in the Euclidean space, where waves have room to propagate.

• The dispersive estimate.

∥ e it∆ u 0 ∥ L ∞ (R d ) ≲ |t| -d 2 ∥u 0 ∥ L 1 (R d ) , ∥ e it∆ u 0 ∥ L 2 (R d ) = ∥u 0 ∥ L 2 (R d ) .
(0.1.1)

• Strichartz estimates. Given 2 ≤ p, q ≤ ∞, satisfying the admissibility condition 2 p + d q = d 2 , with (d, p, q) ̸ = (2, 2, ∞),

∥ e it∆ u 0 ∥ L p t (R,L q x (R d )) ≲ ∥u 0 ∥ L 2 x (R d ) . (0.1.2)
• Bourgain's bilinear estimate.

∥ e it∆ P ≤N u 0 e it∆ P M v 0 ∥ L 2 t,x (R×R d ) ≲ N

1 2 N M d-2 2 ∥P ≤N u 0 ∥ L 2 x (R d ) ∥P M v 0 ∥ L 2 x (R d ) (0.1.3)
• Local smoothing estimate. Let u be solution of the forced Schrödinger equation

(i∂ t + ∆)u = f , u(0) = u 0 .
Then, for every δ > 0,

R×R d ⟨x⟩ -1-δ |⟨∇⟩u(t, x)| 2 dx dt ≲ ∥⟨∇⟩ 1 2 u 0 ∥ 2 L 2 x + R×R d ⟨∇⟩ 1+δ |f (t, x)| 2 dx dt . (0.1.4)
When the dispersion is too weak and does not gain enough integrability, we have no other choice than trading integrability against regularity. To do so, we can use the Sobolev embedding estimates, or, in the Euclidean space, the Bernstein estimate. For randomized initial data, however, we can exploit probabilistic decoupling to gain integrability without trading regularity. We will see in this introduction that there are a number of randomization procedures, depending on the context. The general idea that ties them together is that a typical function has better L p -properties than expected thanks to random averaging effects that cancel out the interference. This is reflected by the Paley Zygmund theorem discussed below. Before going further, let us give an informal statement of a probabilistic well-posedness type of results, which reads as follows.

Theorem 0.1.1 (Probabilistic local well-posedness). Let s d be a deterministic exponent, such as the equation is not uniformly well-posed when s < s d . There exists s 0 such that for all s 0 < s < s d , there exist a probability measure µ supported in H s and a full measure set Σ, such that for all u 0 ∈ Σ, the sequence of regularized solutions (u ε ) ε>0 , with initial data (u 0 * ρ ε ) ε>0 , converges to a limiting object in C([-T , T ], H s ) solution of NLS, for some T = T (u 0 ) > 0.

Often, the probability measure µ is induced by the random Fourier series of type

ω ∈ Ω → u ω = k g n (ω)π n u
where the (π n ) n 's spectral projectors are associated with a spectral resolution of the Laplace operator (λ 2 n , π n ) n , and the (g n ) n 's are, say, a sequence of independent normalized Gaussian variables. Convergence makes sense in L 2 (Ω; H s (M)), when u ∼ n π n u is a fixed function in H s (M). This can be seen with the orthogonality of the elementary pieces (π n u) n 's, and the uniform bound on the variance of the (g n ) n 's:

∥ M n=N g n (ω)π n u∥ 2 L 2 (Ω;H s (M)) = E ∥ M n=N g n (ω) ⟨∆⟩ s 2 (π n u)∥ 2 L 2 (M) = E M n=N |g n (ω)| 2 ⟨λ n ⟩ 2s ∥π n u∥ 2 L 2 (M) ≤ sup n E |g n (ω)| 2 M n=N ⟨λ n ⟩ 2s ∥π n u∥ 2 L 2 (M) -→ N,M →∞ 0 .
Main questions. To conclude this paragraph, we list some questions that motivate the study of dispersive equations at low regularity under stochastic perturbations. Specifically, in the presence of non-trivial dynamics, well understood for smooth solutions, one can consider stochastic perturbations due, for example, to rough microscopic fluctuations, and ask the following questions:

• What is the generic behavior of the solutions in super-critical regimes?

• Does randomness prevent deterministic instabilities?

• How does randomness affect the long-time behavior of coherent solutions?

In this thesis we study different manifestations of these questions, and we develop technics combining deterministic and probabilistic tools, at low regularity. The main challenge we faced, both in short-time and in long-time, is the aforementioned lack of regularizing effect for dispersive flows. Beyond the fact that we are working at supercritical regularities where instabilities occur, we do not have direct access to the conserved coercive energy, which is fundamental to understand the long-time dynamics of the solutions. Actually, the underlying question we faced in this manuscript that goes beyond the probabilistic considerations, can be stated as follows:

• How to exploit dispersive effects instead of spending regularity, especially when there is little of it?

A brief history of probabilistic Cauchy theory

This general introduction to the study of dispersive equation with randomness is not intended to be exhaustive. Instead, we orient the discussion in light of the works presented in this manuscript. In particular, we focus on the long-time dynamics of probabilistic flows, in the context of the Wiener decomposition in the Euclidean space. Nevertheless, we aim to provide a motivating and illustrative background on the rich history of this active research area.

Random Fourier series The use of randomness in harmonic analysis dates back to the early 1930s, when Paley and Zygmund observed in a series of papers [START_REF] Paley | On some series of functions, (1) (2) (3)[END_REF] that a typical function in H s (T) enjoys better L p (T) estimates than what the Sobolev embedding would predict. More precisely, consider a square-summable trigonometric series n c n e inx , with

n |c n | 2 < ∞ ,
which does not belong to L p (T), for p > 2. However, Paley and Zygmund observed that it suffices to randomly change the sign of each coefficient, independently, for the function to belong to L p (T). This is how the story of random Fourier series started. In modern mathematics, the result in [START_REF] Paley | On some series of functions, (1) (2) (3)[END_REF] states as follows.

Theorem 0.1.2 (Paley-Zygmund). Given a sequence of complex numbers (c n ) ∈ ℓ 2 (Z) and a family of independent mean zero Bernoulli (b n (ω)) random variables, the random series defined by This observation has initiated a series of major contributions in harmonic analysis, and we refer to the book [START_REF] Kahane | Some Random Series of Functions[END_REF] for an overview of the subject. The modern proof of Theorem 0.1.2 is brief, and relies on a standard quantitative large-deviation estimate. Nevertheless, this proof is illustrative of the probabilistic arguments used in this manuscript. We encourage interested readers to consult the note [START_REF] Burq | Random Data Cauchy Theory for Dispersive Partial Differential Equations[END_REF] for a precise discussion of the Paley-Zygmund theorem and its applications in the context of dispersive partial differential equations. Of course, the result is valid for more general random variables. We point out that a major difficulty at that time was to make sense of an infinite family of independent Bernoulli variables. The first explicit realizations of infinite families of independent Bernoulli variables are due to Rademacher [START_REF] Rademacher | Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen[END_REF] and Kolmogorov [START_REF] Kolmogorov | Sur la convergence des séries de Fourier[END_REF] with a dyadic decomposition of a number on the line.

The invariance of the Gibbs measure. The historical reason for interest in dispersive equations in the presence of randomness and low regularity comes from statistical mechanics. In the study of microscopic systems containing a large number of particles at equilibrium, macroscopic observables are accessed by integrating over Gibbs probability distributions. In their works [LRS88; LRS89], Lebowitz, Rose and Speer proved the existence of the Gibbs measure for some Hamiltonian systems governed by the nonlinear focusing Schrödinger equation, and the Zakharov system. Given a finite number of particles N in a finite volume Ω, the Gibbs probability measure on a set of microscopic states is written

µ( dX N ) = Z -1 N exp -βH(X N ) dX N ,
where H is the Hamiltonian of the system, β is the reciprocal temperature and Z n is a normalization factor. In order to understand Langmuir waves and plasma turbulence, they considered the focusing case, so that the Hamiltonian is not bounded from below. They overcome this difficulty by using a cutoff of the mass, which interprets as a system with a finite number of particles, without taking the hydrodynamics limit as N goes to infinity. The Gibbs measure is absolutely continuous with respect to a Gaussian measure, and the realization of a Gibbs measure consists in Wiener paths that are quite rough. Indeed, the Gaussian measures are of type

(ω, x) → n∈Z d g n (ω) ⟨n⟩ 1 2 e inx ∈ L 2 (Ω; H 1-d 2 -(T d )) .
Hence, at that time, the existence of a well-defined equilibrium dynamics on the support of the Gibbs measure was hypothetical. For instance, in the case of the cubic Schrödinger equation in dimension d = 2, the Gibbs measure is supported in H 0-but the deterministic threshold for local well-posedness is s c = 0. In a paragraph called "Open Questions" [START_REF] Lebowitz | Statistical mechanics of the nonlinear Schrödinger equation[END_REF], Lebowtiz Rose and Speer raised the following problem: how to prove the existence of a nonlinear dynamics on the support of the Gibbs measure, at least for small-time? In addition, they wrote a short argument proposed by E. Stein to show that the free linear evolution is not bounded in L ∞ (T). The case d = 3 is still an important open problem. 1

The birth of probabilistic Cauchy theory The construction of invariant measures has become a classical problem since the 1990s. Following the early works of Lebowitz, Rose and Speer [LRS88; LRS89], Frielander [START_REF] Friedlander | An invariant measure for the equation u tt -u xx + u 3 = 0[END_REF] and Zhidkov [START_REF] Zhidkov | An invariant measure for a nonlinear wave equation[END_REF] proved the invariance of the Gibbs measure for the one-dimensional nonlinear wave equation, followed by Bourgain [START_REF] Bourgain | Periodic nonlinear Schrödinger equation and invariant measures[END_REF], who proved the invariance of the Gibbs measure for the cubic nonlinear Schrödinger equation on T. In these contributions, even if the initial data in the support of the Gibbs measure is by essence random, the Cauchy theory is deterministic. Indeed, in one dimension, the Gibbs measure is supported at regularity s < 1 2 , while the deterministic critical regularity is s c = 1 2 -2 p-1 . Nevertheless, Bourgain needed to introduce the Fourier restriction norm to establish the local well-posedness in H 1 2 -(T) [START_REF] Bourgain | Periodic nonlinear Schrödinger equation and invariant measures[END_REF]. To prove global well-posedness, however, Bourgain developed in the same paper [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] a new globalization argument discussed in Section 0.1.3. This argument is considered probabilistic since it exploits the formal invariance of the Gibbs measure that serves as a substitute for the conserved energy.

Let us now move to the two-dimensional case that, strictly speaking, was the birth of the probabilistic Cauchy theory. In another celebrated work [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF], Bourgain obtained both the global existence for data in the statistical ensemble, and the invariance property of the Gibbs measure for 1 We refer to [START_REF] Deng | Invariant Gibbs measures and global strong solutions for nonlinear Schröodinger equations in dimension two[END_REF][START_REF] Deng | Random tensors, propagation of randomness, and nonlinear dispersive equations[END_REF], where a notion of probabilistic scaling is introduced. It corresponds to s p = -1 p-1 , which is independent of the dimension. The cubic Schrödinger equation in dimension three is therefore critical for the probabilistic scaling, since the Gibbs measure is supported in H -1 2 -(T 3 ).

the two-dimensional Wick-ordering2 cubic Schrödinger equation (i∂ t + ∆)u =: |u| 2 u : , (t, x) ∈ R × T 2 , u(0) = u ω 0 . Up to a gauge transformation to remove the conserved mass, the initial data can be represented as a Fourier series

u ω 0 (x) = n∈Z 2 g n (ω) ⟨n⟩ e inx ,
where ⟨n⟩ = 1 + |n| 2 and (g n ) n∈Z 2 is a sequence of independent normalized Gaussian variables with complex values. In contrast to the one-dimensional case, the support of the Gibbs measure lies at super-critical regularities H 0-(T 2 ), the critical scaling exponent being s c = 0. To overcome this difficulty, Bourgain searched the solution under the sum of a rough linear evolution, and a smoother nonlinear remainder u = e it∆ u ω 0 + v . (0.1.5) Specifically, the nonlinear Duhamel term v turns out to be at regularity H 1 2 -(T 2 ). Bourgain performed a fixed point argument for v at regularity H ε , for some ε > 0, represented by the Duhamel integral written I : | e it∆ u ω 0 + v| 2 e it∆ u ω 0 + v : .

To run the contraction mapping argument, Bourgain first observed a gain of 1 2 -derivatives in the second Picard iteration. Then, Bourgain exploited multilinear cancellations to analyze the interactions between e it∆ u ω 0 and v 2 , by developing a T T * -probabilistic argument. In this approach, v is treated deterministically, since one does not exploit any particular random structure of the remainder. The linear-nonlinear decomposition (0.1.5) in the context of random initial data is also called the Da Prato-Debussche method [START_REF] Da Prato | Two-dimensional Navier-Stokes equations driven by a space-time white noise[END_REF], which successfully used a similar strategy in the context of fluid mechanics. Subsequently, Burq and Tzvetkov extended Bourgain's strategy to nonlinear wave equations in the case of a general Riemannian compact manifolds (M, g) without boundaries. In this way, they provided a general framework that made possible further developments of the theory. They proceeded as follows. Take s < s c so that the Cauchy problem in H s (M) := H s (M) × H s-1 (M)

(∂ tt -∆ g )u = -|u| 3 , (t, x) ∈ R × M (u(0), ∂ t u(0)) = (f 0 , f 1 ) , (0.1.6)
is ill-posed. Then, consider randomized initial data defined by taking (f 0 , f 1 ) in H s , expanding them in an orthonormal basis of L 2 (M) constructed from the eigenfunctions of the Laplace operator -∆ g and decoupling each mode by independent standard Gaussian variables (h n , l n ):

ω → (f ω 0 , f ω 1 ) = n h n (ω)(f 0 | e n )e n , n l n (ω)(f 1 | e n )e n .
This defines a H s (M)-valued random variable, and we have almost-sure local existence according to the induced measure in the following sense:

Theorem 0.1.3 (Burq-Tzvetkov [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF]). For T > 0, there exists a set Σ T such that for all ω ∈ Σ T , the initial value problem (0.1.6) with initial data (u(0), ∂ t u(0)) = (f ω 0 , f ω 1 ) admits a unique solution in C([-T , T ], H s ) under the form

u(t) = cos(t|∇|)f ω 0 + sin(t|∇|) |∇| f ω 1 + v .
Uniqueness holds for the nonlinear term v in a subspace of C([-T , T ], H σ ), for some s c < σ. Moreover, there exist C, c, δ > 0 such that for all T > 0,

P( c Σ T ) ≤ C exp -cT -δ ∥(f 0 , f 1 )∥ -2 H s .
The proof follows the strategy of [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF], performing a fixed point on v. Here, the authors exploit the L p -regularization of the linear evolution under the effect of the randomization, in the spirit of Paley Zygmund's observation, and the gain of one derivative in the Duhamel formula for the wave equation. For Schrödinger-type equations, however, there is no comparable general result. As we point out in Chapter 5, the behavior of solutions to Schrödinger-type equations on compact manifolds, even in short time, is much more complex because of the infinite speed of propagation. In this perspective, another ensuing series of works on compact manifolds concerns the almost-sure global well-posedness of the Schrödinger equation on two and three-dimensional balls [START_REF] Tzvetkov | Invariant measures for the defocusing Nonlinear Schrödinger equation[END_REF][START_REF] Bourgain | Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3D ball[END_REF][START_REF] Bourgain | Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case[END_REF][START_REF] Bourgain | Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball II: The 3D case[END_REF].

Randomization procedure in the Euclidean space. In the Euclidean space, where the long-time dynamics of Schrödinger equations are completely different, the probabilistic framework is not the same since we do not have a discrete spectral resolution for the Laplace operator. For this reason, there is no canonical randomization procedure in the Euclidean space. One way to mimic the randomization procedures presented above in a compact setting is to use a unit-scale frequency decomposition, leading to the so-called Wiener randomization. The procedure, introduced by [START_REF] Lührmann | Random data Cauchy theory for nonlinear wave equations of power-type on R 3[END_REF] and [START_REF] Oh | Probabilistic global well-posedness of the energycritical defocusing quintic nonlinear wave equation on R 3[END_REF] independently, consists in superposing unit-scale frequency blocks decoupled by independent Gaussian variables (see Figure 0.1.2). Definition 0.1.4 (Wiener randomization). Take a unit-scale partition of unity in the frequency space

k∈Z d ψ(ξ -k) ≡ 1 , ξ ∈ R d ,
for a suitable smooth cutoff function ψ supported in [-1 , 1] d . Then, take u 0 ∈ H s (R d ), and write its Wiener decomposition

u 0 = k∈Z d P 1;k u 0 , P 1;k u 0 = F -1 (ψ(• -k)Fu 0 ) .
Given a probability space (Ω, A, P) and a family of independent complex valued standard Gaussian variables (g k ) k∈Z d , the mapping

ω ∈ Ω → u ω 0 = k∈Z d
g k (ω)P 1;k u 0 (0.1.7) defines a random variable valued in H s (R d ). Note that the regularization procedure does not regularize the initial data. For all p ≥ 2 and σ > 0, if u 0 ∈ W s,p (R d ) \ W σ,p (R d ) then so does u ω 0 , almost-surely as proved in Lemma B.1 from [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF].

This randomization procedure has for side effect to relax the admissibility condition in the Strichartz estimates, imposed by the scaling of the equation (see Figure 0.1.2). Given a function whose frequency support is of size one, we can use the Bernstein inequality to pass from L q to L 2 without losing any derivative. Then, we glue the individual estimates together by the mean of probabilistic decoupling, in the spirit of the proof of Paley-Zygmund's theorem.

Proposition 0.1.5 (Probabilistic Strichartz estimates, Prop. 1.5 [START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF]). Let (p, q) be a Strichartz-admissible pair and s > 0. For every q < q < ∞, there exist C, c > 0 such that for all u 0 ∈ H s (R d ), and λ > 0, we have the large deviation bound

P ω ∈ Ω : ∥ ⟨∇⟩ s e it∆ f ω 0 ∥ L p t (R;L q x (R 3 ) > λ ≤ C exp -cλ 2 ∥u 0 ∥ -2 H s .
Then, almost-sure local well-posedness at supercritical regularities follows from the Da Prato-Debussche trick, as described in the above paragraph. In this case, the nonlinear smoothing effect for the remainder v was evidenced in the analysis of [START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF], combining the probabilistic Strichartz estimates with bilinear estimates in Fourier restriction spaces, or more recently by exploiting the local-smoothing estimate in lateral spaces [START_REF] Dodson | Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation[END_REF]. In the first approach, used in this manuscript in dimension 3, the nonlinear remainder gains s derivatives:

v(t) = u(t) -e it∆ u ω 0 = -i t 0 e i(t-t ′ )∆ | e it ′ ∆ u ω 0 | 2 e it ′ ∆ u ω 0 dt ′ second Picard's iteration + . . . ∈ H 2s-(R 3 ) a.s.
Bényi, Oh and Pocovnicu same authors refined in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF] the analysis in U 2 and V 2 spaces for small data, to prove almost-sure global existence and scattering. Let us conclude by mentioning another randomization procedure used to study the nonlinear Schrödinger equation on the line. It is based on the lens transform, that compactifies time and space by transforming the Laplacian into the harmonic oscillator. We refer to [Poi12; PRT14; BT20]. Notice, however, that in dimension greater than one this procedure gains derivatives in L p -spaces, so we prefer not following this approach.

Quasilinear probabilistic decomposition: the paracontrolled approach. The linearnonlinear decomposition we have discussed so far has shown its limits, and Da Prato-Debussche's trick does not always lead to the desired nonlinear probabilistic regularizing effect. Indeed, in some cases where the dispersion is too small or where the nonlinear interactions are prevalent, singular interactions may appear and limit the possible gain. In the worst cases, bad interactions can even exclude any possible nonlinear smoothing. In such cases, one cannot run the standard probabilistic semilinear resolution scheme à la Bourgain. An illustrative example is the derivative nonlinear wave equation, whose probabilistic study was carried out by Bringmann in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], who by the same time proposed the first probabilistic quasilinear resolution scheme.

The derivative wave equation considered in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] writes 1/q 1/p Figure 2: The purple region indicates the pairs of exponents for which the Strichartz estimate holds almost-surely. We observe that the Wiener randomization procedure relaxes the admissibility condition materialized by the purple line, and gains integrability without trading derivatives.

-∂ 2 tt u + ∆u = |∇u| 2 , (t, x) ∈ R × R 3 , u(0) = u 0 , ∂ t u(0) = u 1 .
In addition to the scaling-critical exponent s c =3 2 , the Lorentz-critical critical regularity s d = 2 dictates the well-posedness threshold: the equation is locally well-posed in H s × H s-1 when s > 2, whereas ill-posedness occurs when s < 2 [START_REF] Lindblad | A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations[END_REF], respectively. Still, Grünrock proved in [START_REF] Grünrock | On the wave equation with quadratic nonlinearities in three space dimensions[END_REF] that well-posed holds in Fourier-Lebesgue spaces, in scaling-subcritical regimes 3 2 < s ≤ 2. At these regularities, Bringmann investigated in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] probabilistic well-posedness in the scale of the Sobolev spaces. Writing the linear ansatz

u = f ω + v , f ω (t) = cos(t|∇|)u ω 0 + sin(t|∇|) |∇| u ω 1 ∈ H s ,
and considering the high-low interactions ∇P ≫1 f ω ∇P 1 f ω in the second Picard's iteration, the author observed that the equation does not experience any smoothing. Indeed, given ν > s-derivatives, we have

|∇| ν t 0 sin((t -τ )|∇|) |∇| ∇P ≫1 f ω (τ )∇P 1 f ω (τ ) dτ ≃ t 0 sin(t|∇|)|∇| ν P ≫1 f ω (τ )∇P 1 f ω (τ ) dτ .
It turns out that the ν-derivatives fall onto the linear term f ω , which can only handle s derivatives.

In contrast to the case of the Schrödinger equation, there is neither bilinear estimate nor local smoothing estimate to compensate for this surge of derivatives. To remove the high-low interactions, Bringmann successfully implemented the paracontrolled calculus framework, developed by Gubinelli, Imkeller and Perkowski [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF] in the context of parabolic equations with stochastic forcing. The key idea is to see the singular interactions, which consist in the paraproduct of a high-frequency function with a low-frequency potential, as part of the linear evolution. In this perspective, Bringmann used a refined probabilistic ansatz of type

u = f ω + C + w ,
where 3 roughly speaking, f ω is the linear evolution of the randomized initial data, and the paracontrolled4 term C reads C = N I(t) P N f ω P ≪N (u -f ω ) , (0.1.8)

where I denotes the Duhamel operator. The nonlinear remainder w is smoother and lives above the deterministic threshold. In particular, the singular interactions are paracontrolled by the linear evolution [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF]. Notice that the proof in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], based on an induction on the size of frequencies of the initial data, is to be compared with quasilinear schemes. However, in contrast with the existing literature on paracontrolled calculus for parabolic equations, Bringmann further exploited the stochastic structure of the solution. Namely, when analyzing the singular interactions, Bringamnn crucially exploited the independence between the high frequencies and the low frequencies of randomized initial data. In the spirit of [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], we used in Chapter 4 a similar paracontrolled approach to prove almost-sure local well-posedness for a weakly dispersive equation, and we refer to the introduction of Chapter 4 and to paragraph 0.2.3 for a more detailed discussion on the paracontrolled probabilistic ansatz.

In the context of dispersive partial differential equations, paracontrolled calculus has also been developed in a spectacular way by Deng, Nahmod and Yue to prove the invariance of the Gibbs measure for the Wick-ordering Schrödinger equation on T 2 (i∂ t + ∆)u =: |u| p-1 u : , (t, x) ∈ R × T 2 , u(0) = u 0 , with an arbitrary odd power p ≥ 5. This is an important breakthrough, and the use of random averaging operators has proven to be extremely powerful and promising in addressing other challenging problems open for many years (see e.g. paragraph 0.2.3 concerning the invariance of the Gibbs measure for NLS on S 2 ). For Schrödinger equations, in contrast with the case of the nonlinear wave equation [START_REF] Oh | Invariant Gibbs measures for the 2-d defocusing nonlinear wave equations[END_REF], the case p ≥ 5 is much harder than the cubic case. The reason being that the high-low-low . . . low -type frequency interactions of the random data have regularity 1 2 -, whereas the deterministic critical threshold has regularity s c = 1 -2 p-1 , which is strictly larger than 1 2 when p is larger than 5. To prove local existence on the support of the Gibbs measure, and to remove these bad interactions, they built upon tools not only from the paracontrolled calculus, but also from random matrix theory, and introduced the random averaging operators, that encapsulate the bad interactions. Subsequently, in other remarkable works, the random averaging operators method has been successfully implemented and refined to obtain strong solutions on the support of the Gibbs measure. For instance, [START_REF] Sun | Refined probabilistic global well-posedness for the weakly dispersive NLS[END_REF] considered fractional Schrödinger equations posed on T, with weak dispersion. Specifically, the linear operator is (-∆) α with 1 2 < α ≤ 1 sufficiently close to 1 2 . They used random averaging operators in the physical space in place of the frequency space where [DNY19; DNY22] employed counting estimates. We also mention important results for wave-type equations. In [START_REF] Bringmann | Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: Measures[END_REF][START_REF] Bringmann | Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics[END_REF], Bringmann proved invariance of the Gibbs measure for NLW with a Hartree nonlinearity. Recently, [START_REF] Bringmann | Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation[END_REF] proved the invariance of the Gibbs measure for the cubic wave equation in T 3 . An important conjecture is the counterpart of this result for the Schrödinger equation.

We now give a very informal statement of the type of result one can hope to obtain from paracontrolled calculus, and we refer to the detailed introductions of [START_REF] Deng | Invariant Gibbs measures and global strong solutions for nonlinear Schröodinger equations in dimension two[END_REF][START_REF] Sun | Refined probabilistic global well-posedness for the weakly dispersive NLS[END_REF][START_REF] Bringmann | Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: Measures[END_REF][START_REF] Bringmann | Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics[END_REF] for in-depth discussion on these technics. Theorem 0.1.6 (Refined probabilistic ansatz (informal version)). There exists a full measure set Σ ⊂ H s such that for all ω ∈ Σ, the sequence of smooth local solutions (u n ) n≥0 to i∂ t u n + Lu n = N (u n ) , N = 2 n , u n (0) = P ≤N u ω 0 converges to a unique limit in u ∈ C([-T , T ], H s ), for some T = T (ω) > 0, which solves the equation with the full initial data. Moreover, u n admits the decomposition u n = e itL P ≤N u ω 0 + P n (u ω 0 ) + w n .

• The sequence of smooth remainders (w n ) converges in a subcritical space H σ .

• The random averaging operator P isolates the singular part, and inherits the randomness structure from the initial data, captured by certain matrix norms. It roughly corresponds to a superposition of paracontrolled terms of type C written in the informal refined ansatz (0.1.8), which solve a linear equation with a low-frequency potential independent of the high frequency initial data.

In Part III, we present two situations in which a refined probabilistic ansatz is required. First, in Chapter 4, we prove probabilistic well-posedness for the Schrödinger-half wave equation, which is a weakly dispersive with rich long-time dynamics but for which the Cauchy theory is yet to be understood. Then, we evidence in Chapter 5 some bad interactions for NLS posed on the sphere S 2 that compromise the regularization of the second Picard's iteration, and indicate therefore the need to use random averaging operators.

Recent progress and perspectives

To end this historical journey in the study of dispersive equations with random initial data, let us mention the spectacular breakthroughs made recently in related problems. We mention the results of Collot and Germain [CG19; CG20] and of Deng and Hani [DH21b;[START_REF] Deng | Full derivation of the wave kinetic equation[END_REF] in the study of wave turbulence, concerning the propagation of statistical quantities (also called propagation of chaos) by wave kinetic equations. Another lines of research concern the study of geometric equations with Brownian paths [START_REF] Bringmann | The wave maps equation and Brownian paths[END_REF], or the statistical study of the formation of rogue waves, related with the derivation of large deviation principles for cubic Schrödinger equations with random initial data [START_REF] Garrido | Large deviations principle for the cubic NLS equation[END_REF]. Very recently, [START_REF] Bringmann | Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation[END_REF] proved in an outstanding work the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation, by introducing a new caloric decomposition of the initial data, and by combining the analytic framework of random tensors with the combinatorial molecule estimates developed in [START_REF] Deng | On the derivation of the wave kinetic equation for NLS[END_REF][START_REF] Deng | Full derivation of the wave kinetic equation[END_REF], respectively.

To conclude, we can say that modern techniques such as paracontrolled calculus and random tensors play an important role in these advances, and that they will continue to be explored and developed in order to solve other problems that were considered out of reach for a long time (such as the invariance of the Gibbs measure for the three-dimensional cubic Schrödinger equation). But these new and exciting stories go far beyond the scope of this manuscript. . .

Globalization of probabilistic flows and generic dynamics.

One of the main question addressed in this dissertation can be formulated as follows. Given a random initial data of the form u(0) = u 0 + f ω 0 , u 0 ∈ H σ , f ω 0 ∈ H s , with s < s c ≤ σ , (0.1.9) and the associated probabilistic local flow u ω , can one prove that u ω extends globally in time, and say something on its long-time dynamics? In this paragraph, we give a non-exhaustive presentation of the existing technics and results in this direction to provide a context for the contributions collected in this dissertation.

Qualitative properties on the support of a Gibbs measure The purpose of this paragraph is to briefly discuss how Gibbs measures can replace conserved energies to obtain global well-posedness. We also give an insight into the qualitative properties of Hamiltonian dynamical systems in the presence of invariant measures, such as the Poincaré recurrence property. As explained in the next paragraph, these dynamics are in sharp contrast with scattering mechanisms that arise in the Euclidean space.

In order to fix the ideas, we consider the one-dimensional periodic Schrödinger equation, with a defocusing cubic nonlinearity:

(i∂ t + ∆)u(t, x) = |u| 2 u(t, x) , (t, x) ∈ R × T .

(NLS)

This is the historical model studied by Bourgain in [START_REF] Bourgain | Periodic nonlinear Schrödinger equation and invariant measures[END_REF]. Besides, the method we outline here is referred to as the Bourgain globalization argument. We recall that the Gibbs measure associated with the one-dimensional (NLS) is supported in H s (T) for all s < 1 2 . Hence, (NLS) is locally well-posed on the support of the Gibbs measure, which is above the deterministic threshold s = 0 given by Galilean invariance. Therefore, to construct local solutions, there is no need for the probabilistic Cauchy theory initiated a little later by Bourgain in order to address the two-dimensional case [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF]. To understand the long-time dynamics, however, Bourgain used the formal invariance of the Gibbs measure as a substitute for the conservation of the energy, for the (finite dimensional) Galerkin approximated system

(i∂ t + ∆)u N = Π N |Π N u N | 2 Π N u N , u(0) = Π N u 0 , (NLS N )
where, for x ∈ T, t ∈ R and N , we write

Π N u 0 (x) = |n|≤N (u 0 | e inx ) e inx , u N (t, x) = |n|≤N u n (t) e inx .
We briefly present the construction of the Gibbs measure associated with the Hamiltonian structure of NLS on T, and the probabilistic globalization argument of Bourgain.

• NLS as a Hamiltonian system. Equation (NLS N ) can be seen as a finite-dimensional Hamiltonian system. Setting p n = Re(u n ) , q n = Im(u n ) , it holds d dt p n q n n=1,...,N = -∂ qn H N (p, q) ∂ pn H N (p, q) n=1,...,N ,

with Hamiltonian

H N (p, q) = 1 2 |n|≤N n 2 (p 2 n + q 2 n ) + nonlinear terms .
The Liouville theorem states that the volume measure dp dq := |n|≤N dp n dq n is invariant under the flow of (NLS N ). As a consequence of the conservation of the Hamiltonian under the flow, the (finite-dimensional) approximate Gibbs measure dµ N (p, q) = Z -1 N exp -βH N (p, q) dp dq is also invariant under the flow. Namely, denoting the flow by Φ N (t) and writing (p(t), q(t)) = Φ N (t)(p 0 , q 0 ), we have

µ N (Φ N (t) -1 (A)) =
A exp -βH N (p(t), q(t)) dp(t) dq(t) = A exp -βH N (p 0 , q 0 ) dp 0 dq 0 = µ N (A) .

Additionally, we observe from the Cauchy-Lipschitz theorem for ODE's and from the mass conservation that the solutions to (NLS N ) are global. This being said, the objective is now to remove the truncation and to construct the dynamics on the support of the (formal) infinite-dimensional Gibbs measure 5 associated with (NLS)

" dµ(u) = Z -1 exp -H(u) -M (u) du" . (0.1.10)

The Bourgain's globalization argument achieves these objectives. Since this method is now standard, we only sketch the contours of the argument, and we refer to the original article [START_REF] Bourgain | Periodic nonlinear Schrödinger equation and invariant measures[END_REF] for further details.

• Construction of the Gibbs measure. Given a probability space (Ω, P, A) and a sequence of independent normalized Gaussian variables, we consider the Gaussian free field ρ induced by the map ω ∈ Ω → n∈Z g n (ω) ⟨n⟩ e inx . (0.1.11)

Exploiting the independence of the Gaussian variables g n , we see that the above series makes sense in L 2 (Ω; H s (T)) as the limit of a Cauchy sequence, for any s < 1 2 . From the Sobolev embedding, we see that ∥u∥ L 4 is finite almost-surely with respect to the Gaussian measure. Then, the Gibbs measure (0.1.10) associated with (NLS) is the density measure with respect to dρ dµ(u) = exp -V (u) dρ(u) , V (u) = 1 2 T |u(x)| 4 dx .

Furthermore, the sequence of finite-dimensional Gibbs measure (µ N ) associated with the truncated system weakly converges to dµ. For every open set U ⊂ H s (T), with s < 1 2 , we have µ(U ) = lim

N →∞ µ N (U ∩ E N ) , with E N = span e inx | |n| ≤ N .
This essentially follows from the almost-sure convergence of the series (0.1.11), and from the integrability of the density.

• Global well-posedness on the support of the Gibbs measure. We recall that (NLS) is locally well-posed on the support of µ, since it is well-posed on H of µ N under the flow Φ N . This leads to uniform bounds on Φ N (t) on full-measure sets and on long-time intervals. The argument is as follows. Set

Ω s,K = u ∈ E N | ∥u∥ H s ≤ K ,
for fixed s < 1 2 and 1 < K. We have the large-deviation estimate, for some constant c = c(s) > 0:

µ N ( c Ω s,K ) ≤ e -cK 2 .

Subsequently, we see from the local Cauchy theory for the approximated equation (NLS N ) that there exists τ ∼ K -θ , for some θ > 0, such that

∥u 0 ∥ H s ≤ K =⇒ sup t∈[-τ ,τ ] ∥Φ N (t)u 0 ∥ H s ≤ 2K .
Next, given T > 1, we set

Ω s,K = ⌈ T τ ⌉ j=0 Φ(-τ j) Ω s,K .
By construction, for all u ω 0 ∈ Ω s,K we have

sup |t|≤T ∥Φ N (t)u 0 ∥ H s ≤ 2K .
Moreover, it follows from the sub-additivity of µ N and its invariance under the flow Φ N and the large-deviation estimate that µ N ( c Ω s,K ) ≤ T K θ e -cK 2 .

As a consequence, for every ε > 0 and T > 1, there exists a set Ω such that for all u 0 in this set,

sup |t|≤T ∥Φ N (t)u 0 ∥ H s ≤ C log T ε 1 2 , µ N ( c Ω s,K ) ≤ ε .
To establish the long-time existence on the support of µ and the invariance of µ, it remains to combine the above uniform bounds with the weak convergence of the measures µ N , together with a deterministic approximation lemma that yields the strong convergence of Φ N (t) to Φ(t) in H s , uniformly in time. The proof of this lemma consists in computing the difference between Φ(t)u 0 and Φ N (t)Π N u 0 by exploiting local well-posedness in X s,b -spaces, and iterating on a finite number of small spacetime slabs.

• Long-time dynamics on the support of the Gibbs measure.

As a byproduct of the existence and invariance of the Gibbs measure, the classical theory of dynamical systems yields asymptotic information about the flow, such as the Poincaré recurrence property. Specifically, by obtaining strong local solutions on the support of a Gibbs measure, and by proving its invariance under the flow-map, we are in a very good position to prove the following statement:

There exists a full µ-measure set Σ in H s , such that for any u 0 ∈ Σ the Cauchy problem with initial data u 0 has a strong global solution in C(R; H s ). Moreover, the flow map Φ satisfies the flow property

Φ(t)(Σ) = Σ , Φ(t 1 + t 2 ) = Φ(t 1 ) • Φ(t 2 ) ,
and the Poincaré recurrence property: for all t ∈ R there is a subsequence of integers (n k ) k going to +∞, such that

lim k→∞ ∥Φ(tn k )u 0 -u 0 ∥ H s = 0 .
We refer, for instance, to Theorem 6 and Corollary 1.2 in [START_REF] Sun | Gibbs measure dynamics for the fractional NLS[END_REF] for a precise formulation.

The Bourgain globalization argument also works in the context of probabilistic Cauchy theory (below the deterministic threshold) provided that we can prove local well-posedness for solutions with random initial data according to the support of the Gibbs measure. An illustrative example is the subsequent series of works of Bourgain and Bulut [BB14c;[START_REF] Bourgain | Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case[END_REF][START_REF] Bourgain | Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball II: The 3D case[END_REF] who proved global existence for NLS and NLW on the ball with Dirichlet boundary conditions. Note, however, that their method does not provide the flow property and the invariance of the Gibbs measure. Furthermore, Bourgain's globalization argument adapts when probabilistic local well-posedness follows from the paracontrolled ansatz discussed in Section 0.1.2 (e.g. [DNY19; DNY22; ST21; Bri20a; Bri20b]). Finally, we mention that one can also systematically construct weak solutions from the existence of a Gibbs measure, by using Prokhorov compactness theorem and Skorokhod convergence theorem. We refer to [START_REF] Burq | Remarks on the Gibbs measures for nonlinear dispersive equations[END_REF] for results in this direction, applied to a number of weakly dispersive models. We also refer to [START_REF] Tzvetkov | Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation[END_REF][START_REF] Deng | Invariance of the Gibbs measure for the Benjamin-Ono equation[END_REF] for some important contributions to the invariance of the Gibbs measure for the Benjamin Ono equation.

Before turning to other dynamical behavior of dispersive nonlinear equations in the presence of randomness, let us point out that the dynamics on the support of a Gibbs measure is not compatible with scattering mechanisms. We propose a proof of this statement, by showing that if a solution of a globally well-posed equation satisfies a certain time decay, materialized by uniform space-time bounds in Lebesgue spaces L q t (R; L q x (R d )), with q < ∞, then the only invariant measure is the Dirac mass at zero. Proposition 0.1.7 (Invariant measure and scattering are not compatible). Let µ be a probability measure on H s , invariant under a flow Φ(t) globally defined on a full µ-measure set Σ. Namely, for every Borelian set A ∈ H s , and t, τ ∈ R,

µ(Φ(t)(A)) = µ(A) , Φ(t)(Σ) = Σ , Φ(t + τ ) = Φ(t) • Φ(τ ) .
If for some 2 < p, q < ∞, we have that for all K > 0, there is a C(K) ≥ K such that for all u 0 ∈ Σ,

∥u 0 ∥ H s ≤ K =⇒ ∥Φ(•)u 0 ∥ L ∞ t (R;H s )∩L p t (R;L q x ) ≤ C(K) , then µ = δ 0 is the Dirac mass at zero. Proof. For λ > 0, we set A λ = u 0 ∈ Σ : ∥u 0 ∥ L q
x > λ , and we prove that µ(A λ ) = 0 for almost every λ. First, we observe from the global well-posedness assumption that for all K > 0,

B K := u 0 ∈ Σ : ∥u 0 ∥ H s ≤ K ⊆ u 0 ∈ Σ : sup t∈R ∥Φ(t)u 0 ∥ H s ≤ C(K) =: B K .
Then, given τ ∈ R, we use the invariance of µ and the flow property to deduce that

µ(A λ ∩ B K ) ≤ µ u 0 ∈ Σ : sup t∈R ∥Φ(t)u 0 ∥ H s ≤ C(K) , ∥u 0 ∥ L q x > λ = µ u 0 ∈ Σ : sup t∈R ∥Φ(τ ) • Φ(t)u 0 ∥ H s ≤ K , ∥Φ(τ )u 0 ∥ L q x > λ = µ B K ∩ Φ(-τ ) A λ .
Since the left-hand side is independent of time, integrating over λ p-1 dλ, and then over τ on an arbitrary long-time interval [0 , T ), we obtain

T 1 0 µ(A λ ∩ B K )λ p-1 dλ ≤ T 0 +∞ 0 µ u 0 ∈ B K : Φ(τ )u 0 ∈ A λ λ p-1 dλ dτ ≲ +∞ 0 T 0 B K 1 Φ(-τ )(A λ ) (u 0 ) dµ(u 0 ) dλ dτ .
We denote the Lebesgue measure on the line by L, and we deduce from the Fubini Tonelli's formula that

+∞ 0 T 0 B K 1 Φ(-τ )(A λ ) (u 0 ) dµ(u 0 ) dλ dτ = B K +∞ 0 T 0 1 Φ(-τ )(A λ ) (u 0 ) dτ dλ dµ(u 0 ) = B K +∞ 0 L [0,T ] τ : ∥Φ(τ )u 0 ∥ L q x > λ λ p-1 dλ dµ(u 0 ) = 1 p B K ∥Φ(•)u 0 ∥ p L p t L q x dµ(u 0 ) ≤ 1 p C(K) p .
Finally, we proved that for all T > 0,

T 1 0 µ(A λ,K )λ p-1 dλ ≤ 1 p C(K) p .
By letting T go to +∞, we obtain that µ(A K,λ ) = 0 for almost every 0 < λ ≤ 1. Since K is arbitrary and Σ has full µ-measure, this proves that µ = δ 0 .

Note however that scattering is not incompatible with the existence of quasi-invariant measures. Moreover, for the defocusing Schrödinger equation on the line with initial data supported just below L 2 (R), Burq and Thomann proved in [START_REF] Burq | Almost sure scattering for the one dimensional nonlinear Schrödinger equation[END_REF] global existence when the nonlinearity p > 1, and scattering when p > 3. This result is a consequence of uniform space-time bounds on the density obtained from quasi-invariant measures, which are constructed through a compactifying procedure called the Lens Transform.

Quasi-invariant measures

We briefly mention the use of quasi-invariant measures to obtain globalization of probabilistic flows, in the absence of invariant measures and conserved energies. This is a promising strategy, though beyond the scope of this dissertation. Given a Gaussian measure µ and a Hamiltonian flow {Φ t } t , we say that µ is quasi-invariant under the flow if for every t,

(Φ t ) # µ ≪ µ ,
where (Φ t ) # denotes the push-forward of the measure by the flow. In such a case, by the Radon-Nikodym theorem, there exists a density f t ∈ L 1 ( dµ) such that

(Φ t ) # µ = f t µ .
Then, one can deduce global well-posedness for the flow (Φ t ) from L p ( dµ)-bounds on the density, combined with the Bourgain globalization argument. This strategy, initiated by Tzvetkov in [START_REF] Tzvetkov | Quasi-invariant Gaussian measures for one dimensional Hamiltonian PDE[END_REF] generalizes the use of invariant Gibbs measure, which are also absolutely continuous measures with respect to a Gaussian measure. Yet, the use of quasi-invariant measure is much more flexible. We refer to [BT20; FT22; GLT21; OTT19] who have successfully implemented and developed this method, and to the references therein.

Deterministic methods

In this section, we present some globalization methods that do not rely on the use of an invariant or quasi-invariant measure. In light of the previous discussion, these methods can be developed to study scattering mechanisms, and are employable at regularities not too far from the regularity required to access a conserved coercive quantity. They consist in obtaining a uniform a priori bound on the nonlinear remainder v = u -u ω L , and in iterating the local existence result through the uniform bounds. They are called deterministic methods, as opposed to the probabilistic Bourgain globalization argument we described in the previous paragraph.

• The high-low method. This method, introduced by Bourgain in [START_REF] Bourgain | Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity[END_REF], consists in decomposing the solution into a low frequency part, and a high frequency part. The low frequency part being smooth, we can use the (almost) conservation of energy to control it. On the other hand, the high frequency part turns out to be well approximated by the linear flow, thanks to a certain nonlinear smoothing effect. Thus, each part of the solution is under control, and we obtain a priori bounds for the solution. In [START_REF] Colliander | Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L 2 (T)[END_REF], Colliander and Oh implemented the high-low method in a probabilistic setting to prove the almost-sure global existence for the renormalized NLS on T, at negative regularities. Note that in such regimes, conservation of mass, almost-surely defined for the nonlinear remainder v = u -u ω L , has been exploited. We also mention the later work [START_REF] Poiret | Probabilistic well-posedness for the supercritical nonlinear harmonic oscillator[END_REF], which applied the high-low method in a probabilistic framework to prove the almost-sure global existence and scattering for NLS in R d , d ≥ 3, in a small data regime.

• The use of modified energies When the remainder v = u -u ω L is in H 1 one can try to obtain a priori bounds on v through a modified non-conserved energy, of type

E[v](t) := 1 2 M ∇v(t, x) 2 dx + 1 2 M ∂ t v(t, x) 2 dx + 1 p + 1 M v p+1 (t, x) dx .
In [START_REF] Burq | Probabilistic well-posedness for the cubic wave equation[END_REF], Burq and Tzvetkov proved almost-sure global existence for the cubic defocusing NLW in T 3 , by the energy method for v together with a Cauchy-Schwarz estimate and a Gronwall argument. A refinement, proposed in [START_REF] Lührmann | On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on R 3[END_REF], consists in putting the linear evolution in the potential energy in order to cancel certain terms. See also the works [OP16; OP17; LM14b; SX16] and the references therein, who proved almost-sure global well-posedness for NLW in different contexts, but in the same spirit.

However, in the case of NLS, the method is less favorable since the energy does not control ∂ t v. Instead, one has to replace it using the equation, and losing therefore ∆v. We mention the work of [START_REF] Oh | On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities[END_REF] that develop from ideas used to treat NLW, but in higher dimensions d ≥ 5. We also mention the conditional global well-posedness result [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF].

Almost-sure scattering When a state evolves under nonlinear Schrödinger flow, with a defocusing self-interacting nonlinearity, its dynamics in long-time are well approximated by a free linear evolution. This scattering phenomenon is well understood in the energy space, when the equation is energy-subcritical, in or energy-critical as in [RV07; Vis12], for the four-dimensional case, [START_REF] Colliander | Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3[END_REF] for the three-dimensional, and [START_REF] Dodson | Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 2[END_REF] for the mass-critical case, respectively. The defocusing cubic Schrödinger equation in R 3 gives an illustrative example of scattering in energy space, and we sketch the proof as a prerequisite to presenting the general strategy leading to probabilistic scattering type of results. The cubic NLS is energy-subcritical, and hence it is locally well-posed in the energy space H 1 (R 3 ). We see from the coercive conserved energy that the solution is global. Moreover, we deduce from Strichartz's estimates that a given solution scatters provided that we have a priori bound on the space-time norm

R×R 3 |u(t, x)| p dx dt < ∞ , for some 3 10 < p < 10 .
Such an a priori decay estimate follows from the so-called interaction Morawetz inequality,

I×R 3 |u(t, x)| 4 dt dx ≤ C sup t∈I ∥u(t)∥ 2 L 2 x ∥u(t)∥ 2 Ḣ 1 2 .
(0.1.12) combined with some conservation laws. Now that we have scattering in the energy space, a natural extension would be to prove that for any 1 2 < s < 1, the local solution in H s (R 3 ) is global and scatters. Unfortunately, this simple question turns out to be a very challenging open problem that is still far from being solved in all generality. Partial results were obtained by employing the celebrated I-method [START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF] and refinements based on long-time Strichartz estimates [START_REF] Dodson | Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when n = 3 via a linear-nonlinear decomposition[END_REF][START_REF] Dodson | Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when d = 2, 3 and u 0 is radial[END_REF]. We refer to paragraph 0.2.2 and to the introduction of Chapter 2 for a detailed discussion on this problem, as well as detailed presentation of the I-method and of the Morawetz estimate. Despite the difficulty of the scattering problem in the subcritical and critical regimes 1 2 ≤ s < 1, it is interesting to achieve a probabilistic result. Besides, there exists a local-probabilistic flow even in supercritical regimes. The following question arises:

• Does a general supercritical initial data of type (0.1.9) scatter at infinity? In recent years, the stability of the scattering mechanism under random perturbations have been extensively studied. In the small data regime, global well-posedness and scattering can be obtained as a by-product of almost-sure local well-posedness. See, for example, [START_REF] Poiret | Solutions globales pour l'équation de Schrödinger cubique en dimension 3[END_REF][START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]. Outside the small data regime, however, the problem is much more complicated and the first class of problems to consider are energy-critical problems, when the deterministic theory in the critical energy space is well understood. In such a context, the probabilistic flows are in H s for s close to 1, but the nonlinear remainder is in H 1 . Therefore, the probabilistic ansatz (0.1.5) should allow analyzing the long-time behavior of the nonlinear term v in the energy space, solution to a nonlinear Schrödinger equation with stochastic forcing terms.

(i∂ t + ∆)v = |v + u ω L | p-1 (v + u ω L ) , (t, x) ∈ R × R d , v(0) = 0 , (NLS f )
where u ω L := e it∆ u ω 0 is the linear evolution of the random initial data. The first step is to use the stability theory of the deterministic equation in H 1 to obtain a conditional scattering result. By doing so, we reduce the problem to obtaining a uniform control on the H 1 -norm of v. Next, in a second step, the strategy is to find appropriate modified energies and modified Morawetz inequalities for v, and to perform a double bootstrap argument on these two quantities.

For this strategy to work, another ingredient is often necessary to absorb the weights that come from the Lin Strauss and Morawetz type inequalities. It consists in concealing a radial symmetry assumption in the randomization procedure, by assuming that u 0 is radially symmetric in (0.1.7). Note that a Wiener randomization of a radially symmetric function need not be radially symmetric. Nevertheless, the solution retains some of the decay properties that radially symmetric functions have from the radial Sobolev embedding theorem. This strategy was first implemented by Dodson, Lührmann and Mendeslon [START_REF] Dodson | Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation[END_REF] to prove almost-sure scattering for the energy critical wave equation in R 4 . Then, Kilipp, Murphy and Visan obtained a similar almost-sure scattering result for the energy critical Schrödinger equation in R 4 [KMV19], which was improved in [START_REF] Dodson | Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation[END_REF]. In the interest of considering the largest possible class of initial data, Bringmann implemented in [START_REF] Bringmann | Almost sure scattering for the energy critical nonlinear wave equation[END_REF] tools from harmonic analysis, such as Bourgain bushes, and solved the scattering problem for the energy-critical wave equation, without any radial assumptions. This approach is less suitable for Schrödinger-type equations, since the wave packet propagates faster. Nevertheless, the transposition of this type of argument to the Schrödinger case is an interesting problem.

In the energy-subcritical framework (e.g., cubic NLS in R 3 ), however, there was no such almost-sure scattering result outside the small data regime. We refer to 0.2.2 and Chapter 2 for the presentation of an almost-sure scattering result in this framework, with radially-symmetric randomized initial data.

Stability of coherent states under random perturbations

The probabilistic Cauchy theory is mainly concerned with the perturbation of dispersive solutions, in the defocusing regime, or for small initial data. However, in the focusing regime where dispersive equations generally exhibit rich dynamical behaviors, much less is known about the effect of random perturbations. Nevertheless, stability results for special coherent solutions of dispersive partial differential equations under rough and random perturbations have been obtained recently. The first long-time result in this direction is due to Kenig and Mendelson, for an energy-critical model. In [START_REF] Kenig | The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data[END_REF], they proved that when s > 5 6 , one can construct a global solution of the focusing quintic wave equation in R 4 that scatters to a modulated soliton, for almost every initial data in H s × H s-1 . The considered solitons form a one-parameter family (ϕ a ) a>0 of stationary radially symmetric solutions to NLW satisfying the elliptic equation ∆ϕ a + ϕ 5 a = 0 .

They have regularity Ḣs , for any s > 1 2 , and are extremizers of the Sobolev embedding Ḣ1 (R 3 ) → L 6 (R 3 ) given by the explicit formula

ϕ a (x) = (3a) 1 4 1 + a|x| 2 1 2
.

The result states as follows.

Theorem 0.1.8 (Almost-sure asymptotic stability for energy critical NLW, [KM19]). Let s > 5 6 , and (f ω 0 , f ω 1 ) a suitably randomized initial data. There exists a set full-measure set Σ ⊂ H s such that for all f ω 0 ∈ Σ, there exist ε > 0, h : Σ → R with |h| ≤ ε 2 and a unique solution u to

-(∂ 2 tt -∆)u = -u 5 , u(0) = ϕ + εf ω 0 + h(εf ω 0 , εf ω 1 )Y , ∂ t u(0) = εf ω 1 + h(εf ω 0 , εf ω 1 )κY , (NLW) such that u(t) = ϕ a(t) + v , with ȧ ∈ L ∞ t ∩ L 1
t is small, and v is a small radiation term in the sense that

∥v∥ L 8 t,x ≲ ε .
The function h ensures an orthogonality condition for the pure point spectrum of the linearized operator around the ground state. Besides, the random perturbations are projected onto a Lipschitz manifold because of an unstable direction, and we refer to the article [START_REF] Kenig | The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data[END_REF] for further details on the exact randomization procedure. To establish this type of result, the main challenge was to deal with a linear source term stemming from the soliton. To overcome this difficulty, the authors decided to adapt the randomization procedure to the linearized operator around the soliton. Instead of using standard Fourier multipliers in the randomization procedure (0.1.7), they used a distorted Fourier transform that diagonalizes the linearized operator around the ground state:

H a = -∆ -5ϕ 4 a .
Unfortunately, this operator has a resonance at zero, which obliges to use weighted spaces, and to derive technical kernel estimates. Note that a weighted Sobolev condition is also required on the initial data. The only existing counterpart of this almost-sure asymptotic stability result for the Schrödinger equation is the matter of Chapter 3. As for the Schrödinger case, we stress out that the Duhamel integral does not gain a derivative, so that a dyadic decomposition and bilinear estimates must be performed to establish the probabilistic local well-posedness. However, this becomes quite difficult in the presence of a potential, since the distorted Fourier transform does not share the multiplicative properties of the Fourier transform. We refer to paragraph 0.2 and to Chapter 3 for a detailed presentation of an almost-sure asymptotic stability result for NLS.

Inspired by the work of Kenig and Mendelson on the focusing energy-critical radial wave equation, Bringmann proved in the [START_REF] Bringmann | Stable blowup for the focusing energy critical nonlinear wave equation under random perturbations[END_REF] the almost-sure stability of blow-up profile for the same wave equation below the energy space. The ODE blow-up profile is

u (T ) (t, x) := κ T -t -1 2 , T > 0 .
Thanks to finite propagation speed, it can be localized in the interior of the light cone

C T := (t, x) ∈ [0 , T ] × R 3 , |x| ≤ T -t .
The almost-sure stability result states as follows.

Theorem 0.1.9 (Stability of the blow-up profile under random perturbations, [START_REF] Bringmann | Stable blowup for the focusing energy critical nonlinear wave equation under random perturbations[END_REF]). For 0 < δ < δ 0 small enough, there exist a set Σ δ such that for all ω in this set, there is a T (ω) ∈ [1 -δ , 1 + δ] and a solution u : C T → R to (NLW) with suitably randomized initial data (f ω 0 , f ω 1 ), satisfying

∥(T -t) -3 4 u -u T ∥ L 2 t L 4
x (C T ) ≲ δ . Moreover, there is C, c > 0 and θ > 0 such that for all 0 < δ < δ 0 ,

P(Ω \ Σ δ ) ≤ C exp -cδ θ .
The proof revisits deterministic methods in the probabilistic setting, such as the Strichartz estimates for one-parameter semigroup S(t) associated to the linearized operator around the blow-up profile in similarity coordinates. The challenge was to obtain probabilistic estimates for the difference operator S(t) -S 0 (t), where S 0 (t) is the free wave propagator. To achieve such a goal, Bringmann proved fine oscillatory integrals estimates. In a subsequent work [START_REF] Fan | Construction of L 2 log-log blowup solutions for the mass critical nonlinear Schrödinger equation[END_REF], Fan and Mendelson proved the almost-sure stability of the log-log blow-up profile for the mass critical Schrödinger equation in R 2 . However, the randomized data have a particular structure, since their Fourier transform is piecewise constant, with some decaying assumption.

To our knowledge, these aforementioned results are the only ones to date to prove the stability of coherent states under supercritical random perturbations.

Presentation of the results in this manuscript

In this section, we present the main contributions of the thesis put in perspective with the research themes discussed in the previous Sections 0.1.1, 0.1.2 and 0.1.3. We focus on the manifestations of the general questions raised at the end of Section 0.1.1. However, we merely state and motivate the theorems, and discuss the main ingredients of the proofs, avoiding technicalities. Thus, we defer the precise description of the arguments and the detailed list of references until the introduction of the individual chapters.

On the Cauchy theory in scaling-supercritical regimes

At scaling-supercritical regularities, the heuristics predict that instabilities will occur, thus undermining the possibility of establishing a general Cauchy theory. The matter of this section is to clarify the genericity of such instabilities, in the context of the random data Cauchy theory. Before going further, let us recall some terminology and basic facts about the local well-posedness for a general dispersive equation, which reads

i∂ t u(t, x) -L(u)(t, x) = N (u)(t, x) , (t, x) ∈ R × M , u(0) ∈ H s (M ) .
Here, (M, g) is a Riemannian manifold, L is a self adjoint operator on H k (M ), for some k large enough, and N denotes the nonlinearity, possibly depending on a finite number of space-derivatives of u. Definition 0.2.1 (Local well-posedness, see Definition 1.1.1). We say that the above Cauchy problem is well-posed in H s (M ) if for every bounded set B ⊂ H s (M ), there exist T = T (B) > 0 and a Banach space X T continuously embedded in C([t 0 , T ], H s (M )) such that for any u 0 ∈ B, there is a solution u ∈ X T satisfying the Duhamel integral representation formula

u(t) = e i(t-t 0 )L u 0 -i t t 0 e i(t-τ )L N (u(τ )) dτ . (0.2.1)
Moreover, the following holds.

1. (Uniqueness) Let u 1 , u 2 ∈ X T satisfying (0.2.1). If, for some

t 0 ≤ t ≤ T , u 1 (t) = u 2 (t), then u 1 = u 2 on [t 0 , T ]. 2. (Continuity) The flow map u 0 ∈ B → u ∈ X T is continuous. 3. (Persistence of regularity) If u 0 ∈ H σ (M ) with σ > s, then u ∈ C([t 0 , T ], H σ (M )).
When the flow-map is uniformly continuous, we say that the Cauchy problem is semi-linearly well-posed. This certifies the absence of instability for the high frequencies of the solution. Usually, we prove uniform well-posedness by performing a Picard's iteration argument, so that the flow-map is actually Lipschitz continuous. Furthermore, the flow-map is analytic when the nonlinearity is polynomial.

In the particular case of nonlinear Schrödinger and wave equations with power-type nonlinearity in the Euclidean space, or in the periodic case, the scaling invariance actually dictates the critical threshold for the uniform local well-posedness theory, at least when s c > 0. Indeed, according to [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF], we have local well-posedness in H s (R d ) as soon as s ≥ max(0, s c ), and norm-inflation occurs when s < max(0, s c ). For the Schrödinger equation on compact manifolds, however, the picture is less clear, since the behavior of the solution depends strongly on the geometry of the manifold, even in short time, as a consequence of the infinite speed of propagation for the solutions of NLS. For instance, the cubic NLS is not uniformly well-posed on the Euclidean sphere S 2 when s < 1 4 , whereas local well-posedness holds up to the scaling threshold 0 < s on the torus T 2 , respectively (see Subsection 0.2.3).

We briefly discuss the instability mechanism. The aforementioned work of Christ, Colliander and Tao [START_REF] Christ | Ill-posedness for nonlinear Schrodinger and wave equations[END_REF] does not only prove that the flow map is not continuous at scaling-supercritical regularity 0 < s < s c , but it also provides explicit sequences of initial data that indicate the mechanism involved. Specifically, they constructed a sequence of concentrating blow-up profiles converging to zero in H s , but for which the H s -norm of the corresponding smooth local solution inflates, in arbitrarily small-time. Namely, by writing Φ(t) the local-in-time solution map at time t on H σ , for σ large-enough, there is a sequence (u 0,n ) n in the Schwarz class and a sequence of times (t n ) going to zero, such that

lim n→∞ ∥u 0,n ∥ H s = 0 , but lim n→∞ ∥Φ(t n )(u 0,n )∥ Ḣs = +∞ .
Such a norm-inflation mechanism indicates the existence of low-to-high frequency cascades in arbitrarily short-time, as mentioned in the general introduction 0.1.1. This is consistent with the scaling heuristic that when s < s c , studying short-time existence for small data amounts to study large data. We refer to the pioneering work of [START_REF] Lebeau | Non linear optic and supercritical wave equation[END_REF], and many other references cited in the introduction of Chapter 1, where we discuss norm-inflation in more details.

In contrast, the probabilistic Cauchy theory, as stated in Theorem 0.1.1, provides a full-measure set of initial data for which a strong solution exists, at least up to a certain time. Furthermore, probabilistic solutions are natural in the sense that they are the limit of the regularized solutions by convolution. Notice, however, that by choosing a bad regularization procedure one can show norm-inflation for any data, as the result of [START_REF] Xia | Generic Ill-posedness for wave equation of power type on three-dimensional torus[END_REF] indicates. Specifically, for any u 0 ∈ H s , the author proved that there exist (u 0,n ) in Schwartz-class, and (t n ) going to zero, such that

lim n→∞ ∥u 0,n -u 0 ∥ H s = 0 , but lim n→∞ ∥Φ(t n )u 0,n ∥ Ḣs = +∞ .
In light of this informal discussion, there are bad regularization procedure that lead to norminflation for any initial data, and good initial data for which the natural regularization by convolution converges to a strong solution. The natural question raised by Sun and Tzvetkov is about the genericity of bad initial data leading to norm-inflation even when regularized by convolution. Such bad initial data form a pathological set, defined as

P = u 0 ∈ H s : lim sup ε,t→0 ∥Φ(t)(ρ ε * u 0 )∥ H s = +∞ .
Here, (ρ ε ) is a fixed approximate identity.

Pathological set of initial data for scaling-supercritical NLS joint work with L. Gassot [START_REF] Camps | Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations[END_REF]. In the case of the wave equation, Sun and Tzvetkov proved in [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] that the pathological set contains a G δ dense set. Hence, two different types of genericity coexist:

• The Baire category genericity, in a complete metric space. It is a topological notion of genericity.

• The probabilistic genericity.

An important subsequent remark is that the full-measure set of data leading to local well-posedness cannot contain a dense G δ set, since G δ sets are stable under intersections.

In Chapter 1, we generalized the result of [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] for Schrödinger-type equations, hence precising the Cauchy theory at scaling supercritical regularities for NLS, whose study is the subject of Part II, and also for the Schrödinger half-wave equation, less studied, which is the matter of Chapter 4. The result can be formulated as follows.

Theorem 0.2.2 (Pathological set of scaling-supercritical data for NLS, [START_REF] Camps | Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations[END_REF]). Let 0 < s < s c . The pathological set P contains a dense G δ set S ⊂ H s (R 3 ): for every u 0 ∈ S, there is a sequence ε n → 0, and a sequence of times t n → 0, such that the local smooth solution u εn n = Φ(t)(u 0 * ρ εn ) of NLS exists in H s (R 3 ) up to the time t n , but the family experiences norm-inflation

lim n→∞ ∥u εn n (t n )∥ Ḣs (R 3 ) = +∞ .
The result of [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] does not only provide a dense G δ set, but it also gives an explicit construction, which we reformulate here in the Schrödinger case.

-The individual inflating bubbles. The elementary building blocks are the inflating bubbles, widely used to prove norm-inflation in a variety of contexts, and introduced in [START_REF] Christ | Ill-posedness for nonlinear Schrodinger and wave equations[END_REF]. Given a fixed, smooth, and radially-symmetric profile φ ∈ C ∞ c (R 3 ), we define the bubble concentrating at scale n -1 by v n (0, x) = log(n) -γ n for some γ > 0. The bubble is convoluted by an approximate identity in order to state the result in the context of probabilistic Cauchy theory (see Theorem 0.1.1). Then, the ODE profile is

v ε n (t, x) := v ε n (0, x)V (t|v ε n (0, x)| p-1 ) , V (t) = e itσ .
It solves the nonlinear ODE corresponding to NLS but without the Laplace operator (which is the dispersive term, that can be replaced by any other reasonable operator here)

i∂ t v ε n + σ|v ε n | p-1 v ε n = 0 , v ε n (0) = ρ ε * v n (0) .
Uniformly in ε, we have the lower bound

∥v ε n (t)∥ H s ≥ c log(n) -γ (n 3-2s t) s . (0.2.2)
As a consequence, after a time t n ∼ log(n) β n -(3-2s) with β > γ, the H s -norm of the profile starts to inflate, as n goes to infinity so that the initial data concentrate.

-The small dispersion analysis. The proof of [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] consists in adapting the small dispersion analysis due to [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF], in the presence of the convolution and with an infinite number of inflating bubbles, sufficiently distant from each other. First, let us outline the standard small dispersion analysis for one bubble. It consists in proving that in small-time, the nonlinear profile v n dominates the effect of the dispersion, so that the solution to the dispersive equation NLS with data v n (0) stays close to the bubble v n (t). This is a small-time analysis. Still, one needs to give the inflating bubble time to burst (that is t ≥ t n ). The control of the difference from the solutions and the bubble follows from energy considerations.

-The "tanghuru" construction. As in the preparation of tanghurus, candied fruit lollipops very popular in some Asian countries, the construction of the dense pathological set consists in putting side by side an infinity of bubbles, concentrated at arbitrarily small scales, and sufficiently distant from each other. In addition, the construction is made around an arbitrary function in C ∞ c (R 3 ) in order to generate a dense set in H s (R 3 ) that is, for some k 0 ≫ 1,

S = u 0 + ∞ k=k 0 v n k (0, x -x k ) , n k = e e k , x k = 1 k , u 0 ∈ C ∞ c (R 3 ) .
Then, the proof consists of two steps. In a first step, Sun and Tzvetkov isolate the n-Th bubble at time t n and scale ε n , and they perform the standard small-dispersion analysis for u solution to NLS with initial data v n (0), which is just a piece of the initial data in S. Subsequently, they use the finite propagation of speed to ensure that the remaining part of the solution does not interfere with the inflating bubble v n , and prove that the full solution u εn stays close to u εn up to the time t n , which is known from the first step to be close to the inflating profile v εn n . At the end of the day, the difference from u εn to v εn n in H s is controlled up to time t n , and the inflation of v n , described by the lower bound (0.2.2) leads to the inflation of u εn .

The second step of the proof strongly relies on the finite speed of propagation, which is a specific feature of wave-type equations. For this reason, it was not clear how to obtain an analogous result in the case of Schrödinger-type equations, or for any other dispersive equation that does not satisfy finite speed of propagation. Our main observation is the following. Given a fixed regularization scale ε > 0, the convolution by the profile ρ ε has for side effect to collapse the bubbles that concentrate at scale ≪ ε. Therefore, at some appropriate time t n , the bubble concentrated at scale ∼ ε n is the only one to inflate. Consequently, when isolating the bubble that inflates at time t n , there is no need to use finite speed propagation to make sure that the evolution of the other bubbles, more concentrated, does not interfere, and rules out the inflation. Let us now give some comments and consequences of this observation that is the key of our proof.

• In [START_REF] Camps | Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations[END_REF] we actually proved the result for any power-type nonlinearity |u| p-1 u, with p ≥ 3 an odd integer, as stated in Chapter 4. This assumption is technical, and guarantees that the inflating bubbles are smooth. It should be possible to remove it.

• When p ≥ 7, in the energy supercritical case, we only proved that P contains a dense set. In order to get a dense G δ -set, however, we need to uniformly control the time of existence of the smooth solution, say in H 2 , and the supercritical nature of the energy prevents us from doing so by basic energy considerations. Still, being able to treat the energy-supercritical case is an improvement of the method in [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] that requires global existence of smooth solutions.

• A similar comment is about the focusing or defocusing nature of the nonlinearity. We prove that there is a G δ dense subset when we have access to a conserved coercive energy, and at least a dense set of inflating data in the focusing case, which is also a novelty. Note however that ill-posedness results in the focusing case are of different nature, since there are special stationary solutions that may be used to evidence instabilities.

• Another consequence of not exploiting finite propagation speed is the freedom in the choice of the position of the bubbles in the construction. Indeed, the absence of interaction among them does not depend on their spacial localization. Hence, we can take bubbles centered at zero and obtain a pathological set that is radially symmetric.

We refer to the introduction of Chapter 1 for further remarks, detailed explanations and references. Let us just mention that a possible perspective would be to use the construction to describe a loss of regularity mechanism, as evidenced by [Leb01; AC09; CR09] for the energy-supercritical wave equation, and for Schrödinger equations, by the mean of nonlinear geometric optics.

Long-time dynamics under stochastic perturbations

In this part, we investigate the long-time behavior of the local probabilistic flow solution to the cubic Schrödinger equation in R 3 , presented in Section 0.1.2. First, we address in Chapter 2 the scattering issue for the defocusing equation, outside the small data regime. In order to do so, we propose a way to revisit in a probabilistic setting the I-method with a Morawetz bootstrap. Then, we consider in Chapter 3 the stability of small ground states that emerge from the presence of an external short-range potential. By performing a "critical-weighted" strategy, we prove that these ground states are asymptotically stable even if the perturbations have infinite energy.

Almost-sure scattering for energy-subcritical Schrödinger equation We refer to the above Paragraph in Section 0.1.2 called "almost-sure scattering", and to the introduction of the Chapter 2 for more detailed background and motivations. The model is the cubic Schrödinger equation with a defocusing nonlinearity, in the Euclidean space of dimension 3.

i∂ t u(t, x) + ∆u(t, x) = |u(t, x)| 2 u(t, x) , (t, x) ∈ R × R 3 . u(0, x) = u 0 (x) , (NLS)
This equation is energy subcritical in the sense that the critical scaling, which is s c = 1 2 , lies between two conserved quantities, the L 2 -mass and the energy, at the level of H 1 . Hence, we have local solutions in H s (R 3 ) for any 1 ≤ s. As explained in 0.2.2, the repulsive nature of the nonlinearity induces a linear behavior of the solution, and scattering is expected to occur. We do know that scattering holds in the energy space and above since the work of Ginibre and Velo [START_REF] Ginibre | Scattering theory in the energy space for a class of nonlinear Schrödinger equations[END_REF]. However, at the level of H s when 1 2 ≤ s < 1 we have no conserved coercive quantity, and we cannot say for sure that every solution in the scaling-subcritical regime extend globally in time, and scatter at infinity. Conjecture 0.2.3. Let 1 2 ≤ s < 1. Local solutions to NLS initiated from initial data in H s (R 3 )extend globally in time, and scatter at infinity: for all u 0 ∈ H s (R 3 ), there exist

u ± ∈ H s (R 3 ) such that lim t→±∞ ∥u(t) -e it∆ u ± ∥ H s = 0 .
This conjecture has attracted a lot of attention over the last two decades. For now on, the conjecture is proved up to 2 3 < s. The optimal result is due to [START_REF] Su | Global well-posedness and scattering for defocusing, cubic NLS in R 3[END_REF], pursuing an important series of fruitful works. By important, we mean that these works have induced a number of major developments that are now essential in the study of dispersive differential equations. Let us briefly discuss some existing results below the energy space.

• The first global well-posedness and scattering results below the energy space is due to Bourgain [START_REF] Bourgain | Scattering in the energy space and below for 3D NLS[END_REF] who developed by the way the high-low method in order to capture smoothing of the nonlinear term. Namely, Bourgain proved that u(t)-e it∆ u 0 ∈ C(R; H 1 (R 3 ))6 for s > 11 13 . In addition, Bourgain proved scattering in the radial case when s > 5 7 by employing Lin Strauss Morawetz estimate and radial Sobolev embedding.

• Then, the I-team [START_REF] Colliander | Almost conservation laws and global rough solutions to a Nonlinear Schrödinger Equation[END_REF][START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF] developed the I-method, which is roughly speaking a smooth version of the high-low method that exploits as wall the conservation of the energy. More importantly, they obtained scattering for any data when s > 5 6 by combining the I-method with a new Morawetz-type estimate, called the interaction Morawetz estimate, which yields a priori space-time L 4 t,x -bounds.

• Subsequently, Dodson improved in [START_REF] Dodson | Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when n = 3 via a linear-nonlinear decomposition[END_REF] the scattering result for any data when s > 5 7 , refining the I-method with a double layer decomposition (which we discuss below), and a linear-nonlinear decomposition.

• The best result at this day is due to Su [START_REF] Su | Global well-posedness and scattering for defocusing, cubic NLS in R 3[END_REF], proving scattering for s > 2 3 . The proof uses long-time iteration (meaning that the size of the interval increases at each set of the iteration), made possible by the use of second generation I-method (consisting in adding a correction to the energy function).

• Combining the long-time Strichartz estimates with local-smoothing and radial Sobolev embedding, Dodson [START_REF] Dodson | Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when d = 2, 3 and u 0 is radial[END_REF] proved the conjecture in the radial case all the way down to s > 1 2 . • In the critical case, Kenig and Merle [KM10] used a rigidity argument, based on the concentration-compactness method, to show that if the Ḣ 1 2 -norm of the solution is uniformly bounded then the solution is global, and there is scattering.

• Recently, Dodson [START_REF] Dodson | Scattering for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space[END_REF] proved that there is scattering for initial data in a critical Besov space, with nice polynomial bounds.

It seems that this story is far from over, and that new techniques are needed to go further in understanding this problem. More modestly, we can try to show that scattering occurs for almost all initial data. This is all the more justified since a local probabilistic flow exists in H s for 1 4 < s < 1 2 , but its long-time behavior is not understood outside the small-data regime. Indeed, almost-sure global well-posedness and scattering type of results outside the small-data regime only existed in the energy-critical case. The result presented in Chapter 2 proves almost sure scattering below the deterministic scaling. It reads as follows Theorem 0.2.4 (Almost-sure scattering for energy-subcritical NLS [START_REF] Camps | Scattering for the cubic Schrödinger equation in 3D with randomized radial initial data[END_REF] and [START_REF] Shen | Almost sure well-posedness and scattering of 3D cubic nonlinear Schrödinger equation[END_REF]). Take f 0 ∈ H s (R 3 ) for 3 7 < s be a radial function. We denote its corresponding Wiener randomization ω → f ω 0 , which is not radial. There exists a full-measure set Σ in H s (R 3 ) such that for all f ω 0 ∈ Σ, the corresponding probabilistic local strong solution extends globally in time and scatters at infinity: there exists v ± ∈ H σ (R 3 ), such that

lim t→±∞ ∥u(t) -e it∆ (f ω 0 + v ± )∥ H σ = 0 , σ = 2s -.
We give some comments on the result and on the main difficulties.

• The exact same result was proved independently by Shen, Soffer and Wu [START_REF] Shen | Almost sure well-posedness and scattering of 3D cubic nonlinear Schrödinger equation[END_REF]. The method is a bit different, and relies on the use of the high-low method. Surprisingly, they obtained the same lower bound 3 7 < s, but we think that there is nothing special with this threshold. By the way, we think that the method proposed in Chapter 2 is clear and robust, so that we can implement other deterministic refinements of the I-method and lower the threshold, hopefully reaching 1 3 < s, which corresponds to the best known result 2 3 ∼ σ = 2s-. • As usual, we do a linear-nonlinear decomposition u = e it∆ f ω 0 + v. The key observation is that the nonlinear term is in H σ , for σ < 1, but still close enough, so that the deterministic conjecture is known at this regularity. Yet, v solves a perturbed Schrödinger equation, with stochastic forcing terms. Subsequently, the challenge (but also the motivation) was to combine the probabilistic method with the I-method and a Morawetz bootstrap. The difficulty is that the stochastic forcing terms break the gauge-invariant structure of the nonlinearity. Hence, we have extra terms without frequency-cancellation to deal with. Instead, we can have lots of derivatives falling onto the rough, stochastic term.

• For this reason, and as in the energy-critical case, the concealed radial-symmetry assumption, or any other similar assumption, seems to be required. The idea is to exploit some mild radial Sobolev embedding in order to implement the local smoothing estimate, and overcome the loss of derivatives in the energy method. We recall that the randomized data f ω 0 , on the other hand, is not likely to be radially-symmetric. As mentioned in Section 0.2.2 the only work avoiding such an assumption is due to Bringmann, for the energy-critical wave equation in R 4 [START_REF] Bringmann | Almost sure scattering for the energy critical nonlinear wave equation[END_REF].

• The interaction Morawetz estimate needs at least 1 2 derivatives, and we are tempted to only prove a Morawetz estimate for v. Though, such an estimate strongly relies on monotonicity formula that are sensitive to the structure of the nonlinearity of NLS. To overcome this difficulty, we proved a modified version of the interaction Morawetz estimate, where we consider Iu in place of u. This is possible since

If ω 0 makes sense in H 1 2 .
• As pointed out by N. Tzvetkov, it is remarkable that the convergence of the remainder holds in a space H σ close to H 1 even if the initial is in H s for some s < 1 2 . the fact that the scattering result for the nonlinear remainder holds in H σ for some σ close to H 1 . This property might also be true for the deterministic results with 2 3 < s < 1, but it is less clear. Let us now give an outline of the proof.

Blow up criterion. The first part of the analysis follows pre-existing lines. We consider the forced equation

(i∂ t + ∆)v = |f ω + v| 2 (f ω + v) , f ω = e it∆ f ω 0 . v(0) = v 0 , v 0 ∈ H σ (R 3 ) , (NLS f )
Thanks to the refined probabilistic Strichartz estimates, we have access to space-time bounds for f ω , almost-surely in the randomization. We proceed pathwise in ω, and fix ω in a full measure set such that

F ω (R) := ∥⟨∇⟩ s f ω ∥ L 10 t,x + ∥⟨∇⟩ s f ω ∥ L 4 t,x + ∥⟨∇⟩ s f ω ∥ L 4 t L 12
x < +∞ . In particular, this quantity controls the scaling-critical L 5 t,x -norm, that requires loosing 1 2 -derivatives to access the Strichartz-norm L 5 t L 30 11

x . In the argument, we are frequently led to reduce the analysis to a finite number of space-time slabs where we impose a smallness condition on the forcing term:

F ω (J j ) ≤ ε 0 , j ≤ L , L ≲ ε 0 ,ω 1 .
First, we set up a refined probabilistic Cauchy theory for (NLS f ), with

v ∈ X 1 2 (J) ⊂ C(J; H 1 2 (R 3 )) ,
and where X is based on the U 2 and V 2 spaces. Since the probabilistic structure of the linear evolution is preserved under the evolution 7 , the hope is to iterate the local probabilistic Cauchy theory, provided we are able to control the H σ -norm of v. By doing so, we obtain the blow-up criterion, stated in Proposition 2.3.5: if v is a maximal lifespan solution to (NLS f ) on J * , with T * = sup J * , then

∥v∥ L 5 t,x (J * ×R 3 ) < ∞ =⇒ T * = +∞ , and v scatters in H σ (R 3 ) .
The scattering result in H σ , for 2 3 < σ,, also come with the uniform space-time bound for the solution

∥u∥ L 5 t,x (R×R 3 ) ≤ C(∥u 0 ∥ H σ (R 3 ) ) ,
that we use to develop a stability theory for (NLS f ), in the functional space X(J). Specifically, we reduce the blow-up criterion to

sup t∈J * ∥v(t)∥ H σ (R 3 ) < ∞ =⇒ ∥v∥ L 5 t,x (J * ×R 3 ) < ∞ .
A priori bounds. As explained above, Theorem 0.2.4 reduces to obtain the a priori uniform bound for the maximal lifespan solution to (NLS f ) in H σ (R 3 ), with 2 3 < σ < 2s:

sup t∈J * ∥v(t)∥ H σ (R 3 ) < ∞ .
Since we have no conservation law at the level of H σ (R 3 ), we need to use modified energies. Given a large integer N , the I-operator I : H σ (R 3 ) → H 1 (R 3 ) is the Fourier multiplier given by the symbol

m N (ξ) = 1 , |ξ| ≤ N , N |ξ| 1-σ , |σ| > 2N . with ∥f ∥ H σ ≲ ∥If ∥ H 1 ≲ N 1-σ ∥f ∥ H σ .
It also maps

H s (R 3 ) to H 1 2 (R s ), with ∥f ∥ H s ≲ ∥If ∥ H 1 2 ≲ N 1 2 -s ∥f ∥ H s ,
where we used that s < 1 2 and σ < 2s. We see from (0.2.2) that it suffices to control ∥Iv∥ H 1 . To recall the method, let us consider u the (deterministic) local solution to NLS, in H σ for some 1 2 < σ < 1. Basically, since u has an infinite energy, one can fear that

∥Iu∥ H σ → N →∞ +∞ ,
On the other hand, E(Iu) is all the more close to the conserved energy since N is large. Hence, we expect that the time derivative of truncated energy decays. Basically, [START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF] proved that when σ is close enough to 1, and if J = [a , b] is a time interval with E(u(a)) ≲ 1, such that we have a smallness condition on ∥u∥ L 4 t,x (J×R 3 ) , then

d dt E(Iu) ≲ 1 N 1-.
Yet, the overall goal is to prove that the conservation law eventually wins, so that uniformly in N , sup

t≤T * E(Iu(t)) ≲ 1
In our context, where the energy of v solution to (NLS f ) is not conserved due to the presence of the stochastic forcing terms, we need to control the time-increments of the modified energy

E(v)(t) = 1 2 R 3 |∇Iv(t, x)| dx + 1 4 R 3 |Iu(t, x)| 4 dx .
Notice that we decided to place u = v + f ω in the potential energy, in order to preserve the structure of the cubic Schrödinger equation as much as possible, and to exploit frequency cancellations. It holds

d dt E(v)(t) = Re R 3 ∂ t (Iv)(t, x) |Iu(t, x)| 2 Iu(t, x) -I(|u(t, x)| 2 u(t, x)) :=H dx -Im R 3 ∇If (t, x)∇(|Iu(t, x)| 2 Iu(t, x)) dx .
The first term in the right-hand side contains the commutator H, which displays some frequency cancellations and induces a gain of N -1+ , in suitable space-time norms. On the other hand, the second term is a remainder, coming from the stochastic perturbations. In order to be able to choose s < 1 2 , we have to gain derivatives. One way to do so is to use local smoothing effect for the linear evolution of randomized data. This is the place where we use the radial symmetry assumption, and we discuss the probabilistic estimate:

∥⟨∇⟩If ω ∥ L 2 ω L 2 t L ∞ x ≲ N 1-σ 2 (0.2.3)
in the next paragraph. We stress out that on the right-hand side of the energy estimate of the energy estimate, we keep track of sub-additive quantities (namely, L 2 -norm in time), following a nice idea from [START_REF] Dodson | Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when n = 3 via a linear-nonlinear decomposition[END_REF][START_REF] Dodson | Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when d = 2, 3 and u 0 is radial[END_REF]. In these two works, Dodson used the refined the energy estimate (under the same assumption as in [START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF]):

J d dt E(Iu(τ )) dτ ≲ ∥P >cN u∥ 2 L 2 t (J;L 6 x ) N 1- + O 1 N 2 .
The above quantity is controlled on J as soon as E(Iu(t)) ≲ 1 for all t ∈ J (this is the long-time Strichartz estimates). By doing so, we avoid loosing too many powers of N when summing over the L = L(N ) intervals where the L 4 -norm is small. To conclude, we perform a double bootstrap on the modified energy E combined with the modified interaction Morawetz from Proposition 2.4.6, of type

J×R 3 |Iu(t, x)| 4 dx dt ≲ ∥Iu∥ 2 L ∞ (J; Ḣ 1 2 ∥u∥ 2 L ∞ (J;L 2 ) + R(∇Iu, H) , H := |Iu| 2 Iu -I(|u| 2 u) ,
where R is a remainder controlled thanks to the estimates on the commutator H and to the probabilistic estimates for f ω . Then, we bootstrap the following estimates:

8 sup t∈J E(v)(t) ≤ N 2(1-σ) , J×R 3 |Iu(t, x)| 4 dt dx ≤ M N 1-σ .
Thanks to these a priori bounds, to the energy increment, to the modified interaction Morawetz and to the probabilistic estimate discussed below, we are able to prove that these bounds holds up to T * . Key probabilistic estimate. We discuss estimate (0.2.3). This is a consequence of a square function decay estimate that reads as follows: for all δ > 0, and all u 0 ∈ H δ radially-symmetric, there holds

⟨x⟩ k∈Z 3 |P 1;k u 0 (x)| 2 1 2 L ∞ x ≲ δ ∥u 0 ∥ H δ (R 3 ) .
This estimate is inspired from the stronger one used by Dodson, Lührmann and Mendeslon [DLM19; DLM20] in dimension four, to address the energy-critical case. It drastically improves the radial Sobolev embedding, ∥|x|f ∥ L ∞ x ≲ ∥f ∥ H 1 , which losses one derivative. The strategy consists in study the following oscillatory integral. Up to changing the coordinates, we assume x = |x|(1, 0, 0) and we use spherical coordinates for the frequency variable. It holds

P 1;k u 0 (x) = ∞ 0 2π 0 π 0 e i|x|ρ cos θ ψ(ξ(ρ, θ, φ)) sin(θ) dθ oscillatory integral u 0 (ρ) dφρ 2 dρ .
Notice that the oscillatory integral does not depend on U 0 , thanks to the radial-symmetry assumption. Then, we use the frequency localization of the Fourier multiplier P 1;k on the unit-scale cube centered around k, and we integrate by parts.

Next, as always when we estimate the moments of a random series, we have access to the square function thanks to probabilistic decoupling (see Section 0.1.2 for the discussion on the Paley-Zygmund's theorem): for all 2 ≤ p < ∞,

E k g k (ω)P 1;k u 0 (x) p 1 p ≲ √ p k |P 1;k u 0 (x)| 2 1 2
.

Then the probabilistic estimate (0.2.3) for the free Schrödinger evolution of Wiener randomization of radial functions follows from the local smoothing effect and probabilistic decoupling.

Almost-sure asymptotic stability for small ground states In Chapter 3 we study the asymptotic stability of small ground states of the Nonlinear Schrödinger equation, in the presence of a short range potential

(i∂ t + ∆)u(t, x) = V (x) external potential + µ|u(t, x)| 2 nonlinear pertrubation u(t, x) , (t, x) ∈ R × R 3 .
Under some spectral assumptions on the Schrödinger operator that ensure the existence of linear bound states, the natural question is to study the persistence of these bound states under small nonlinear Hamiltonian perturbations. In a concrete situation, the understanding of such a model would allow to capture the stability of localized waves, emerging from the inhomogeneities of the medium in which they propagate under the effect of nonlinear repulsive interactions. This model has been studied by Soffer and Weinstein [START_REF] Soffer | Multichannel nonlinear scattering for nonintegrable equations[END_REF], where they described a multichannel nonlinear scattering mechanism, showing that the asymptotic behavior is given by a linear combination of a localized standing wave, which is periodic in time, and a dispersive part that is radiative, respectively. At that time, little was known about the stability of coherent solutions with nonlinear Hamiltonian perturbations, for equations that are not integrable. In this sense, the result was very innovative. In particular, they performed a rigorous and fruitful analysis of the so-called modulation equations discussed below.

The spectral assumptions on the Schrödinger operator H := -∆ + V are the following. First, for simplicity, we assume that V is in the Schwartz class, but the result would persist under mild decay and regularity assumptions. Then, we assume that -∆ + V has a unique simple eigenvalue -e 0 < 0, with eigenfunction ϕ 0 , and absolutely continuous spectrum on [0 , +∞). In addition, we assume that there is no resonance at zero. This is a technical assumption that holds for a generic choice of V . Then, the one-complex parameter family of bound state solutions

z → Q(z, x) e -itE(z)
are obtained by a bifurcation argument from the linear ground states z e -ite 0 ϕ 0 , parametrized by z ∈ C. At fixed z, the nonlinear ground state Q satisfies the elliptic equation

∆ -V (x) + |Q(z, x)| 2 Q(z, x) = E(z)Q(z, x) , (z, x) ∈ C × R 3 ,
where, at the vicinity of z = 0, there are some nonlinear corrections e and q obtained from inversion function theorem, such that Q(z) = zϕ 0 + q(z) , E(z) = e 0 + e(z) .

After contributions of Pillet and Wayne [START_REF] Pillet | Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations[END_REF], revisiting the result of [START_REF] Soffer | Multichannel nonlinear scattering for nonintegrable equations[END_REF] with center stable manifold technics from dynamical systems, Gustafson, Nakanishi and Tsai proved in [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF] that asymptotic stability holds for any sufficiently small solution in the energy space H 1 (R 3 ). Namely, small solutions uniquely resolve into the sum of a modulated bound state and a radiative term that scatters at infinity. Meanwhile, the modulation parameter converges to a fixed parameter. However, since the effect of the external potential is negligible in small-time, local well-posedness holds up to critical regularity s c = 1 2 , but there is no reason for the bound state to remain stable under rough perturbations in H s for 1 2 < s < 1. Similarly, what happens to the probabilistic flow, locally defined in H s for some scaling-supercritical 1 4 < s < 1 2 ? The concern of Chapter 3 is to address these two questions. Specifically, we consider random perturbations injected in the initial data, which are of type u(0

) = u 0 + f ω 0 , u 0 ∈ H 1 2 , f ω 0 ∈ H s ∩ Ran P c (H)
, where P c (H) denotes the projection onto the continuous spectral subspace. The initial data are randomized according to the Wiener procedure (see Subsection 0.1.2), but they are adapted to the linear inhomogeneous evolution e itH in the sense that the Fourier multipliers are replaced by distorted Fourier multipliers. Under these perturbations, we prove asymptotic stability for these random perturbations, almost-surely with respect to the probability measure induced by the randomization. As a consequence, we also answer the first question by taking f 0 = 0. We give here an informal statement of the main theorem, and we refer to Chapter 3 for a more precise statement. Theorem 0.2.5. Let 1 4 < s < 1 2 . There exist δ 0 > 0 and a full-measure set Σ in H s (R ω ) such that for all u 0 ∈ H 1 2 with norm less than δ 0 , and for all f ω 0 ∈ Σ, there exist ε = ε(ω) > 0, for which the Cauchy problem

(i∂ t + ∆)ψ(t, x) = µ|ψ(t, x)| 2 ψ(t, x) + V (x)ψ(t, x) , (t, x) ∈ R × R 3 , ψ(0) = εf ω 0 + v 0 ,
admits a local solution under the form

ψ(t, x) = ε e -itH f ω 0 + v(t, x) , v ∈ C(R, H 1 2 (R 3 )) .
Moreover, this solution extends globally in time, and resolves into the sum of a modulated ground state and a radiative term:

ψ(t, x) = Q(z(t)) + η(t, x) .
As time goes to infinity we have asymptotic stability in the sense that there exist z ± ∈ C and final states η ± ∈ H

1 2 such that lim t→±∞ z(t) exp -i t 0 E(z(τ )) dτ = z ± , lim t→±∞ ∥η(t) -e -itH (εf ω 0 + η ± )∥ H 1 2 = 0 .
• As mentioned above, when f 0 = 0, Theorem 0.2.5 extends the deterministic result of [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF] up to the critical regularity s = 1 2 . • This work was inspired by the contribution of Kenig and Mendelson presented in Section 0.1.2 for the energy critical wave equation, contributing to the understanding of the persistence of coherent states under rough perturbations. As explained in Section 0.1.2, in the case of the Schrödinger equation, some challenging difficulties appear. The most serious difficulty is that the Duhamel integral term does not display any smoothing, and the linearized operator around the ground states is not self-adjoint. Hence, considering small ground states is a reasonable first step towards the goal of adapting the result of Kenig and Mendelson for Schrödinger equations.

Instead, we pass the linear potential on the left-hand side of the equation, and we consider the recentered solution around the inhomogeneous linear evolution of the randomized initial data

u = e -itH f ω 0 + v .
The Wiener randomization, however, does not commute with e itH , compromising therefore the probabilistic improvement of the Strichartz estimates. To overcome this issue, we follow [START_REF] Kenig | The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data[END_REF] and we adapt the randomization procedure to the inhomogeneous case by employing the distorted Fourier transform in place of the free Fourier transform in order to define the unit-scale frequency localization that is at the basis of the procedure. This transformation is a partial isometry from L 2 onto the continuous spectral subspace Ran(P c ), and it formally writes

F V f (ξ) ∼ R 3 e(x, ξ)f (x) dx .
The generalized plane waves e(x, ξ) formally solve the Helmholtz equation

(-∆ + V ) e(ξ, •) = |ξ| 2 e(ξ, •) ,
with the Sommerfeld radiation condition e(ξ, x) -

e iξ•x = O |x| -1 .
Their construction is based upon the limiting absorption principle, and the study of such generalized plane waves and of the associated distorted Fourier date back to the early works of [START_REF] Ikebe | Eigenfunction expansions associated with the Schrödinger operators and their applications to scattering theory[END_REF][START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF]. We refer to the introduction of Chapter 3 for detailed exposition and references. Roughly speaking, at the level of this present introduction, we should keep in mind that the linear theory of the Fourier transform adapts without any major modification to the distorted Fourier transform, at least when we assume no resonance at the bottom of the continuous spectrum. In our context, it is crucial to ensure that the Fourier multiplier theorem holds true:

∥F -1 V mF V f ∥ L p ≲ p |E| 1 2 -1 p ∥f ∥ L 2 , p ≥ 2 , supp m ⊂ E .
This is a key ingredient in the proof of the probabilistic Strichartz estimates from Proposition 0.1.5.

In addition, we mention that we use the L p -bound of the wave operator obtained by Yajima in a seminal paper [START_REF] Yajima | The W k,p -continuity of wave operators for Schrödinger operators[END_REF] ∥W ± ∥ L p →L p ≲ p 1 .

We stress out that the use of the distorted Fourier transform in the randomization procedure is crucial when we study the modulation equations and perform the critical weighted argument. Basically, the adapted randomization procedure preserves the continuous spectral subspace and allows us to trace the structure of the different terms during the evolution. However, the distorted plane waves do not satisfy the group property e iξ 1 x e iξ 2 x = e i(ξ 1 +ξ 2 )x .

For this reason, the nonlinear theory is more involved in the inhomogeneous case. For instance, the bilinear Bourgain estimates should not be taken for granted in this case. Nevertheless, in contrast to the study of the energy-critical wave equation addressed in [START_REF] Kenig | The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data[END_REF], we do require such bilinear estimates to overcome the lack of smoothing of the Schrödinger evolution, and to obtain the desired nonlinear probabilistic nonlinear smoothing effect.

Bilinear estimates for Schrödinger operators with a smooth potential. The subtle study of the quadratic interactions of distorted plane waves has attracted a lot of interest recently. Basically, the use of the deformed Fourier transform at the nonlinear level requires a fine analysis of certain bilinear operators with singular kernels. We refer to [START_REF] Pusateri | Bilinear estimates in the presence of a large potential and a critical NLS in 3d[END_REF] and the references therein for important progresses in this direction. In Section 3.3.2 of Chapter 3, however, we consider the case when the potential is, say, in the Schwarz class. In this case, we propose a proof of the bilinear estimate that merely uses the L p -bounds on the wave operators, proving that

∥ e it(∆-V ) P ≲N u 0 e it(∆-V ) P M v 0 ∥ L 2 t,x (R×R 3 ) ≲ N M -1 2 ∥ P ≲N u 0 ∥ L 2 x (R 3 ) ∥ P M v 0 ∥ L 2 x (R 3 ) , (0.2.4)
where P K stands for the Littlewood-Paley multiplier associated with the functional calculus on H. The overall strategy consists in reducing the inhomogeneous case, to the homogeneous case by proceeding in two steps. First, we intertwine the Littlewood-Paley projectors with the distorted ones. Specifically, we deduce from some semi-classical functional calculus that for all K, L with K = 2 k , L = 2 l , |k -l| ≥ 3, and all every α > 0,

∥P K P L ∥ L 2 x →L 2 x ≲ α max K, L -α .
Thus, roughly speaking, we can replace the inhomogeneous projector P N by the flat Littlewood-Paley projector P N in (0.2.4). Then, in order to reduce the matter to the flat Bourgain's bilinear estimate (0.1.3), we need to replace the inhomogeneous linear evolution e -itH by the free one e it∆ . To do so, we put back the potential term in its place of a source term, at the right-hand-side of the equation. Namely, given a dyadic number K, we write

P K e it(∆-V ) u 0 = e it∆ P K u 0 -i t 0 e i(t-τ )∆ P N V e it(∆-V ) u 0 dτ :=F (t)
.

Similarly,

P M e it(∆-V ) v 0 = e it∆ P M v 0 -i t 0 e i(t-τ )∆ P M V e it(∆-V ) v 0 dτ :=G(t)
.

Then, after developing the product P K e it(∆-V ) u 0 P M e it(∆-V ) v 0 , we are led to study the interaction of terms e it∆ P K u 0 and F (t), with the terms e it∆ P K v 0 and G(t). Hence, we are reduced in some sense to apply the bilinear estimate of Bourgain (0.1.3). This is enough to handle the free linear evolutions e it∆ u 0 , e it∆ v 0 . However, to control the source terms F (t) and G(t) we also need to use the local smoothing effect (0.1.4), and to exploit the localization of the external potential V . Overall the analysis is a bit technical, but in essence it combines the bilinear with the local smoothing effect in a suitable way.

Modulation equation. We adapt the framework developed in [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF] to settle the asymptotic stability for small data in the energy space. Their key idea is to use a time-dependent orthogonality decomposition of the phase space, in order to cancel the terms of order one in the modulation equation. Basically, they decouple the local solution into

ψ(t, x) = Q(z(t), x) + η(z(t), x) , η(z) ∈ H c (z) ,
where the space H c (z) associated with the modulation parameter z encodes the time-dependent orthogonality condition

H c (z) = η ∈ L 2 : R 3 ⟨iη(x), ∂ z 1 Q(z, x)⟩ dx = R 3 ⟨iη(x), ∂ z 2 Q(z, x)⟩ dx = 0 .
Thanks to this time-dependent orthogonality condition, they are able to control the projection onto the continuous spectrum of the nonlinear radiative interactions, by merely using the endpoint Strichartz estimate L 2 t L 6

x . As a consequence, they removed the weights assumptions made in [SW90; PW97].

In our setting, on the other hand, we have to use the local smoothing estimate since we cannot obtain nonlinear probabilistic smoothing for the linear term Q 2 ψ. The ansatz is as follows. From the continuous spectral comparison stated in Lemma 3.5.4 it suffices to control the continuous spectral component of the solution. To achieve such a goal, we decompose the local solution into

ψ(t, x) = Q(z(t), x) + η(t, x) = Q(z(t), x) + R(z(t)) ν + e -itH f ω 0 ,
where R(z) is the isometry from Ran P c to H c (z), and ν(t, x) = P c (H)η(t, x) -e -itH f ω 0 ∈ Ran P c (H) .

By doing so, we expect nonlinear probabilistic smoothing effect, so that ν is in H 1 2 . It satisfies the dispersive equation

       (i∂ t + ∆ -V )ν = P c (H) |η| 2 η -D z Q ṁ :=f , nonlinear, decay in time + Q 2 η + Qη 2 :=g , localized ν(0) = 0 .
The main difficulty is to simultaneously control in critical spaces the higher order stochastic terms, collected in f , and the lower order localized terms, collected in g. To achieve this, we proposed a critical-weighted approach, inspired from the work of [START_REF] Koch | Small data scattering and soliton stability in ( Ḣ-1/6 for the quartic KdV equation[END_REF] for KdV equation which is, to the best of our knowledge, the only work in the spirit of Chapter 3 addressing asymptotic stability by employing critical spaces based upon U 2 and V 2 .

The critical weighted strategy. In order to encapsulate the probabilistic nonlinear smoothing effect and the linear local smoothing effect under the same norm, we do a continuity argument in the intersection of the two relevant spaces: given a time-interval J, possibly unbounded, we set

X (J) := V 2 H (J; H 1 2 (R 3 )) ∩ L 2 t (J; H 1,-1 2 -(R 3 )) .
The space V 2 H is the space of bounded 2-variation, and is close to the critical space used in [START_REF] Bényi | Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on R 3[END_REF] to prove almost-sure global existence and scattering for scaling-supercritical NLS with small initial data. We refer to Section 4 in Chapter 3 for some discussions on U 2 and V 2 spaces.9 Second, the weighed space captures the local-smoothing estimate. Given δ > 0 arbitrarily small, the norm writes ∥u∥

L 2 t (J;H 1,-1 2 -δ ) := J×R 3 ⟨x⟩ 1-2δ |⟨∇⟩u(t, x)| 2 dt 1 2
.

By suitably combining the transference principle in U 2 with the local smoothing estimate and the duality between U 2 and V 2 , we end up with the key estimate ∥ν∥

V 2 H (J;H 1 2 )
+ ∥ν∥

L 2 t (J;H 1,-1 2 -δ ≲ ∥ν 0 ∥ H 1 2 + sup N N 1 2 ∥I(t, P N f )∥ DU 2 H + ∥g∥ L 2 t (J;L 2, 1 2 +δ x ) .
Here, I denotes the Duhamel integral term. The rest of the proof consists in proving uniform a priori estimates on each term in the modulation equation, and to perform a continuity argument.

Over the long term, it would be of great interest to obtain a similar result outside the small data regime. In such a situation, the large ground states are not perturbations anymore. Instead, one would have to absorb them to the linear operator and revisit the analysis with some matrix-operators.

A perspective: almost-sure modified scattering Let us now consider the general defocusing Schrödinger equation, for some power p ≥ 1

i∂ t u(t, x) + ∆u(t, x) = |u(t, x)| p-1 u(t, x), (t, x) ∈ R × R d . u(0, x) = u 0 (x) , (NLS p )
If p is larger than a certain threshold p * , depending on the dimension d, the nonlinear interaction is short range and there is scattering. However, when p = p * , the solution keeps exhibiting long range effects, and we expect scattering but with a logarithmic-in-time phase correction. Such a modified scattering result holds in the small data regime, for solutions with decaying assumptions. It is a great challenge to widen our understanding of this phenomenon outside the small and decaying data regime. In this perspective, it would be meaningful to establish modified scattering on generic sets of large initial data. A good model for this is the nonlinear Schrödinger equation on the line, where p * = 3. Indeed, for the cubic Schrödinger equation on the line, Burq and Thomann [START_REF] Burq | Almost sure scattering for the one dimensional nonlinear Schrödinger equation[END_REF] recently evidenced the quasi-invariance of the nonlinear flow solution of (NLS p ) for any p > 1, with respect to some measures supported on a dense subset of H -ε (R), where ε > 0 is arbitrarily small. Moreover, they proved almost-sure scattering for short-range interactions, i.e. when p > 3. In the cubic case, when p = p * = 3, the interaction is long-range, and NLS displays modified scattering for any small and decaying initial data in the energy space H 1 (R) [START_REF] Kato | A new proof of long-range scattering for critical nonlinear Schrödinger equations[END_REF][START_REF] Hayashi | Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations[END_REF]. In a future work with N. Burq we would like to extend this result almost-surely to the probabilistic flow constructed in [START_REF] Burq | Almost sure scattering for the one dimensional nonlinear Schrödinger equation[END_REF]. The main difficulty is the presence of stochastic forcing terms that do not decay in time, and that rule out the strategy of [KP11; HN98]. Instead, one would have to absorb these terms in the effective dynamics, and to proceed as in [START_REF] Hani | Growing Sobolev norms for the cubic defocusing Schrödinger equation[END_REF].

Probabilistic Cauchy theory for weakly dispersive equations

There has been a rich activity on the qualitative behavior of solutions to nonlinear evolution equations with weak dispersion. , where the flow-map is not uniformly continuous. When the weakly dispersive equation is neither completely integrable nor semilinearly well-posed, it is relevant to develop a random data Cauchy theory inspired from quasilinear resolution schemes. Indeed, to run the probabilistic semilinear resolution scheme à la Bourgain one needs to combine the Paley-Zygmund observation from Theorem (0.1.2) with dispersive effects as in (0.1.3), (0.1.4), in order to exhibit probabilistic nonlinear smoothing that may not be sufficient in quasilinear regimes. 10The two articles presented in this section illustrate this point.

Refined probabilistic well-posedness for NLS half-wave equation joint work with L. Gassot and S. Ibrahim [START_REF] Camps | Refined probabilistic local well-posedness for a cubic Schrödinger half-wave equation[END_REF]. In the perspective of studying weakly dispersive equations, we consider the Schrödinger half-wave equation with a power-type nonlinearity: for µ ∈ R, and (t, x, y) ∈ R 1+2 ,

i∂ t u + ∂ xx -|D y | u = µ|u| 2 u , with |D y |u(ξ, η) = |η| u(ξ, η) . u t=0 = u 0 , (NLS-HW)
In this anisotropic model11 , we have a Schrödinger equation in the x-direction, and a half-wave equation in the y-direction, respectively. Hence, in certain regimes with low frequency in the xdirection, and high frequencies in the y-direction, the transport dominates and there is no dispersion. As discussed at the end of this section, the long-time dynamics of smooth solutions to (NLS-HW) is quite rich (see [Xu17; BIK20; BIK21]), despite the fact that the Cauchy theory for this equation is poorly understood. To investigate the Cauchy theory for (NLS-HW) we measure the regularity in the scale of the anisotropic Sobolev spaces:

H s (R 2 ) = L 2 x H s y (R 2 ) ∩ H 2s x L 2 y (R 2 ) .
The natural choice of these space is dictated by the Hamiltonian structure of the equation and by the invariance of the equation under the scaling symmetry u → u λ (t, x, y) = λψ(λ 2 t, λx, λ 2 y) .

In particular, the critical exponent associated with the natural scaling symmetry of the equation is s c = 1 4 . The conserved energy is

H(u) = 1 2 R 2 |D x u(x, y)| 2 + ||D y | 1 2 u(x, y)| 2 + ν 4 R 2 |u(x, y)| 4 dx dy ,
Local well-posedness holds at regularities s > 1 2 . This was proved in [START_REF] Bahri | Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations[END_REF] from a fixed-point argument and Strichartz estimates with derivative loss. However, to the best of our knowledge, a satisfactory Cauchy theory in the energy space and below has not been established yet. The energy space corresponds to the endpoint s = 1 2 , and there are severe obstructions to settle a Cauchy theory in this space. These obstructions are raised in the introduction of Chapter 4, where we suggest adopting a rather sophisticated functional framework, used to tackle similar critical problems as in [START_REF] Bejenaru | The Cubic Dirac Equation: Small Initial Data in H 1 2 (R 2 )[END_REF] for instance.

At scaling-supercritical regularities 0 < s < 1 4 , equation (NLS-HW) is ill-posed. More specifically, we have norm-inflation (see [START_REF] Kato | Ill-posedness for the Half wave Schrödinger equation[END_REF]), in a very strong sense. Indeed, the pathological set construction from Chapter 1, which is quite flexible, can be easily adapted here. Furthermore, we prove in Chapter 4 that the solution map is not of regularity C 3 when 1 4 < s < 1 2 . As a consequence, the equation is semi-linearly ill-posed, and we have no hope to solve the Cauchy problem by performing a fixed point argument. We have to search for other types of solutions, obtained by a quasilinear scheme, a probabilistic method or in a Fourier-Lebesgue space (see [START_REF] Grünrock | On the wave equation with quadratic nonlinearities in three space dimensions[END_REF] for instance).

We prove almost-sure local well-posedness for (NLS-HW) by adapting the refined probabilistic ansatz of [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], in the spirit of the informal Theorem 0.1.6. In doing so, we indicate that such a probabilistic method can overcome the lack of dispersion, and that it could be adapted to other weakly dispersive models in order to generate a statistical ensemble of initial data leading to strong solutions, when semi-linear well-posedness does not hold. For instance, [START_REF] Oh | Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegő equation[END_REF] evidenced singular interactions for models such as Szegő or KdV, that we are very confident to overcome by the paracontrolled approach used in the present paper.

In addition, the results from Chapter 4 precise the Cauchy theory for (NLS-HW), summarized in figure 0.2.3.

-Weak dispersion and lack of nonlinear smoothing. Due to the absence of dispersion in the Half-wave direction, the linear evolution does not gain integrability, and we need to trade regularity against integrability. This is materialized by the Strichartz estimates with derivative loss, used in [START_REF] Bahri | Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations[END_REF] to prove local well-posedness strictly above the energy space. The proof follows from applying the Bernstein estimate in the y-direction, and the one-dimensional Strichartz estimate in the x-direction. Proposition 0.2.6 (Strichartz estimate for Schrödinger half-wave equation). Let (p, q) be a Schrödinger admissible pair in one dimension. For all f 0 ∈ L 2

x,y (R 2 ), if there exists a set E with finite Lebesgue measure such that then

supp F y→η f 0 ⊆ E , Norm-inflation [Kat21] Flow map not C 3 [BGT05a] Local well-posedness [BIK20] H s I L 2 I H 1 4 I H 1 2
∥ e it(∂ 2 xx -|Dy|) f 0 ∥ L p t L q x,y ≲ |E| 1 2 -1 q ∥f 0 ∥ L 2 x,y . (0.2.5)
In particular, we deduce from a Littlewood Paley decomposition that

∥ e it(∂ 2 xx -|Dy|) f 0 ∥ L 4 t L ∞ x,y ≲ ∥|D y | 1 2 + f 0 ∥ L 2
x,y . The probabilistic method can overcome this loss of derivative in the Strichartz estimates. Indeed, if we consider the Wiener randomization in the y-direction (see figure 4.2), which basically consists in superimposing unit-scale frequency localized initial data, decoupled by normalized independent Gaussian variables, one can easily show that the Strichartz estimates for the one-dimensional NLS hold almost-surely, without any derivative loss. Note, however, that we have no dispersive regularization effect, such as local smoothing or bilinear estimates in the spirit of (0.1.3) or (0.1.4), respectively. As a consequence, the second Picard's iteration has the same regularity as the initial data. To see this, we observe that the Duhamel integral of high-low-low type frequency interactions of a function localized at frequencies ≲ 1 in the x-direction, is essentially the transport of the high frequency component in the y-direction. See (4.1) for a more precise statement. This phenomenon must be compared with Bringmann's observation for the derivative wave equation in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], discussed in Section 0.1.2. As a consequence, we follow [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] and we use a refined probabilistic ansatz to prove almost-sure local well-posedness below the deterministic semi-linear threshold, for some s < 1 2 . The statement of our main theorem is as follows. Given a function f 0 ∈ L 2 (R 2 ), we associate a random function obtained by the Wiener randomization

f ω 0 = k∈Z g k (ω)f 0 (k) , f 0 (k) = P 1;k f 0 = F y→η -1 φ(η -k)F y→η (f 0 ) (0.2.6)
where (g k ) is a sequence of independent mean zero complex standard Gaussian variables,

P 1;k is defined from a partition of unity φ ∈ C ∞ c (R, [0, 1]) supported in [-1 2 , 1 2 ] as in Figure 4.2.
Given n ≥ 0 and N = 2 n , we define the projection Q N f ω 0 onto the frequencies localized around N as

P 1 f ω 0 = g 0 (ω)P 0 f (0) , P N f ω 0 = N 2 ≤|k|<N g k (ω)f 0 (k) , P ≤N f ω = M ≤N Q M f ω 0 .
Theorem 0.2.7 (Probabilistic local well-posedness). Let s ∈ (13/28, 1/2], and f 0 ∈ H s . There exist T 0 > 0 and a sequence

(u n ) n≥1 ∈ C([-T 0 , T 0 ] ; H ∞ ) N converging in expectation to a limiting object denoted u lim n→∞ E ∥u n -u∥ 2 L ∞ ([0,T 0 ];H s = 0 , (0.2.7)
in such a way that, almost-surely in ω ∈ Ω, there exists T ω > 0 such that for all n ∈ N, u n and u exist in C([-T ω , T ω ]; H s ) and solve (NLS-HW) with initial data P ≤n f ω 0 and f ω 0 , respectively.

A common feature of the derivative wave equation studied by Bringmann in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] shared by the Cauchy problem addressed here and many other weakly dispersive problems is that the second Picard's iteration of a randomized data does not data does not become smoother.12 Let us comment Theorem 0.2.7.

• In [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], the presence of a derivative in the nonlinearity rules out the possibility of merely employing the recentering method of Bourgain u = S(t)f ω 0 + v to prove probabilistic wellposedness. Indeed, the derivative counteracts the gain of one derivative in the Duhamel formula for the linear wave equation. In the absence of bilinear estimates, this gain of one derivative is a key ingredient to show that v is smoother than f ω 0 , and to prove probabilistic well-posedness for wave-type equations. For equation (NLS-HW), however, the derivative loss comes from the lack of dispersion in the linear part of the equation. We are convinced that the refined probabilistic ansatz should be adapted to many weakly dispersive equations that are semi-linearly ill-posed, such as the Szegő equation [START_REF] Oh | Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegő equation[END_REF].

• For wave-type equations (see for instance [DLM20; Bri21; Spi22]), the probabilistic Strichartz estimate for Wiener randomized initial data are refinements of the Klainerman-Tataru dispersive estimate from A.66 in [START_REF] Klainerman | On the optimal local regularity for the Yang-Mills equations in R 4+1[END_REF]. For Schrödinger type equations, however, we just combine unit-scale Bernstein estimates with the Strichartz estimates.

• In contrast with the usual probabilistic Cauchy theory, the convergence does not hold pathwise in ω ∈ Ω, but in L 2 (Ω). The reason being that we need to perform multiple contraction mapping arguments (one at each step n in the iteration scheme). Instead, we use the truncation method of De Bouard and Debussche discussed in the next paragraph.

• In the iteration scheme, we need to ensure frequency localization of the terms involved. This requires to make slight modifications in the framework of [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF]. Namely, in the absence of a convenient conserved energy, we use the conservation of the mass (which is not true for wave-type equations) to prove the almost frequency localization of the adapted linear at frequencies ∼ N . Moreover, we replace the Gronwall arguments by using the T T * Strichartz estimates.

• Since the standard Strichartz estimates for the one-dimensional free Schrödinger evolution provide control of the L ∞ x (R), we only need to consider randomized initial data in the y-direction, as in Figure 4.2. Hence, our result actually holds in L 2

x H s y .

• Unfortunately, our result does not cover the whole range 1 4 < s < 1 2 , where the equation is quasi-linear. This would require to remove other bad frequency interactions from the second Picard's iteration, such as F ω n P >N γ u n-1 u n-1 , but this would deteriorate the frequency structure of F ω n , ruling out the probabilistic Strichartz estimates. One possibility to lower the regularity threshold can be to adapt the random-tensors approach, developed in [START_REF] Deng | Random tensors, propagation of randomness, and nonlinear dispersive equations[END_REF].

• In Theorem 0.2.7, we give a different formulation of the result obtained from the paracontrolled approach by Bringmann in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF]. Indeed, we emphasize that the solution is defined as the limit of the smooth solutions with frequency-truncated data13 . This is to put in perspective with the informal Theorem 0.1.1, and with the situation described in Chapter 1. Besides, we explain how to easily adapt to (NLS-HW) the result of Chapter 1 on the existence of a pathological set of inflating data for NLS.

-The iteration scheme and the paracontrolled approach. Take n ≫ 1, and set N = 2 n . We know from [START_REF] Bahri | Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations[END_REF] that (NLS-HW) is locally well-posed in H 1 2 + . In particular, the smooth solution initiated from the truncated initial data

i∂ t + ∂ xx -|D y | u n = µ|u n | 2 u n , u n (0) = Q ≤N f ω 0 ,
exists and is unique in C([-T , T ], H 1 2 + ), given a certain time T > 0 depending on ω and on n. Our goal is to prove the convergence of (u n ) in C([-T 0 , T 0 ], H s ), for s 0 < s < 1 2 and a uniform time T 0 . To achieve this goal, we consider the increment v n = u n -u n-1 , solution of

i∂ t + ∂ xx -|D y | v n = µ|v n + u n-1 | 2 (v n + u n-1 ) -µ|u n-1 | 2 u n-1 , v n (0) = Q N f ω 0 ,
As explained above, the standard probabilistic method fails due to high-low-low -type interactions. Instead of merely employing the recentering method from Bourgain around the linear evolution, we introduce the adapted linear evolution F ω n . Dropping the complex conjugate from the notation, it solves

i∂ t + ∂ 2 xx -|D y | F ω n = µF ω n (P ≤N γ u n-1 ) 2 , F ω n (0) = Q N f ω 0 .
(0.2.8)

Here, 0 < γ < 1 is a parameter to be adjusted to minimize the regularity s, and P ≤N γ is the frequency cutoff in the y-direction. The adapted linear evolution F ω n encapsulates the bad interactions, and solves a linear equation with the high frequency initial data, multiplied by the low frequency components of the solution constructed at the previous steps of the iteration. Note that the potential (P ≤N γ u n-1 ) 2 is measurable in the sigma algebra generated by g k : |k| < 1 2 N . As a consequence, they are independent of the sigma algebra generated by g k | 1

2 N ≤ |k| < N . Then, we use the linearity of (0.2.8) together with the random structure (0.2.6) of the initial data, to decompose . Next, we use the aforementioned probabilistic independence between the high-frequency initial data and the low-frequency potential in order to transfer the probabilistic Strichartz estimate from the individual F n,k to the adapted linear evolution F ω n . This is in the spirit of the proof of the Paley Zygmund theorem, but with a conditioning on the sigma algebra F n-1 ). With these probabilistic Strichartz estimates at hand, we decompose

F ω n = N 2 ≤|k|<N g k (ω)F n,
v n = F ω n + w n ,
where w n satisfies

i∂ t + ∂ 2 xx -|D y | w n = µ|F ω n + w n + u n-1 | 2 (F ω n + w n + u n-1 ) -µ|u n-1 | 2 u n-1 -µF ω n (P ≤N γ u n-1 ) 2 ,
with zero initial condition. Since the worst nonlinear interactions are removed from the above equation, we expect some nonlinear smoothing to occur for w n . Specifically, we run a contraction mapping argument on w n at regularity H ν smoother than the deterministic threshold (ν > 1 2 ). There are two main ingredients.

First, we need to use the truncation method from De Bouard and Debussche [START_REF] Bouard | A stochastic nonlinear Schrödinger equation with multiplicative noise[END_REF]. The reason being that, in contrast with the standard probabilistic Cauchy theory, we need to do the contraction mapping argument at each step n, so that the time of existence might not be in uniform in n. Instead, we do the contraction-mapping argument for w n in a L r ω L p t L q x,y -type space, on a fixed time-interval, not depending on n. However, since there is no smoothing in L r ω , one needs to truncate the nonlinearity. The truncation is subsequently removed, by implementing some a priori estimates for the relevant norms of (w n ), on a uniform time-interval. We refer to Section 3 in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], to [START_REF] Bouard | A stochastic nonlinear Schrödinger equation with multiplicative noise[END_REF] and to Chapter 4 for a precise description of the truncation method.

Second, we prove some paracontrolled trilinear estimates on the nonlinear remainder w n , in suitable functional spaces that encode the frequency localizations. Specifically, F ω n and w n have to be localized at high frequencies of size ∼ N . Roughly speaking, the general strategy consists in putting the stochastic terms F ω n in L ∞ in order to benefit from the refined probabilistic Strichartz estimates. Surprisingly, one must also proceed in this way in some cases when F ω n has the highest frequency. The reason being that the rough linear evolution F ω n has more derivatives in L ∞ than w n . This is a consequence of the nice probabilistic structure of F ω n , and of the frequency localization of the unit blocks F n,k . As explained below, however, this only holds in small-time, and it makes it even more difficult to iterate the local well-posedness result to extend the solutions globally in time.

-Comments on the globalization of this refined probabilistic flow. Unfortunately, we are not able to extend the solutions globally in time, say in the defocusing case (µ > 0). Understanding the long-time behavior of the probabilistic solutions to (NLS-HW) obtained in Theorem 0.2.7, would be significant progress. In the presence of an invariant Gibbs measure, Deng, Nahmod and Yue [START_REF] Deng | Invariant Gibbs measures and global strong solutions for nonlinear Schröodinger equations in dimension two[END_REF], and Sun and Tzvetkov [START_REF] Sun | Refined probabilistic global well-posedness for the weakly dispersive NLS[END_REF] used the so-called Bourgain globalization argument to extend the local solution obtained by the paracontrolled strategy globally in time. Without any invariant, or quasi-invariant measure, however, we are not able yet to iterate the local well-posedness result. This is in sharp contrast with the standard linear-nonlinear probabilistic ansatz, as a discussion from paragraph 0.1.3 attests. Let us briefly discuss the main obstructions to iterate the local well-posedness result. First, at each step n of the recurrence, the adapted linear evolution F ω n has better L ∞ properties than the nonlinear term w n (in the y-variable, the adapted linear evolution is in W σ ′ ,∞ (R y ), whereas w n is in W σ,∞ (R y ) for some σ < σ ′ , respectively). In [START_REF] Sun | Gibbs measure dynamics for the fractional NLS[END_REF], the authors faced a similar issue where the nonlinear remainder is not in the same space as the rough linear evolution that is controlled in some Fourier-Lebesgue spaces. They used summed spaces to overcome this difficulty. In our case, however, the linear rough evolution has more derivatives in L ∞ than the nonlinear remainder term has, even if the latter is smoother in the scale of Sobolev spaces. More importantly, as time progresses, the frequency components intermingle so that the key frequency localization of the unit blocks F n,k and the random structure of F ω n have no reason to persist in long-time. Finally, let us point out that the deterministic Cauchy theory for (NLS-HW) is poorly understood, especially in long-time. For instance, we do not know yet if the smooth solutions extend globally in time.

-Perspectives: long-time dynamics. It is also of interest to study the long-time dynamics of (NLS-HW), at least for smooth solutions. In the defocusing case, we expect modified scattering as in [START_REF] Xu | Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation[END_REF], but with different asymptotic effective dynamics. Meanwhile, for (NLS-HW) on the waveguide R x × T y with a focusing nonlinearity, there exists a family of ground state traveling waves. In [START_REF] Bahri | Transverse Stability of Line Soliton and Characterization of Ground State for Wave Guide Schrödinger Equations[END_REF], the authors characterized the ground states u = e itω Q ω at low frequency ω, and proved that the traveling waves only depend on the x-variable, and correspond to the NLS line solitons. Then, they establish orbital stability properties, and prove transversal instability at high frequencies. Changing the geometry and considering the wave-guide T x × R y , we expect to obtain similar results, but using the relevant family of half-wave traveling waves, that are characterized in [START_REF] Gérard | A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line[END_REF]. Specifically, at high frequencies we expect transverse instability, together with a possible bifurcation at the stability threshold.

Almost-sure well-posedness for NLS on S 2 (joint work with N.Burq, M. Latocca, C. Sun and N. Tzvetkov ) In an effort to understand the effect of the background geometry on the dynamics of nonlinear Schrödinger equations, Burq Gérard and Tzvetkov proved in a series of works [BGT02; BGT04; BGT05b; BGT05a; BGT10] that the Cauchy problem for the cubic Schrödinger equation on the two-dimensional sphere S 2 is semi-linearly well-posed in H s (S 2 ) if an only if s > 1 4 (see also [START_REF] Banica | On the nonlinear Schrödinger dynamics on S d[END_REF]). As a consequence, the gap to bridge between the deterministic threshold and the space where the corresponding Gibbs measure lives, which is H 0-(S 2 ), is significant. It is therefore extremely challenging to obtain the counterpart on the sphere S 2 to the celebrated result of Bourgain [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] on the two-dimensional torus T 2 . Still, setting up a probabilistic Cauchy theory just below the deterministic threshold s = 1

4 is yet to be done. Let us briefly discuss the difficulties. The most important one is the degeneracy of the spherical harmonics. Namely, given the energy level λ 2 n = n(n + 1), the multiplicity of the corresponding eigenspace, generated by the spherical harmonics of degree n, is of dimension 2n + 1. Moreover, some spherical harmonics exhibit concentration around the geodesics of the sphere, which is not the case for the plane waves e in•x on the torus. Last but not least, and in sharp contrast with the case of the torus, high-low -type frequency interactions can generate low frequencies.

To initiate the study of NLS on S 2 , we obtained a preliminary result by isolating the worst singular interaction, and proving that the Wick-ordering cubic Schrödinger equation posed on S 2 with random data is still quasilinear. Namely, the first nontrivial Picard's iteration has no better regularity than the randomized initial data.

Before diving into the probabilistic problem, let us mention the result of [START_REF] Burq | High frequency solutions of the nonlinear Schrödinger equation on surfaces[END_REF]. It describes the nonlinear character of NLS on S 2 by considering H s (S 2 )-normalized sequences (u n ) in H 1 (S 2 ) that are oscillatory in the sense that

lim n→∞ ∥u n ∥ L 2 = 0 , ∥∇u n ∥ L 2 ≤ C∥u n ∥ -1-s s L 2
.

The oscillations are all the more important when n is large. When s ≤ 1 4 , and in the asymptotic regime n → ∞, [START_REF] Burq | High frequency solutions of the nonlinear Schrödinger equation on surfaces[END_REF] proved that the nonlinear evolution of strongly oscillating data (u n ) is not well approximated by their linear evolution:

lim n→∞ ∥u n (t) -e it∆ u n 0 ∥ H s = 0 ,
if and only if s > 1 4 . For the two-dimensional torus, however, the above convergence holds up to s > 0. This result indicates that the high oscillations are not linearly propagated, and it strongly suggests that a semilinear scheme is not appropriate. To exhibit the instabilities, Burq Gérard and Tzvetkov considered stationary solutions that are nonlinear versions of the spherical harmonics with the highest weight that concentrate on a great circle. In Chapter 5, we also use these spherical harmonics to prove that a probabilistic semilinear approach fails.

We need to introduce some notations and basic properties of the spherical harmonics, and we refer to the book [START_REF] Stein | Introduction to Fourier Analysis on Euclidean Spaces[END_REF] for interesting results and discussions on spherical harmonics through the perspective of harmonic analysis. We write (Y n,k ) 1≤n ,|k|≤n the canonical orthonormal basis of L 2 (S 2 ) made of spherical harmonics, defined in spherical coordinates by

Y n,k (θ, φ) = e ikφ L n,k (cos(θ)) ,
where L n,k is the associated Legendre function of degree n and order k. They satisfy

-∆ S 2 Y n,k = n(n + 1)Y n,k .
For n ≥ 1, we write λ 2 n := n(n + 1) , the n th energy level, of multilplicity [n] := 2n + 1 .

We keep in mind that for large n, we have λ n ∼ [n] ∼ n up to an insignificant multiplicative constant. The space of spherical harmonics of degree n writes

E n = span Y n,k : |k| ≤ n , and
π n := ker -∆ S 2 -λ 2 n Id .
It is important to observe that E n is invariant under the isometries of R 3 . As a result,

|k|≤n |Y n,k (x)| 2 = [n]
, for all x ∈ S 2 , (0.2.9) up to a normalization constant that does not depend on n. To see this, we observe that the integral kernel of the operator π n is

K n (x, y) = |k|≤n Y n,k (x)Y n,k (y) .
Since E n is invariant under the isometries, we have that for all R ∈ O 3 , and for all x, y ∈ S 2 ,

K n (x, R -1 y) = K n (x, y) .
Since the rotations act transitively on S 2 , we see that the quantity K n (x, x) is constant on S 2 . Then (0.2.9) follows by integrating over S 2 . To finish with the notations, and to give some meaning to the mass of an initial data on the support of the Gibbs measure, which is below L 2 (S 2 ), we need to introduce a truncation at a large frequency N , and we set

Π N = n≤N π n , P N = Π N -Π N 2 .
Let us now set up the probabilistic framework. We fix α ∈ [ 1 2 , 3 4 ), and a sequence of independent normalized Gaussian variables with complex values (g n,k ) 1≤n ,|k|≤n . We set,

ϕ ω = 1≤n e ω n λ α n , e ω n (x) = |k|≤n g n,k (ω)
[n]

1 2 Y n,k (x) . (0.2.10)
The above series converges in L 2 (Ω;

H α-1 2 -(S 2 )), so that ϕ ω lives in H α-1 2 -(S 2 )) almost-surely. When α = 1
2 , the initial data is distributed according to the Gibbs measure associated with the Hamiltonian equation NLS, as in the case of the torus (see Section 0.1.3 for discussion on Gibbs measures). However, the deterministic threshold is H 1 4 (S 2 ). The Wick-ordering Schrödinger equation with a cubic nonlinearity reads

i∂ t u + ∆u = |u| 2 -2∥u∥ 2 L 2 (S 2 ) u , (t, x) ∈ R × S 2 . u(0) = ϕ ω , (NLS W )
Passing from (NLS) to (NLS W ) amounts to do the gauge transformation

v(t) = e 2it∥u(t)∥ 2 L 2 u(t) ,
and to use the mass conservation. Our result states as follows.

Theorem 0.2.8 (Lack of regularization of the second Picard's iteration, [START_REF] Burq | Lack of regularization in the first Picard iteration for NLS on S 2[END_REF]). Let α ≥ 1 2 , and

ϕ ω ∈ H α-1 2 -(S 2
) be the randomized initial data as in (0.3.2). There exist η > 0, N 0 > 0 such that for every N ≥ N 0 and t,

|t|η log(N ) ≤ E ∥ t 0 e i(t-τ )∆ Π N N (Π N e iτ ∆ ϕ ω ) dτ ∥ 2 H α-1 2 (S 2 )
,

where N denotes the nonlinear term of the Wick-ordering cubic (NLS W ).

Let us briefly comment this statement.

• As N goes to infinity, we see that the quadratic moment of the H α-1 2 (S 2 )-norm of the second Picard's iteration of ϕ ω has a log(N ) divergence. Since the induced measure lives in H α-1 2 -(S 2 ), this shows that there is no probabilistic regularizing effect.

• For technical reasons, we could not prove that the divergence occurs almost-surely in ω.

It is not clear whether there exists an elementary proof of this fact that does not rely on sophisticated tools from stochastic analysis.

• In sharp contrast, the second Picard's iteration on T 2 gains 1 2 derivatives. This is also true in the case of irrational tori, and we give a self-contain proof of this result, which was previously proved in the analysis of [START_REF] Fan | 2D-defocusing nonlinear Schrödinger equation with random data on irrational tori[END_REF].

We now outline the proof of Theorem 0.3.3, and see how to exhibit one singular interaction that causes the obstruction. For convenience, we do as if

[n] = λ n = n.
-Exploiting some properties of the Gaussian spherical harmonics. We shall essentially use two important properties of (e ω n ) n≥1 . The first one is that

E |e ω n (x)| 2 = 1 for all x ∈ S 2 .
This follows from (0.2.9):

E |e ω n (x)| 2 = E   1 [n] n k,k ′ =-n g n,k (ω)g n,k ′ (ω)Y n,k (x)Y n,k ′ (x)   = 1 [n] |k|≤n |Y n,k (x)| 2 = 1 ,
where we used that (g n,k ) |k|≤n are independent normalized Gaussian variables. Besides, as a consequence of the orthogonality of the spherical harmonics (Y n,k ) |k|≤n , we have that

E ∥e ω n ∥ 2 L 2 (S 2 ) = 1 .
Subsequently, the expectation of the renormalized potential in the nonlinearity (NLS W ) applied to the term e ω n vanishes pointwise in x:

E |e ω n (x)| 2 -∥e ω n ∥ 2 L 2 (S 2 ) = 0 for all x ∈ S 2 . (0.2.11)
In sharp contrast with the case of the torus, however, the above estimate does not hold pointwise in ω. 14 By the way, we point out that a key feature is that the law induced by e ω n is invariant under the action of the rotations that act transitively on S 2 . Then, in the spirit of Paley-Zygmund's theorem, Gaussian spherical harmonics have better integrability properties than expected by the Sobolev embedding. For 2 < q < +∞ arbitrarily large, there exists C q > 0 such that for all n, ∥e ω n ∥ L q (S 2 ) ≤ C q a.s. .

Once again, this follows from (0.2.9). We refer to [START_REF] Burq | Probabilistic Sobolev embeddings, applications to eigenfunctions estimates[END_REF] for general study of such a probabilistic gain of integrability on the sphere. To put this in perspective, we note that a given spherical harmonic can have its L ∞ -norm of size ∼ n. To give a simple example, we can think of the zonal spherical harmonic (see e.g. [START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF][START_REF] Burq | An instability property of the nonlinear Schrödinger equation on S d[END_REF]).

-Dissociating the different regimes. Let us now isolate a singular interaction. We formally decompose the nonlinearity N (u) := (|u| 2 -2∥u∥ 2 L 2 )u into three different regimes:

N 1 (u) = n 1 ,n 2 ,n 3 n 2 ̸ =n 1 ,n 3 π n 1 uπ n 2 uπ n 3 u , N 2 (u) = 2 n 1 ,n 2 n 2 ̸ =n 1 |π n 2 u| 2 -∥π n 2 u∥ 2 L 2 (S 2 ) π n 1 u , N 3 (u) = n 1 |π n 1 u| 2 π n 1 u -2 n 1 ∥π n 1 u∥ 2 L 2 π n 1 u .
Given a time t > 0 and a function F ∈ C([0 , t], L 2 (S 2 ) , we denote the Duhamel integral term as

I(t, F ) = t 0 e i(t-τ )∆ S 2 F (τ ) dτ ,
so that given a large N , the truncated first nontrivial Picard's iteration is

u (1) N (t) := e it∆ ϕ ω N + I t, Π N N (e it∆ ϕ ω N )
. where

e it∆ ϕ ω N (t) = Π N e it∆ S 2 ϕ ω = N n=1 e -itn 2 e ω n n α
is the truncated free Schrödinger evolution of ϕ ω . With these notations at hand, we write

I := I(t, Π N N 1 (ϕ ω N )) , II := I(t, Π N N 2 (ϕ ω N )) , III := I(t, Π N N 3 (ϕ ω N ))
. The singular interaction we study in Chapter 5 is II. Note, however, that the term I contains a singular interaction as well. Nevertheless, we prove from probabilistic independence of the Gaussian variable (g n,k ) that

E [(I | II) S 2 ] = E [(I | III) S 2 ]
= 0 , so that the two singular interactions cannot cancel each other, and

E ∥ II ∥ 2 L 2 (S 2 ) -E ∥ III ∥ 2 L 2 (S 2 ) ≤ E ∥I(t, Π N N (ϕ ω N ))∥ 2 L 2 (S 2 ) .
As for the third interaction III, we easily prove that it gains one derivative. As a consequence, the contribution of this term is uniformly bounded with respect to N , so that it suffices to obtain a logarithmic divergence for the contribution of the singular interaction II.

-Estimating from below the contribution of the singular interaction. From probabilistic independence and (0.2.11), combined with the orthogonality of the spherical harmonics of different degrees, we obtain the lower bound

E ∥ II ∥ 2 L 2 (S 2 ) ≳ t 2≤n≤N |k|≤n 1 n 2 E S 2 |e ω 1 (x)| 2 -∥e ω 1 ∥ 2 L 2 (S 2 ) |Y n,k (x)| 2 dσ(x) 2 .
Thus, we are reduced to prove that there is η > 0 such that for every n sufficiently large

|k|≤n E S 2 |e ω 1 (x)| 2 -∥e ω 1 ∥ 2 L 2 (S 2 ) |Y n,k (x)| 2 dσ(x) 2 ≥ ηn . (0.2.12)
First, we observe that up to a normalization constant,

e ω 1 (x) = g 1,1 (ω)β(x 1 + ix 2 ) + g 1,1 (ω)Y 1,0 + g -1,1 (ω)Y -1,1
, where β > 1 is a normalization constant. By taking ω in the set of positive measure

S ε,A = ω ∈ Ω : |g 1,1 (ω)| > A , |g 1,0 (ω)| < ε , |g -1,1 (ω)| < ε ,
for some suitable choice of ε ≪ 1 and A > 0, we obtain

|e ω 1 (x)| 2 -∥e ω 1 ∥ 2 L 2 (S 2 ) = |g 1,1 (ω)| 2 β 2 1 -x 2 3 -1 + O(ε)
, Next, we observe that in (0.2.12), the summands are all the larger as the spherical harmonics Y n,k are concentrated on the equator. Indeed, the function β 2 (1 -x 2

3 ) -1 concentrates its mass in the region |x 3 | ≪ 1. Subsequently, our key ingredient is a quantitative estimate that measures the number of spherical harmonics concentrated on the equator, in the semiclassical regime. Roughly speaking, we prove that ∝ n spherical harmonics with the highest weight |k| concentrate on the equator. This yields the lower bound (0.2.12), which is sufficient to evidence the logarithmic divergence.

-Obtaining quantitative concentration estimates for a number of spherical harmonics. Take 0 < δ ≪ 1, and n large enough. The key quantitative estimate reads

∥1 |x 3 |>2δ Y n,k ∥ L 2 (S 2 ) ≲ δ 1 n for all k with 1 -δ 2 n < |k| ≤ n .
We sketch the proof, written in Section 5.3. Recall that in spherical coordinates the spherical harmonics write

Y n,k (θ, φ) = e ikφ v n,k (θ) .
Notice, however, that the spherical coordinates are singular at the poles. To overcome this issue, and to be able to use semiclassical functional calculus, we apply a pseudo-conformal transformation in order to send the singularities at infinity. Specifically, the map

y = arctanh(cos(θ)) . is a C ∞ -diffeomorphism from (0 , π) to R.
We do the change of variable, and write

v n,k (y) = v n,k (θ). It turns out that P h (y, hD y , α) v n,k (y) = 0 , y ∈ R ,
where h ∼ n -1 is the semiclassical parameter and α = |k|h. The differential operator has a symbol of order 2

p h (y, ξ, α) := p 0 (y, ξ, α) + hp 1 (y, ξ) + h 2 , p 0 (y, ξ, α) := ξ 2 + α 2 -cosh(y) -2 .
When αn < |k|, this operator is elliptic near in the region |x 3 | ≤ δ, in the sense that

Char P h,α = (y, ξ) : p 0 (y, ξ, α) = 0 ⊆ (y, ξ) : tanh(y) ≤ δ ,
The quantitative estimate follows. To prove this standard result, we performed self-contained basic computations in the spirit of semiclassical analysis, in order to track the dependence on the parameter α.

Paracontrolled approach for NLS on the sphere Our result also come with some positive observations, and does not preclude any local well-posedness statement. Indeed, we have not only identified the obstruction for performing the Bourgain strategy, but we have also proved that other interactions display a smoothing effect. In a long-term program N. Burq, C. Sun and N. Tzvetkov we develop a probabilistic Cauchy theory, employing paracontrolled calculus and the random averaging operators introduced in Section 0.1.2. Specifically, we overcome the lack of nonlinear smoothing by refining the probabilistic ansatz and exploiting the independence of high-frequency initial data and low-frequency remainders, employing the random averaging operators mentioned in Section 0.1.2. Our use of the random averaging operators is inspired by the works of Bringmann [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], Deng, Nahmod, Yue [START_REF] Deng | Invariant Gibbs measures and global strong solutions for nonlinear Schröodinger equations in dimension two[END_REF], and Sun, Tzvetkov [START_REF] Sun | Refined probabilistic global well-posedness for the weakly dispersive NLS[END_REF], but it will come with novelties. The most important one is that we exploit the propagation by the flow map of the invariance of the Gaussian measures under the action of the isometries. This key feature allows us to significantly break the deterministic regularity threshold s = 1 4 , but it is not enough to go all the way down to the Gibbs measure. To achieve such a goal, we shall have a better understanding of the properties of the colored Gaussian, and their random structure. For now on, we merely use rough L ∞ estimates to handle these terms. To address the deterministic smoother remainders, we use the bilinear estimates form [START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF] and prove their convergence in H 1 4 + (S 2 ). A first step towards the goal of reaching the Gibbs measure would be to consider the equation with a Hartree nonlinearity instead. Indeed, for this equation, we prove deterministic local well-posedness in H 0+ (S 2 ). Hence, the situation looks similar to the case of the two-dimensional torus, where we merely have to gain 2ε-derivatives to reach the support of a Gibbs measure. Nevertheless, it seems that singular high-high-to-low -type frequency interactions are ruling out the regularizing effect of the Hartree nonlinearity, and makes it hard to gain these 2ε derivatives. . .

Synthèse en Français

Cette thèse a pour objet l'étude d'équations de Schrödinger non-linéaires dans des régimes singuliers où les solutions sont fortement oscillantes, en présence de perturbations stochastiques. Avant de discuter de certaines motivations, on souligne que contrairement aux solutions d'équations paraboliques les solutions des équations dispersives ne bénéficient généralement pas d'effets régularisant. Ainsi, lorsqu'on travaille à une régularité donnée, il faut isoler les différentes échelles d'oscillations et exploiter finement les mécanismes dispersifs qui régissent leur propagation. En outre, au niveau non-linéaire, il est nécessaire de comprendre comment les différentes ondes interagissent entre elles.

Par ailleurs, les équations dispersives non-linéaires présentent des instabilités qui ont lieu dans certains régimes et qui compromettent la description de l'évolution de toutes les données initiales. Néanmoins, en dotant l'ensemble des données initiales d'une mesure de probabilité pertinente, on injecte naturellement un aléa dans l'étude. L'objectif est alors d'exhiber un ensemble statistique de données initiales qui donnent lieu à des solutions fortes, limites de solutions approchées qui sont elles régulières.

En présence d'une dynamique non triviale, bien comprise pour les solutions lisses, il est également pertienent de considérer des perturbations stochastiques dues par exemple à des fluctuations microscopiques irrégulières, et de se poser les questions suivantes :

• Quel est le comportement générique des solutions dans des régimes sur-critiques ?

• L'approche statistique permet-elle d'aller au delà des instabilités déterministes ?

• Comment l'aléatoire affecte-t-il la dynamique asymptotique de systèmes non-linéaires ? Ce manuscript se propose d'explorer différentes manifestations de ces questions, et d'y répondre en développant des techniques qui combinent outils déterministes et probabilistes, le tout à basse régularité. Le principal défi auquel ces travaux se sont confrontés, tant en temps court qu'en temps long, est le manque d'effet régularisant des systèmes dispersifs. Outre le fait que nous travaillons à des régularités sur-critiques où des instabilités se produisent, l'on ne dispose pas de lois de conservations qui nécessitent des solutions une certaines régularité. Hélas, l'utilisation de ces lois de conservation est indispensable si l'on veut comprendre la dynamique à long terme des solutions. Ainsi, la question sous-jacente aux travaux qui composent ce manuscrit, et qui va au-delà des considérations probabilistes, peut être énoncée comme suit :

• Comment exploiter les effets dispersifs lorsque les solutions sont très irrégulières ? Le manuscript de thèse comporte trois parties, dont les enjeux et les conclusions sont résumés dans les paragraphes qui suivent.

Les régimes sur-critiques: entre instabilités et ensemble statistique de données initiales

Dans l'étude de la théorie de Cauchy pour les équations dispersives, on rencontre généralement un seuil de régularité critique qui détermine si le problème de Cauchy est bien posé ou non. Étant donné un instant T > 0, rappelons qu'une courbe continue u : t ∈ [-T, T ] → u(t) ∈ H s est une solution forte au problème de Cauchy avec la condition initiale u(0

) = u 0 ∈ H s i∂ t u + ∆u = µ|u| 2 u , (t, x) ∈ R × M . u |t=0 = u 0 .
lorsqu'elle satisfait la formule de représentation intégrale de Duhamel

u(t) = e it∆ u 0 -i t 0 e i(t-τ )∆ |u(τ )| 2 u(τ )dτ . (0.3.1)
Dans le contexte des équations aux dérivées partielles dispersives, la définition d'un problème de Cauchy bien posé est rappelée dans la Définition 1.1.1. Lorsque l'on sait montrer que le problème de Cauchy est bien posé pour des données initiales très régulières, disons dans H σ pour un certain σ suffisamment grand, il est alors possible de définir un flot dans H σ via l'application

u 0 ∈ H σ -→ u ∈ C([-T, T ]; H σ ) .
Pour montrer le caractère mal posé du problème de Cauchy dans un espace de faible régularité H s , pour s ≪ σ, il suffit alors de montrer que le flot, définit sur H σ , ne s'étend pas à H s . Cette stratégie est motivée par la persistance de la régularité du flot, qui permet de voir les solutions dans H s , si elles existent, comme une limite de solutions régularisées, vivant dans l'espace H σ . Dans le cas des équations du type Schrödinger ou des ondes non-linéaires, la discontinuité du flot à des régularités sur-critiques se déduit de l'existence d'une suite de données initiales régulières qui convergent vers 0 pour la norme H s , mais telles que les normes H s des solutions locales correspondantes, définies dans H σ , vont à l'infini en des temps arbitrairement courts. Un tel mécanisme d'inflation des normes a été mis en évidence par [Leb01; Leb05; Lin93] pour des équation d'ondes nonlinéaires, puis étendu à l'équation de Schrödinger dans [START_REF] Christ | Ill-posedness for nonlinear Schrodinger and wave equations[END_REF]. La stratégie générale est d'effectuer une analyse de petite dispersion. On génère dans un premier temps des données initiales lisses (ψ n ) n à partir d'une fonction test ré-échelonée, qui convergent vers zéro dans H s tout en se concentrant spatialement. Ensuite, on considère la solution lisse u n dans H σ associée à la donnée initiale ψ n , et on étudie l'évolution du profil v n , solution l'équation différentielle sans dispersion

i∂ t v n = |v n | p-1 v n , v n (0) = ψ n .
Le

∥ψ n -u 0 ∥ H s = 0 , lim n→∞ ∥u n ∥ H s = +∞ ,
où u n est la solution lisse associée à ψ n . Mais que se passe-t-il lorsque la procédure de régularization est choisie judicieusement ? En outre, la théorie de Cauchy probabiliste étudiée dans cette thèse permet d'établir l'existence d'un ensemble statistique qui fournit des données initiales donnant lieu à des solutions fortes. De plus, ces solutions fortes sont les limites des solutions lisses associées à la suite des données initiales régularisées par convolution avec une approximation de l'identité. Ainsi, pour cette procédure de régularization, on dispose d'un ensemble de mesure pleine de données initiales pour lesquelles il n'y a pas d'inflation des normes.

A la lumière de cette discussion, il est pertinent de s' intéresser à l'ensemble pathologique des données initiales qui, lorsque régularisées par convolution, donnent quand même lieu à des solutions lisses approchées pour lesquelles l'inflation de la norme se produit. En d'autres termes, cet ensemble est contenu dans le complémentaire de l'ensemble statistique qui est dense et de mesure pleine. Néanmoins, Sun et Tzvetkov ont prouvé dans [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] que pour les équations d'ondes non-linéaires l'ensemble pathologique est également générique (au sens de la topologie) puisqu'il contient un ensemble G δ -dense. L'énoncé précis est écrit dans le Théorème 1.1.4.

Solutions fortes dans R d issues d'un ensemble statistique de données initiales A la suite des contributions pionnières de Bourgain [Bou94; Bou96] pour l'équation de Schrödinger sur le tore et des travaux de Burq,Tzvetkov [BT08a;[START_REF] Burq | Random data Cauchy theory for supercritical wave equations II: a global existence result[END_REF] pour l'équations des ondes sur des variétés compactes sans bords, de nombreux auteurs ont adopté une approche statistique pour étudier le problème de Cauchy en régularité sur-critique. Plus précisément, étant donné un espace de Sobolev H s sur lequel le flot n'est pas prolongeable par continuité, il est possible de trouver un sous-ensemble statistique de H s qui est de mesure pleine, pour une la mesure image d'une série de modes propres du Laplacien (dans le cas où il existe une résolution spectrale) découplés par des varaibles gaussiennes indépendentes. Les fonctions qui constituent l'ensemble statistique donnent lieu à des solutions fortes, . De plus, l'ensemble statistique est dense et la régularité qui n'est pas meilleure que H s . Une telle approche est motivée paar l'étude des mesures de Gibbs [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF], et a été largement développée dans d'autres contextes depuis. On renvoie à [BOP19a; BT08a] pour une introduction à la théorie de Cauchy dite probabiliste.

Dans l'espace euclidien où il n'y a pas de résolution spectrale pour l'opérateur de Laplace, il n'existe pas de procédure de randomisation canonique. Une façon de faire est d'utiliser une décomposition de l'espace des fréquences en des cubes unitaires (dits de Wiener). La randomisation qui en découle a été introduite par [START_REF] Lührmann | Random data Cauchy theory for nonlinear wave equations of power-type on R 3[END_REF] et [START_REF] Oh | Probabilistic global well-posedness of the energycritical defocusing quintic nonlinear wave equation on R 3[END_REF] indépendamment (voir la figure 0.1.2). Cette procédure a pour effet de relaxer la condition d'admissibilité dans les estimations de Strichartz (voir Figure 0.1.2). En effet, pour une fonction dont le support en fréquence est unitaire, nous pouvons utiliser l'inégalité de Bernstein pour passer de L q à L 2 sans perdre de dérivée. Ensuite, les estimations individuelles obtenues sur les blocs unitaires sont rassemblées au moyen d'un découplage probabiliste, dans l'esprit du théorème de Paley-Zygmund.

A l'aide d'inégalités de Strichartz probabilistes, Bényi, Oh et Pocovnicu [START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF] ont démontré l'existence d'un ensemble statistique Σ de données initiales à régularité sur-critique qui donnent lieu à des solutions locales fortes. Avec Louise Gassot, nous précisons ce résultat dans le Chapitre 1 en montrant qu'étant donnée une approximation de l'identité (ρ ε ) ε>0 définie par

ρ ε (x) := ε -3 ρ x ε , ρ ∈ C ∞ c (R d ) ,
si u 0 ∈ Σ alors la solution probabiliste issue de u 0 est la limite dans C([-T, T ]; H s ) des solutions approchées u ε , associées aux données initiales régularisées u 0;ε := u 0 * ρ ε .

Ensembles de données initiales pathologiques pour NLS. Le résultat principal du Chapitre 1 est la description de l'ensemble pathologique pour les équations de type Schrödinger. Pour cela, on utilise la même construction que [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] qui consiste à mettre côte à côte une infinité de profils concentrés à des échelles arbitrairement petites. La preuve proposée dans [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] revisite l'analyse de petite dispersion de [START_REF] Christ | Ill-posedness for nonlinear Schrodinger and wave equations[END_REF] en régularisant les profils par convolution avec ρ ε . Les auteurs utilisent ensuite le principe de propagation à vitesse finie de l'équation des ondes pour s'assurer que les différents profils n'interagissent pas avant le temps nécessaire à l'inflation des normes. Dans le cas des équations de type Schrödinger, on ne dispose pas d'un tel principe de propagation à vitesse finie. Pour remédier à cela, on exploite plutôt l'effet de la convolution sur les profils trop concentrés. Plus précisément, on observe qu'à une échelle de régularisation donnée ε > 0, la convolution par ρ ε casse les profils qui se concentrent à l'échelle λ -1 ≪ ε. Cette observation indique que le phénoème ne repose pas sur le principe de propagation à vitesse finie. Par conséquent, la même approche s'étend très certainement à de nombreuses autres équations non-linéaires dispersives.

Dynamiques en temps long pour des données initiales dans l'ensemble statistique

L'une des principales questions abordées dans cette thèse peut être formulée comme suit. Étant donné une donnée initiale typique (i.e. dans l'ensemble statistique) de la forme

u(0) = u 0 + f ω 0 , u 0 ∈ H σ , f ω 0 ∈ H s , avec s < s c ≤ σ ,
et la solution au problème de Cauchy qui lui est associée, que dire de son existence globale et de sa dynamique ?

Lorsque la mesure de probabilité est invariante ou quasi-invariante par le flot non-linéaire, l'argument de globalization de Bourgain (présenté dans le paragraphe 0.1.3) permet de déduire l'existence globale des solutions probabilistes ainsi que des propriétés dynamiques en temps long (propriété de récurrence de Poincaré par exemple). En effet, la présence d'une mesure invariante ou quasi-invariante permet de transporter une information de nature probabiliste par le flot non-linéaire, nécessaire à la résolution du problème de Cauchy local. Au delà de cet argument de globalization très général, et en l'absence de mesures invariantes, il s'agit de combiner des outils déterministes avec la théorie de Cauchy probabiliste afin de comprendre la stabilité de mécanisme dynamiques (scattering non-linéaire, stabilité asymptotique, croissance des normes de Sobolev,...) sous l'effet de perturbations aléatoires, très irrégulières. Les résultats qui constituent la partie II de ce manuscript illustrent cette démarche.

Scattering Lorsqu'une donnée initiale évolue par une équation de Schrödinger non-linéaire dont la non-linéarité est défocalisante (c'est à dire répulsive), sa dynamique en temps long est parfois approchée par l'évolution linéaire d'une autre donnée initiale. Ce phénomène de scattering nonlinéaire est bien compris dans l'espace d'énergie, lorsque l'équation est énergie sous-critique, ou énergie critique (voir par exemple [RV07; Vis12] en dimension 4, [START_REF] Colliander | Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3[END_REF] en dimension 3 et [START_REF] Dodson | Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrödinger equation when d = 2[END_REF] pour le cas masse-critique pour NLS cubique en dimension 2).

On considère dans le Chapitre 2 l'équation NLS avec une non-linéarité cubique défocalisante. Ce modèle est énergie sous-critique ce qui fait que le problème de Cauchy est bien posé dans l'espace d'énergie H 1 (R 3 ). Il découle de la coercivité de l'énergie conservée que la solution est globale. De plus, on déduit des estimations de Strichartz qu'il suffit d'obtenir une estimation a priori du type

R×R 3 |u(t, x)| p dx dt < ∞ , pour certains 3 10 < p < 10
pour prouver qu'il y a scattering dans H 1 (R 3 ). Une telle estimation a priori de la décroissance découle d'inégalités de flux de type Morawetz

I×R 3 |u(t, x)| 4 dt dx ≤ C sup t∈I ∥u(t)∥ 2 L 2 x ∥u(t)∥ 2 Ḣ 1 2 .
combinée à certaines lois de conservation. Une fois le scattering établi dans l'espace d'énergie, il s'agit de montrer que toutes les solutions locales (définies dans H s avec 1 2 < s < 1) existent elles aussi globalement, et qu'il y a scattering. Malheureusement, cette question simple s'avère être un problème ouvert très difficile, qui est encore loin d'être résolu en toute généralité. Des résultats partiels ont été obtenus via la célèbre I-method [START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF] et des raffinements basés sur long-time Strichartz estimates [Dod13; Su12; Dod19b]. On réfère au paragraphe 0.2.2 et à l'introduction du Chapitre 2 pour une discussion à propos de cette conjecture. Compte tenu de sa difficulté, il est intéressant d'aborder ce problème avec une approche statistique. En outre, il existe des ensembles statistiques de données initiales même dans les régimes sur-critiques, mais leur comportement en temps long n'est compris que sous des hypothèses de petitesse. En effet, les résultats de type scattering presque-sûr en dehors du régime des petites données n'existaient que dans le cas d'équations énergie critique.

Dans ce chapitre, on démontre qu'il y a scattering presque sûrement dans H s pour certains s < 1/2 en procédant comme suit. À partir de la décomposition de Da Prato et Debussche (0.1.5)

u = e it∆ f ω 0 + v ,
on analyse le comportement à long terme du terme non-linéaire v dans l'espace d'énergie H 1 qui est solution d'une équation de Schrödinger non-linéaire avec des termes de forçage stochastiques :

(i∂ t + ∆)v = |v + e it∆ f ω 0 | p-1 (v + e it∆ f ω 0 ) , (t, x) ∈ R × R d , v(0) = 0 , (NLS f )
La première étape consiste à utiliser la théorie de la stabilité de l'équation déterministe dans H 1 pour obtenir un résultat de scattering conditionnel. Ce faisant, on réduit le problème à l'obtention d'un contrôle uniforme sur la norme de H 1 de v. Ensuite, dans une deuxième étape, la stratégie consiste à trouver des énergies modifiées appropriées et des inégalités de Morawetz modifiées pour v puis à effectuer un double-argument de bootstrap à partir de ces deux quantités. Le résultat présenté dans le Chapitre 2 prouve le scattering presque-sûr en pour des régularités sous le seuil critique.

Theorem 0.3.1 (Scattering presque-sûr pour NLS dans R 3 [START_REF] Camps | Scattering for the cubic Schrödinger equation in 3D with randomized radial initial data[END_REF] et [START_REF] Shen | Almost sure well-posedness and scattering of 3D cubic nonlinear Schrödinger equation[END_REF]). Pour 3 7 < s on fixe f 0 ∈ H s (R 3 ) une fonction radiale et nous notons ω → f ω 0 la randomisation de Wiener qui lui est associée. Il existe un ensemble statistique de de mesure pleine Σ dans H s (R 3 ) tel que pour tout f ω 0 ∈ Σ, la solution probabiliste correspondante existe globalement en temps, et il y a scattering : il existe

v ± ∈ H σ (R 3 ), tel que lim t→±∞ ∥u(t) -e it∆ (f ω 0 + v ± )∥ H σ = 0 , σ = 2s -.
Voici quelques commentaires sur le résultat et sur les principales difficultés.

• On effectue une décomposition linéaire-non-linéaire u = e it∆ f ω 0 + v. L'observation clé est que le terme non-linéaire est dans H σ , pour σ < 1, mais suffisamment proche de 1 pour que la conjecture déterministe soit connue à cette régularité (à savoir 2/3 < s). Le défi (mais aussi la motivation) est de combiner la méthode probabiliste avec la I-method pour analyser le comportement asymptotique de v. La difficulté réside dans le fait que les termes de forçage stochastiques brisent la structure de la non-linéarité. Par conséquent, des termes supplémentaires apparaissent, pour lesquels il n'y a pas d'annulation de fréquence. Au lieu de cela, beaucoup de dérivés peuvent tomber sur le terme stochastique irrégulier.

• Pour cette raison, et comme dans le cas habituellement étudié des équations énergie critique, une hypothèse de symétrie radiale cachée, ou toute autre hypothèse similaire, semble être nécessaire. L'idée est d'exploiter une injection de Sobolev radiale afin de tirer profit de la décroissance locale de l'énergie (0.1.4) et de surmonter la perte de dérivées inhérente aux méthodes d'énergie. On rappelle que les données aléatoires f ω 0 ne sont en général pas radiales. Comme mentionné dans la section 0.2.2, le seul travail évitant une telle hypothèse est dû à Bringmann, pour l'équation d'onde énergie critique dans R 4 [START_REF] Bringmann | Almost sure scattering for the energy critical nonlinear wave equation[END_REF].

Stabilité asymptotique de petits états fondamentaux La théorie probabiliste de Cauchy s'intéresse principalement à la perturbation des solutions dispersives, dans le régime défocalisant ou pour de petites données initiales. Cependant, dans le régime focalisant où les équations dispersives présentent des dynamiques très riches, l'effet de perturbations aléatoires est bien moins compris. Le premier résultat dans cette direction est dû à Kenig et Mendelson, pour l'équation d'ondes énergie critique. Dans [START_REF] Kenig | The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data[END_REF], les autheurs ont prouvé que l'on peut construire une solution globale de l'équation d'onde quintique focalisante dans R 4 qui se décompose comme un soliton modulé et un terme dispersif, pour un ensemble statistique de données initiales dans H s × H s-1 lorsque s > 5 6 . Au lieu d'utiliser des multiplicateurs de Fourier classique, ils ont utilisé une transformée de Fourier dite distordue dans la procédure de randomisation (0.1.7), qui diagonalise l'opérateur linéarisé autour du soliton -∆ + V .

Le seul équivalent existant de ce résultat de stabilité asymptotique presque sûr pour l'équation de Schrödinger fait l'objet du Chapitre 3. Dans le cas des équations de type Schrödinger, soulignons que l'intégrale de Duhamel ne gagne pas de dérivée de sorte qu'une décomposition dyadique et des estimations bilinéaires doivent être effectuées pour établir un schéma de point fixe. Cependant, cela devient assez difficile en présence d'un potentiel, puisque la transformée de Fourier déformée ne partage pas les propriétés multiplicatives de la transformée de Fourier classique.

Dans le Chapitre 3, on étudie la stabilité asymptotique des petits états fondamentaux de l'équation de Schrödinger non-linéaire, en présence d'un potentiel de courte portée V ∈ S(R 3 )

(i∂ t + ∆)u(t, x) = V (x) potentiel externe + µ|u(t, x)| 2 pertrubation non-linéaire u(t, x) , (t, x) ∈ R × R 3 ,
où l'on suppose que l'opérateur de Schrödinger -∆+V a une valeur propre simple négative. Ce modèle a d'abord été étudié par Soffer et Weinstein [START_REF] Soffer | Multichannel nonlinear scattering for nonintegrable equations[END_REF], qui montrent que le comportement asymptotique des petites solutions, sous des hypothèses de décroissance, est donné par la superposition d'une onde stationnaire localisée, qui est périodique dans le temps, et d'une partie dispersive. Gustafson, Nakanishi et Tsai ont ensuite prouvé dans [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF] que la stabilité asymptotique a lieu pour toute solution suffisamment petite dans l'espace d'énergie H 1 (R 3 ), sans aucune hypothèse de décroissance. Qu'advient-il des états fondamentaux sous l'effet de perturbations plus irrégulières, qui ont une énergie infinie ? Plus précisément, étant donnée une petite donné initiale u 0 dans l'espace critique H 1 2 , on considère dans le Chapitre 3 des perturbations aléatoires de la forme

u(0) = u 0 + f ω 0 , u 0 ∈ H 1 2 , f ω 0 ∈ H s ∩ Ran P c (H) ,
où P c (H) désigne la projection sur le sous-espace spectral continu. Les données initiales sont randomisées selon la procédure de Wiener (voir la sous-section 0.1.2), mais elles sont adaptées à l'évolution linéaire inhomogène e itH dans le sens où les multiplicateurs de Fourier sont remplacés par des multiplicateurs de Fourier distordus. On démontre la stabilité asymptotique des états fondamentaux sous l'effet de ces perturbations.

Theorem 0.3.2. Soit 1 4 < s < 1 2 . Il existe δ 0 > 0 et un ensemble de mesure pleine Σ dans H s (R ω ) tel que pour tout u 0 dans H 1 2 dont la norme inférieure à δ 0 , et pour tout f ω 0 dans Σ, il existe ε = ε(ω) > 0 pour lequel le problème de Cauchy

(i∂ t + ∆)ψ(t, x) = µ|ψ(t, x)| 2 ψ(t, x) + V (x)ψ(t, x) , (t, x) ∈ R × R 3 , ψ(0) = εf ω 0 + v 0 ,
admet une solution locale sous la forme

ψ(t, x) = ε e -itH f ω 0 + v , v ∈ C(R, H 1 2 (R 3 )) .
De plus, cette solution s'étend globalement dans le temps, et se décompose en la somme d'un état fondamental modulé et d'un terme radiatif :

ψ(t, x) = Q(z(t)) + η(t, x) .
En temps long, les états fondamentaux sont asymptotiquement stables, dans le sens où il existe

z ± ∈ C et des états finaux η ± ∈ H 1 2 tels que lim t→±∞ z(t) exp -i t 0 E(z(τ )) dτ = z ± , lim t→±∞ ∥η(t) -e -itH (εf ω 0 + η ± )∥ H 1 2 = 0 .
• On réfère au Chapitre 3 pour un énoncé plus précis.

• Comme mentionné ci-dessus, lorsque f 0 = 0, le théorème 0.2.5 étend le résultat déterministe de [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF] jusqu'à la régularité critique s = 1 2 en prenant f 0 = 0. • Ce travail a été inspiré par la contribution de Kenig et Mendelson présentée dans la section 0.1.2 pour l'équation d'onde énergie critique. Comme expliqué dans la Section 0.1.2, dans le cas de l'équation de Schrödinger, certaines difficultés apparaissent. La difficulté la plus sérieuse est que le terme intégral de Duhamel ne présente aucun gain de dérivée, et que l'opérateur linéarisé autour des états fondamentaux n'est pas auto-adjoint. Par conséquent, la prise en compte de petits états fondamentaux est un premier pas raisonnable vers l'objectif d'adapter le résultat de Kenig et Mendelson aux équations de Schrödinger.

• La principale nouveauté est le fait d'avoir combiné l'utilisation d'espace fonctionnels critiques avec le principe de décroissance locale de l'énergie, pour la première fois dans le contexte de la théorie de Cauchy probabiliste.

Equations faiblement dispersives : vers une théorie de Cauchy probabiliste quasilinéaire

Dans certains cas où la dispersion est trop faible ou lorsque les interactions non-linéaires sont prépondérantes, des interactions singulières peuvent apparaître et limiter le gain possible de régularité des itérations de Picard. Dans le pire des cas, les mauvaises interactions peuvent même exclure tout gain de régularité. Dans de tels cas, on ne peut pas utiliser le schéma de résolution probabiliste semi-linéaire de Bourgain. Bringmann dans [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] a étudié une équation d'onde quasi-linéaire, et a proposé le premier schéma de résolution probabiliste quasi-linéaire . Il s'agit d'un schéma itératif sur la taille des fréquences de la donnée initiale, qui s'inspire du calcul para-contrôlé, dans un contexte dispersif. L'idée clé est d'utiliser l'indépendance probabiliste entre les hautes fréquences de la donnée initiale et la solution approchée, qui est construite à partir des basses fréquences de la donnée initiale. Ce schéma itératif quasi-linéaire a ensuite été développé de manière spectaculaire par Deng, Nahmod et Yue pour prouver l'invariance de la mesure de Gibbs associée à l'équation de Schrödinger semi-linéaire (avec une renormalisation de Wick) sur T 2 ,

(i∂ t + ∆)u =: |u| 2p u : (t, x) ∈ R × T 2 , u(0) = u 0 ,
où la puissance p ≥ 2 de la non linéarité est arbitraire. Il s'agit d'une percée importante, dans laquelle les auteurs ont introduits des opérateurs de moyennisation aléatoires dont l'utilisation s'est avérée extrêmement puissante et prometteuse, dans la perspective de résoudre d'autres problèmes difficiles ouverts depuis de nombreuses années. On renvoie à l'introduction du Chapitre 4 et au paragraphe 0.2.3 pour une discussion plus détaillée à propos du schéma de résolution quasi-linéaire.

Théorie de Cauchy probabiliste raffinée pour des équations faiblement dispersives Dans l'esprit de [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], on utilise dans le Chapitre 4 un schéma itératif quasi-linéaire similaire pour résoudre le problème de Cauchy local associé à une équation faiblement dispersive qui est par ailleurs anisotrope.

Le comportement qualitatif des solutions aux équations d'évolution non-linéaires à faible dispersion fait l'objet d'une activité intense. Des exemples éloquents sont l'équation de demi-ondes (HW) [Poc11; Gér+18a], l'équation de Szegő [GG10; GG12b] (Sz), l'équation de Schrödinger sur une variété compacte [START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schröinger equation on surfaces[END_REF] ou sur le groupe de Heisenberg [Gér06; Gas21a], les équations de Schrödinger fractionnaire [FL13; GPH15; ST20c] (FNLS), les équations de Kadomtsev-Petviashvili [START_REF] Hadac | Well-posedness and scattering for the KP-II equation in a critical space[END_REF] (KPI et KPII) ainsi que de nombreux autres modèles. En général, il est très difficile de résoudre le problème de Cauchy pour ces équations à faible régularité, la raison étant que dans de tels contextes le manque de dispersion se traduit par une perte de dérivées dans les estimations de Strichartz (0.1.2). Lorsque l'équation faiblement dispersive n'est ni complètement intégrable ni semi-linéairement bien posée, il est pertinent de développer une théorie de Cauchy à données aléatoires inspirée des schémas de résolution quasi-linéaires. En effet, pour faire fonctionner le schéma de résolution probabiliste semi-linéaire de Bourgain, il faut combiner l'observation de Paley-Zygmund du théorème (0.1.2) avec des effets dispersifs comme dans (0.1.3), (0.1.4).

Dans la perspective d'étudier les équations faiblement dispersives à faible régularité, on considère dans le Chapitre 4 l'équation de Schrödinger demi-onde (NLS-HW) avec une non-linéarité de type puissance :

pour µ ∈ R, et (t, x, y) ∈ R 1+2 , cette équation s'écrit i∂ t u + ∂ xx -|D y | u = µ|u| 2 u , où |D y |u(ξ, η) = |η| u(ξ, η) . u t=0 = u 0 ,
Il s'agit là d'un modèle anisotrope, où on a une équation de Schrödinger dans la direction x et une équation de demi-onde dans la direction y, respectivement. Par conséquent, dans les régimes à basse fréquence dans la direction x et à haute fréquence dans la direction y, le transport domine et il n'y a pas de dispersion. Comme discuté à la fin de cette section, la dynamique asymptotique des solutions lisses de cette équations est très riche (voir [Xu17; BIK20; BIK21]), malgré le fait que la théorie de Cauchy pour cette équation est mal comprise. Pour étudier la théorie de Cauchy pour (NLS-HW) on mesure la régularité dans l'échelle des espaces de Sobolev anisotropes :

H s (R 2 ) = L 2 x H s y (R 2 ) ∩ H 2s x L 2 y (R 2
) . Le choix naturel de ces espaces est dicté par la structure hamiltonienne de l'équation et par l'invariance de l'équation par changement d'échelle :

u → u λ (t, x, y) = λψ(λ 2 t, λx, λ 2 y) .
En particulier, l'exposant critique associé cette symétrie est s c = 1 4 . L'énergie conservée s'écrit

H(u) = 1 2 R 2 |D x u| 2 + ||D y | 1 2 u| 2 dx dy + µ 4 R 2 |u| 4 dx dy .
Le problème de Cauchy est bien posé lorsque s > 1 2 . Ceci a été prouvé dans [START_REF] Bahri | Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations[END_REF] à partir d'un point fixe et d'estimations de Strichartz avec perte de dérivée. Cependant une théorie de Cauchy satisfaisante dans l'espace d'énergie et en dessous n'a pas encore été établie. L'espace d'énergie correspond au point limite s = 1 2 , et il existe de sévères obstructions pour établir une théorie de Cauchy dans cet espace. Ces obstructions sont soulevées dans l'introduction du Chapitre 4.

On démontre que le problème de Cauchy est presque-sûrement bien posé dans H s pour des 1/4 < s < 1/2 en adaptant le schéma itératif de [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF]. On montre à cette occasion qu'une telle méthode probabiliste peut surmonter le manque de dispersion, et qu'elle pourrait être adaptée à d'autres modèles faiblement dispersifs afin de générer des grands ensembles de données initiales qui donnent lieu à des solutions fortes. Par exemple, [START_REF] Oh | Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegő equation[END_REF] a mis en évidence des interactions singulières pour des modèles tels que Szegő ou KdV, qui seront surmontées par l'approche paracontrôlée utilisée dans le présent article. On précise par ailleurs dans le Chapitre 4 la théorie de Cauchy pour (NLS-HW), résumée dans la figure 0.2.3.

Invariance de la mesure de Gibbs pour NLS sur la sphère S 2 ? Afin de comprendre l'effet de la géométrie ambiante sur la dynamique des équations de Schrödinger non-linéaires, Burq, Gérard et Tzvetkov ont prouvé dans une série de travaux [BGT02; BGT04; BGT05b; BGT05a; BGT10] que le problème de Cauchy pour l'équation de Schrödinger cubique sur la sphère S 2 est semi-linéairement bien posé dans H s (S 2 ) si et seulement si s > 1 4 . Il y a par conséquent un écart très important à combler entre le seuil déterministe et l'espace où vit la mesure de Gibbs qui est H 0-(S 2 ). Il est donc extrêmement difficile d'obtenir l'analogue sur la sphère S 2 du célèbre résultat de Bourgain [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] sur le tore T 2 . En outre, la théorie de Cauchy probabiliste juste en dessous du seuil déterministe s = 1 4 reste à établir. Discutons brièvement des difficultés. La plus importante est la dégénérescence des harmoniques sphériques. En effet, étant donné le niveau d'énergie λ 2 n = n(n + 1), la multiplicité de l'espace propre correspondant, généré par les harmoniques sphériques de degré n, est de dimension 2n + 1. De plus, certaines harmoniques sphériques se concentrent autour des géodésiques de la sphère (ce qui n'est pas le cas pour les ondes planes e in•x sur le tore), et cela engendre des instabilités non-linéaires.

Pour amorcer l'étude probabiliste de l'équation NLS sur la sphère, nous avons obtenu un résultat préliminaire où l'on isole une interaction singulière dans la première itération de Picard, et on démontre que l'équation de Schrödinger cubique posée sur S 2 avec des données aléatoires est quasilinéaire (malgré la renormalization de Wick et le découplage probabiliste). En particulier, la seconde itération de Picard n'est pas plus régulière (en espérance) que les données initiales aléatoires. Avant de discuter du résultat principal et de ses conséquences, présentons le cadre probabiliste. On fixe α ∈ [ 1 2 , 3 4 ) ainsi qu'une suite de variables gaussiennes normalisées indépendantes à valeurs complexes (g n,k ) 1≤n ,|k|≤n . On définit ensuite

ϕ ω = 1≤n e ω n λ α n , e ω n (x) = |k|≤n g n,k (ω) [n] 1 2 Y n,k (x) . (0.3.2) La série ci-dessus converge dans L 2 (Ω; H α-1 2 -(S 2 )), de sorte que ϕ ω vit dans H α-1 2 -(S 2 )) presque surement. Lorsque α = 1
2 , les données initiales sont distribuées selon la mesure de Gibbs associée à NLS vu comme un système hamiltonien (on renvoie à la section 0.1.3 pour une discussion sur les mesures de Gibbs). Cependant, le seuil déterministe pour que l'équation soit bien posée est H 1 4 (S 2 ). Le résultat est le suivant.

Theorem 0.3.3 (Absence de régularisation de la deuxième itération de Picard, [START_REF] Burq | Lack of regularization in the first Picard iteration for NLS on S 2[END_REF]

). Soit α ≥ 1 2 , et ϕ ω ∈ H α-1 2 -(S 2 ) les données initiales randomisées comme dans (0.3.2). Il existe η > 0, N 0 > 0 tels que pour chaque N ≥ N 0 et t, |t|η log(N ) ≤ E ∥ t 0 e i(t-τ )∆ Π N N (Π N e iτ ∆ ϕ ω ) dτ ∥ 2 H α-1 2 (S 2 )
, où N désigne le terme non-linéaire de l'équation cubique renormalisée (NLS W ).

Commentons brièvement ce résultat.

• Lorsque N tend vers l'infini, le moment quadratique de la norme H α-1 2 (S 2 ) de la seconde itération de Picard de ϕ ω a une divergence logarithmique. Puisque la mesure induite vit dans H α-1 2 -(S 2 ), cela montre qu'il n'y a pas d'effet de régularisation probabiliste.

• Pour des raisons techniques, nous n'avons pas pu prouver que la divergence se produit presquesurement en ω. Il n'est pas clair s'il existe une preuve élémentaire de ceci, qui ne repose pas sur des outils sophistiqués d'analyse stochastique. On montre simplement une divergence logarithmique en espérance.

• A l'opposé, la seconde itération de Picard sur T 2 gagne 1 2 dérivées. On démontre que ceci reste vrai dans le cas des tores irrationnels, et on donne une preuve auto-contenue de ce résultat, déjà évoqué dans l'article [START_REF] Fan | 2D-defocusing nonlinear Schrödinger equation with random data on irrational tori[END_REF].

Le résultat s'accompagne également de quelques observations positives, et n'exclut pas la résolution du problème de Cauchy en dessous de l'espace critique H 1 4 (S 2 ). En effet, on a non seulement identifié l'obstacle à l'exécution d'un schéma semi-linéaire, mais on a également prouvé que d'autres interactions ont un effet régularisant.

Dans un programme à long terme avec N. Burq, C. Sun et N. Tzvetkov, nous développons une théorie probabiliste de Cauchy à l'aide du schéma itératif quasi-linéaire décrit précédemment, en employant le calcul paracontrôlé. Plus précisément, nous surmontons l'absence d'effet régularisant en affinant l'ansatz probabiliste et en exploitant l'indépendance des données initiales à haute fréquence et des solutions approchées construites à partir des basses fréquences, en utilisant les opérateurs de moyenne aléatoire mentionnés dans la section 0.1.2. L' analyse reposera ainsi sur l'utilisation des opérateurs de moyenne aléatoire, s'inspirant des travaux de Bringmann [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], Deng, Nahmod, Yue [START_REF] Deng | Invariant Gibbs measures and global strong solutions for nonlinear Schröodinger equations in dimension two[END_REF], et Sun, Tzvetkov [START_REF] Sun | Refined probabilistic global well-posedness for the weakly dispersive NLS[END_REF], et elle devra comporter des nouveautés importantes. Il nous faudra en outre exploiter la propagation par le flot de l'invariance des mesures gaussiennes sous l'action des isométries de la sphère. Cette caractéristique clé nous permet de briser significativement le seuil de régularité déterministe s = 1 4 . Afin de descendre jusqu'à la mesure de Gibbs, il nous faudra être capable de traiter un terme complètement résonnant, qui est de nature critique.

Bilan Dans le régime sur-critique, nous avons deux comportements différents pour le flot non-linéaire approché issu de données initiales régularisées par convolution. D'une part, le théorème 1.1.4 met en évidence un ensemble pathologique de données initiales où se produit l'inflation des normes, qui contient un sous-ensemble G δ -dense. D'autre part, l'approche statistique initiée par Bourgain puis Bruq, Tzvetkov fournit un ensemble statistique de données initiales de mesure pleine décrit dans le théorème 1.1.3, pour lequel les solutions régularisées convergent vers un objet limite qui est une solution forte de l'équation. Il y a ainsi deux notions différentes de généricité dans les régimes sur-critiques. Alors que l'ensemble pathologique décrit dans le Théorème 1.1.4 est générique pour la topologie, il est négligeable pour la mesure. De plus, nous déduisons du théorème de Baire que l'ensemble statistique donné par le théorème 1.1.3 ne peut pas contenir d'ensemble G δ -dense.

L'étude du comportement en temps long de ces solutions est illustrée dans la partie II, où nous avons obtenus deux résultats de natures différentes. Rappelons que les solutions probabilistes issues de données initiales dans l'ensemble statistique s'obtiennent à partir de la décomposition de Da Prato et Debussche

u(t) = e it∆ f ω 0 + v , (0.3.3)
où la partie non-linéaire v est suffisamment régulière pour être obtenue par un schéma itératif de Picard. D'une part, nous établissons le scattering presque sûr pour l'équation NLS défocalisante dans un régime où l'espace d'énergie est sous-critique. D'autre part, on démontre la stabilité asymptotique de petits états fondamentaux sous l'effet de perturbations irrégulières. Cela montre qu'il est possible d'étudier l'évolution non-linéaire en temps long d'une donnée initiale dans l'ensemble statistique, en adoptant une approche perturbative qui combine des outils probabilistes et des méthodes d'analyse des équations dispersives non-linéaires. Une perspective est l'étude de l'évolution de la mesure elle-même par le flot non-linéaire, afin de comprendre finement la propagation de l'information probabiliste des données initiales (invariance, quasi-invariance)... L'on a ensuite étudié des modèles faiblement non-linéaires, pour lesquels l'effet régularisant des itérations de Picard n'est pas suffisant pour établir la convergence de ce schéma, bien que l'on soit dans un cadre probabiliste. Il n'est alors pas possible d'obtenir une solution de la forme (0.3.3). Pour obtenir des solutions locales, il faut plutôt utiliser un schéma itératif quasi-linéaire qui exploite la structure probabiliste de la solution, dans la continuité des développements récents Bringmann puis Deng, Nahmod, Yue et Sun, Tzvetkov. Se pose alors la question du comportement en temps long de ces solutions. Pour espérer y répondre, il faut comprendre comment se propage l'information probabiliste des données initiales. Ainsi, les techniques perturbatives déterministes développées dans la partie II ne sont pas suffisantes pour pouvoir itérer en temps le schéma quasi-linéaire. Il faudrait démontrer une forme d'invariance ou de quasi-invariance de la mesure de probabilité puis adapter l'argument de globalisation de Bourgain. Comme l'attestent les chapitres qui constituent la partie III, soulignons pour conclure que l'étude des mesures invariantes (e.g. Gibbs) ou quasi-invariantes dans de tels régimes non-linéaires et faiblement dispersifs, et à basse régularité, constitue un domaine de recherche difficile et actif, dont les défis nécessiteront le développement de nouvelles approches aussi bien stochastiques que déterministes.

Part I Supercritical regimes: instabilities and well-posedness

Chapter 1

Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations joint work with Louise Gassot [START_REF] Camps | Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations[END_REF].

Résumé. Ce travail vise à mettre en évidence un ensemble pathologique de données initiales, pour lesquelles la norme de Sobolev des solutions régularisées par convolution croit en un temps arbitrairement court. Le résultat est dans l'esprit de la construction de Sun et Tzvetkov [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF], où l'ensemble pathologique contient une superposition de profils qui se concentrent en différents points. Leur argument repose sur le fait que, grâce au principe de propagation à vitesse finie de l'équation d'onde, au plus un profil présente une croissance significative en un temps t suffisamment petit. Cependant, pour les équations de type Schrödinger, nous ne pouvons pas exclure que les profils interagissent entre eux. La méthode que nous proposons exploite l'effet régularisant de l'approximation de l'identité par laquelle les données initiales sont lissées, qui permet d'éviter la croissance de la norme des profils qui sont concentrés à des échelles plus petites.

Abstract. The purpose of this work is to evidence a pathological set of initial data for which the regularized solutions by convolution experience a norm-inflation mechanism, in arbitrarily short time. The result is in the spirit of the construction from Sun and Tzvetkov [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF], where the pathological set contains superposition of profiles that concentrate at different points. Thanks to finite propagation speed of the wave equation, and given a certain time, at most one profile exhibits significant growth. However, for Schrödinger-type equations, we cannot preclude the profiles from interacting between each other. Instead, we propose a method that exploits the regularizing effect of the approximate identity which, at a given scale, rules out the norm inflation of the profiles that are concentrated at smaller scales.

i∂ t u + ∆u + σ|u| p-1 u = 0 , (t, x) ∈ R × R 3 , p > 1 , (NLS)
where σ = 1 in the focusing case and σ = -1 in the defocusing case, respectively. Our main result is the description of a pathological set for equation NLS, based on the approach of Sun and Tzvetkov [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] for the nonlinear wave equation. We prove that if we regularize the initial data using a given convolution profile, there exists a dense set of initial data in H s (R 3 ) such that the regularized solutions exhibit norm inflation. The precise statement can be found in Theorem 1.1.4 below. Such a result aims at precising the probabilistic well-posedness theory, summarized in our context in Theorem 1.1.3. Let us first recall the motivation for this result. In the study of the Cauchy theory for dispersive equations, one generally encounters a critical regularity threshold that dictates whether the equation is well-posed or not, in the Hadamard sense. We recall that a general dispersive equation on a Riemannian manifold (M, g) can be written under the form

i∂ t u + Lu = F (u) , u(0) ∈ H s (M ) , (1.1.1)
where L is a well-defined operator on H s (M ) such that the linear propagator exp(itL) t∈R is a group of isometries of H s (M ), whereas the nonlinear interaction, denoted F (u), has to make sense in

H s (M ). Given a time interval I ⊂ R containing t 0 , the curve t ∈ I ⊂ R → u(t) ∈ C([t 0 , T ], H s (M ))
is a strong solution to (1.1.1), with initial condition u(t 0 ) = u 0 ∈ H s (M ), if it satisfies the following Duhamel integral representation formula, in the distributional sense

u(t) = e i(t-t 0 )L u 0 -i t t 0 e i(t-τ )L F (u(τ ))dτ . (1.1.2)
For dispersive equations, a suitable notion of well-posedness, that we recall here, can be found in [START_REF] Tzvetkov | Ill-posedness issues for nonlinear dispersive equations[END_REF].

Definition 1.1.1 (Local well-posedness). Given a general dispersive equation as (1.1.1), we say that the Cauchy problem is well-posed in H s (M ) if for every bounded set B ⊂ H s (M ), there exist T = T (B) > 0 and a Banach space X T continuously embedded in C([t 0 , T ], H s (M )) such that for any u 0 ∈ H s (M ), there exists a solution u ∈ X T satisfying the Duhamel integral representation formula (1.1.2). Moreover, the following holds.

1. (Uniqueness) 1 Let u 1 , u 2 ∈ X T satisfying (1.1.2). If, for some t 0 ≤ t ≤ T , u 1 (t) = u 2 (t), then u 1 = u 2 . 2. (Continuity) The flow map u 0 ∈ B → u ∈ X T is continuous. 3. (Persistence of regularity) If u 0 ∈ H σ (M ) with σ > s, then u ∈ C([t 0 , T ]), H σ (M )).
In addition, when the flow-map is uniformly continuous, we say that the Cauchy problem is semi-linearly well-posed.

This definition deserves some comments, especially in the case of critical regimes where the time of existence does not only depend on the bounded set B but also on the profile of the initial data. In Definition 1.1.1, we ask for some persistence of regularity in order to see the solutions in H s (M ) as limits of regularized smooth solutions, and this motivates the study of the Cauchy problem at low regularity. A natural question that arises is whether the flow map is always uniformly continuous, as soon as it is continuous. In other words, is the critical threshold for the well-posedness equal to the critical threshold for the semi-linear well-posedness ? In general, the answer is no, especially in the context of KdV type equations, see [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations II. The KdV equation[END_REF][START_REF] Takaoka | Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary conditions[END_REF]. We refer to [START_REF] Tzvetkov | Ill-posedness issues for nonlinear dispersive equations[END_REF] for discussions on this question, and for further comments on Definition 1.1.1.

Sometimes, an invariance property of the equation under the action of a group of symmetries indicates the aforementioned well-posedness threshold. The first example, that we exploit in this work, is the scaling symmetry. If an equation is invariant under a scaling symmetry

u(t, x) -→ u λ (t, x) := λ α u(λ β t, λ γ x) , λ > 0 ,
then, in the scale of homogeneous Sobolev spaces, its critical regularity s c corresponds to the Sobolev norm left invariant under the action of the scaling

∥u λ (0, x)∥ Ḣsc = ∥u(0, •)∥ Ḣsc .
We expect well-posedness at scaling-subcritical regularities s ≥ s c , and ill-posedness at scalingsupercritical regularities s < s c . Note, however, that this heuristic is quite informal and in general, there might exist other properties of the equation that actually dictate the well-posedness threshold.

For instance, in the case of integrable equations, the scaling, semi-linear and well-posedness thresholds often differ. A classic example is the KdV equation on T, which is continuously well-posed in H s (T) if s ≥ -1, but the solution map is locally uniformly continuous only when s ≥ -1 2 (in which case it is real analytic). Besides, the next equation in the KdV hierarchy is continuously well-posed for s ≥ -1 2 , but the solution map is nowhere locally uniformly continuous for this range of exponents [START_REF] Kappeler | On the wellposedness of the KdV/KdV2 equations and their frequency maps[END_REF], whereas these equations are invariant under the scaling transformation with critical Sobolev index s c = -3 2 . Similarly, the Benjamin-Ono equation on T is continuously well-posed in H s (T) for s > -1 2 [START_REF] Gérard | Sharp well-posedness results of the Benjamin-Ono equation in H s (T, R) and qualitative properties of its solution[END_REF], the flow map is locally uniformly continuous only when s ≥ 0 (in which case it is real analytic) [START_REF] Gérard | On the analyticity of the nonlinear Fourier transform of the Benjamin-Ono equation on T[END_REF],. The next equation in the Benjamin-Ono hierarchy is continuously well-posed for s ≥ 0 [START_REF] Gassot | The third order Benjamin-Ono equation on the torus: well-posedness, traveling waves and stability[END_REF], but an adaptation of [START_REF] Kappeler | On the wellposedness of the KdV/KdV2 equations and their frequency maps[END_REF] implies that the flow map is nowhere uniformly continuous. For all the equations in the Benjamin-Ono hierarchy, the scaling-critical Sobolev exponent is s c = -1 2 . Geometry also influences the dispersion property of the equation, for instance the NLS equation on T, with scaling-critical Sobolev exponent s c = -1 2 , is ill-posed for s < 0 [CCT03c] but continuously well-posed when s ≥ 0 with a smooth solution map [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schrödinger equations[END_REF]. In dimension 2, NLS is well-posed on T2 for s c = 0 < s, whereas the flow map of NLS on the sphere S 2 is not uniformly locally well-posed as soon as s < 1 4 , see [START_REF] Banica | On the nonlinear Schrödinger dynamics on S d[END_REF][START_REF] Burq | An instability property of the nonlinear Schrödinger equation on S d[END_REF]. We also point out that the flow map for the NLS equation on the Heisenberg group cannot be C 3 in H s (H 1 ) for s < 2, whereas the scaling-critical exponent is s c = 1, see for instance [START_REF] Gérard | The cubic Szegő equation[END_REF].

For the nonlinear Schrödinger equation NLS, as well as for the nonlinear wave equation in R 3 or T 3 , the scaling symmetry

u -→ u λ (t, x) = λ 2 p-1 u(λ 2 t, λx)
actually dictates the local well-posedness threshold, which is independent of the focusing or defocusing nature of the equation, see [START_REF] Christ | Ill-posedness for nonlinear Schrodinger and wave equations[END_REF].

Theorem 1.1.2 (Well-posedness [CW90; CCT03a]). Let p > 1. 2
• The Cauchy problem associated to NLS is well-posed in H s (R 3 ) in the sense of Definition 1.1.1 when max(0, s c ) < s.

• When 0 ≤ s c , the Cauchy problem is well-posed in H sc (R 3 ) but the time of existence does depend on the initial data and not just on its H sc -norm.

• When s < s c or when s < 0, the Cauchy problem is ill-posed.

We comment on the ill-posedness result due to [START_REF] Christ | Ill-posedness for nonlinear Schrodinger and wave equations[END_REF] in the next paragraph. Before that, we recall that the mass and the energy are formally conserved under the flow, we denote them

M (u) = 1 2 R 3 |u(x)| 2 dx , H(u) = 1 2 R 3 |∇u(x)| 2 dx - σ p + 1 R 3 |u(x)| p+1 dx .
Note that these conservation laws ensure the existence of smooth global solutions in the energysubcritical regime p ≤ 5, when the energy is coercive.

Ill-posedness issues. Assume that the Cauchy problem is known to be well-posed for smooth initial data in the sense of Definition 1.1.1, say in H σ for some σ large enough. Then, one can try to prove ill-posedness in a low-regularity space H s by showing that the flow-map, defined on H σ for some σ > s, does not extend continuously to C([0 , T ], H s ), with T being arbitrarily small. This strategy is motivated by the persistence of regularity of the flow, that enables one to see the solutions in H s , if they exist, as a limit of smoother solutions in H σ . Before going further, let us point out that it is easier to prove ill-posedness in the presence of special nonlinear solutions. For instance, for focusing Schrödinger equations where nonlinear solutions are accessible, we have the following [START_REF] Sulem | The nonlinear Schrödinger equation: Self-focusing and wave collapse[END_REF]. If s c < 0, the Schrödinger equation is mass-subcritical and there exist blow-up solutions that, combined with scaling arguments and virial identities, lead to solutions blowing up in arbitrarily short time in H s when s < s c . In the mass-critical case, a blow-up solution is obtained by applying a pseudo-conformal transformation to the ground state solution, and for the mass-supercritical case [START_REF] Kenig | On the ill-posedness of some canonical dispersive equations[END_REF] proved that the flow map is not uniformly continuous, respectively. Meanwhile, in the defocusing case, we do not have explicit nonlinear profiles generating blow-up solutions, except in the Schrödinger case with p = 3, d = 1 where one can make use of modified scattering solutions [START_REF] Christ | Asymptotics, Frequency Modulation, and Low Regularity Ill-Posedness for Canonical Defocusing Equations[END_REF]. The norm inflation instability we describe in the next paragraph is by no mean the only way to indicate that a dispersive PDE is ill-posed, and we refer to [START_REF] Kenig | On the ill-posedness of some canonical dispersive equations[END_REF][START_REF] Tzvetkov | Ill-posedness issues for nonlinear dispersive equations[END_REF] for the description of ill-posedness results in a broader context.

Norm inflation

In the case of the Schrödinger or wave type equations, the discontinuity of the flow-map at scaling-supercritical regularities is stronger. Specifically, there exists sequences of smooth initial data going to zero in H s , such that the H s -norms of the corresponding local solutions go to infinity, for arbitrarily short times. Such a norm inflation mechanism was evidenced by [Leb01; Leb05; Lin93] for the wave equation3 , and extended to the Schrödinger equation in [START_REF] Christ | Ill-posedness for nonlinear Schrodinger and wave equations[END_REF].

The general strategy to prove that norm inflation occurs in the scaling-supercritical regime is to perform a small dispersion analysis. Namely, by rescaling an arbitrary bump function, one generates a sequence of smooth initial data (ψ n ) n that go to zero in H s , while spatially concentrating around a point. Then, one considers the smooth solution u n with initial data ψ n , and studies the bubble solution v n to the dispersionless ODE

i∂ t v n = σ|v n | p-1 v n , v n (0) = ψ n .
The ODE profile v n , that captures the oscillations of u n , grows in H s after a time t n going to zero as n goes to infinity. Then, when 0 < s c and s < s c , one uses a priori energy estimates up to time t n and establishes that uniformly in n,

∥u n (t n ) -v n (t n )∥ H s ≲ 1 .
Therefore, the oscillations dominate the dispersion so that the instability stems from a frequency cascade from low to high Fourier modes.

We note that such a mechanism is independent of the focusing or defocusing nature of the equation. Specifically, in the case of Schrödinger equation, the results from [START_REF] Christ | Ill-posedness for nonlinear Schrodinger and wave equations[END_REF] state as follows. Norm inflation occurs when s < -d 2 , and also in the mass-supercritical case when 0 < s < s c . Furthermore, by using the frequency modulation method, Christ Colliander and Tao evidenced in [START_REF] Christ | Ill-posedness for nonlinear Schrodinger and wave equations[END_REF] the high to low frequency cascade when s < -d 2 , and the low to high frequency cascade when 0 < s. They also proved that the flow map fails to be uniformly continuous for s < 0 by using the Galilean symmetries of NLS. Note that a similar small dispersion analysis has been performed earlier, in [START_REF] Kuksin | On squeezing and of energy for nonlinear wave equations[END_REF]. We refer to [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF][START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF] for some subsequent norm inflation results on compact manifolds, with similar proofs that rely on concentrating profiles in a point. We also mention the norm inflation results in negative Sobolev spaces initiated in [START_REF] Bejenaru | Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation[END_REF], then further developed in [CDS12; CK17], see also [START_REF] Kishimoto | A remark on norm inflation for nonlinear Schrödinger equations[END_REF][START_REF] Oh | A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces[END_REF].

Some variations

The norm inflation does not only occur around the zero solution, but also around any solution, as shown in [START_REF] Xia | Generic Ill-posedness for wave equation of power type on three-dimensional torus[END_REF] for the wave equation. The proof does not rely on finite propagation speed, so that it can be adapted to NLS, see also [START_REF] Xia | Ill-posedness of quintic fourth order Schrödinger equation[END_REF] establishing the norm inflation at any point for the fourth-order Schrödinger equation. The general idea is that if the regularizing sequence is arbitrary, norm inflation is very likely to occur for a dense set of initial data. However, as presented in the next paragraph, the picture is more intricate when one uses the convolution by an approximate identity, which is the most natural regularization procedure. It has the property to commute with the linear flow and to be uniformly bounded on L p when p < ∞.

To push this result further, a key construction consists in gluing together blow-up profiles that concentrate at different points. Such a construction is inspired from the work of Lebeau [START_REF] Lebeau | Non linear optic and supercritical wave equation[END_REF], and was used in [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF][START_REF] Xia | Generic Ill-posedness for wave equation of power type on three-dimensional torus[END_REF] for wave equations to generate initial data from which any solution u satisfying the finite propagation speed inflates instantaneously lim sup

t→0 + ∥u(t)∥ H s = +∞ .
To give a precise meaning of the above inflating solution associated with such a pathological initial data, [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] adapted the construction in the presence of a regularizing convolution. As described in the next paragraph, this is also motivated by the probabilistic Cauchy theory at supercritical regularities.

Let us conclude this paragraph by mentioning the result of [START_REF] Lebeau | Non linear optic and supercritical wave equation[END_REF], that evidenced a loss of regularity mechanism for the energy-supercritical wave equation by using some nonlinear geometric optics. For Schrödinger equation we refer to [AC09; CR09], that indicate as well loss of regularity.

Pathological set in the context of probabilistic well-posedness After the pioneering contributions from Bourgain [START_REF] Bourgain | Periodic nonlinear Schrödinger equation and invariant measures[END_REF][START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] for the cubic Schrödinger equation on T 2 , followed by the works from Burq and Tzvetkov for the nonlinear wave equation on compact Riemannian manifolds without boundary [BT08a; BT08b], many authors have investigated the well-posedness issue at scaling-supercritical regularities by using a statistical approach. Specifically, given a Sobolev space H s (M ) where the equation is known to be ill-posed, or semi-linearly ill-posed, one can search for a non-degenerate probability measure supported on a dense subset of H s (M ), and prove that the equation is well-posed for initial data on the support of this measure. Such an approach stems from the study of Gibbs measures [Bou96; BTT18], and has been extensively developed in other contexts since then. We refer to [START_REF] Bényi | Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on R 3[END_REF] for a comprehensive survey on the so-called probabilistic Cauchy theory. Concerning the cubic Schrödinger equation in the Euclidean space R d , with d ≥ 3, Bényi, Oh, and Pocovnicu [START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF] proved generic well-posedness by using a Wiener randomization procedure, based on a unit-scale decomposition of the frequency space that leads to refined Strichartz estimates. To state the result in the context of the present work, we fix an approximate identity (ρ ε ) ε>0 defined as

ρ ε (x) := 1 ε 3 ρ x ε , (1.1.3) where ρ ∈ C ∞ c (R 3 ) is valued in [0, 1] and satisfies R 3 ρ(x) dx = 1.
Let us consider the cubic NLS equation, with p = 3, at a scaling-supercritical regularity s < s c = 1 2 .

Theorem 1.1.3 (Local well-posedness on a full-measure set for cubic NLS). Let 1 4 < s < 1 2 . There exist a probability measure µ supported on H s (R 3 ) and a dense set Σ ⊂ H s (R 3 ) with full µ-measure, such that the following holds. For all f 0 in Σ, the smooth local solutions (u ε ) to the cubic NLS with initial data (f 0 * ρ ε ) are well-defined up to a time T = T (f 0 ) > 0, and they converge to a limiting object u lim

ε→0 ∥u ε -u∥ L ∞ ([0,T ];H s (R 3 )) = 0 , u ∈ C([0 , T ]; H s (R 3 )) . (1.1.4)
Moreover, u solves NLS in the distributional sense (1.1.2) on [0 , T ], with initial data f 0 .

The local well-posedness result is due to [START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF] in dimension 4, and [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF] in dimension d ≥ 3 for cubic NLS, written in the functional framework of U p , V p spaces in order to prove scattering for small data. However, the convergence of the approximate solutions (1.1.4) is not written in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF], and we prove it in Section 1.3 by adapting the analysis performed in [START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF][START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]. Note that the regularization by convolution is essential since, as pointed out in [START_REF] Xia | Generic Ill-posedness for wave equation of power type on three-dimensional torus[END_REF] for the wave equation, norm inflation occurs over a dense set of initial data when the regularization procedure is arbitrary. The convergence of the regularized solutions is in the spirit of Theorems 2.6 and 2.7 from [START_REF] Tzvetkov | Random data wave equations[END_REF] for the wave equation. Let us mention that the large-time behavior of the probabilistic solution has been investigated by many authors, by combining the probabilistic local well-posedness result of Theorem 1.1.3 with deterministic methods. Concerning the cubic Schrödinger equation in R 3 , [BOP15a; PRT14] established almost-sure scattering for small data, and [START_REF] Camps | Asymptotic stability of small ground states for NLS under random perturbations[END_REF] proved almost-sure asymptotic stability for small ground states. Outside the small data regime, [KMV19; DLM20; Spi21] proved almost-sure scattering for the cubic NLS in R4 , and [Cam23; SSW21] extended the scattering result in R 3 , where the cubic NLS is energy-subcritical.

Outside this full-measure set where the statistical approach provides a strong notion of wellposedness, there exist initial data that concentrate in a point so that a low to high frequency cascade occurs. From the result of [START_REF] Xia | Generic Ill-posedness for wave equation of power type on three-dimensional torus[END_REF], we know that if the regularization procedure is arbitrary, such an instability phenomenon may appear in the vicinity of any initial data. In addition, Sun and Tzvetkov proved in [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] that for nonlinear wave equations, when the initial data are regularized by convolution, norm inflation occurs on a pathological set containing a dense G δ set, even when a probabilistic well-posedness result in the spirit of Theorem 1.1.3 holds.

Main result Following the approach of Sun and Tzvetkov [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] for the nonlinear wave equation, we describe the pathological set for equation NLS, which is the counterpart of the probabilistic well-posedness theory presented in Theorem 1.1.3. We show that even if we naturally regularize the initial by convolution, there still exist inflating solutions arising from a dense set of initial data in H s . Theorem 1.1.4 (Pathological set of scaling-supercritical data for NLS). Let p ≥ 3 be an odd integer 4 and 0 < s < s c = 3 2 -2 p-1 . There exists a dense set S ⊂ H s (R 3 ) such that for every f 0 ∈ S, the family of local solutions (u ε ) of NLS with initial data (f 0 * ρ ε ) does not converge because of norm inflation. More precisely, there exist ε n → 0 and t n → 0 such that u εn exists in C([0 , t n ], H s (R 3 )), and lim

n→∞ ∥u εn (t n )∥ H s (R 3 ) = ∞ . (1.1.5)
Moreover, in the defocusing energy-subcritical and critical cases p ∈ 3, 5 , where NLS is known to be globally5 well-posed in H 1 , the pathological set P contains a dense G δ set.

To exhibit the pathological set, we use in part 1.2.2 the same construction as in [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] that consists in putting side by side an infinity of inflating bubbles concentrated at arbitrarily small scales.

As discussed above, this construction is contained in [BT08a; CCT03a], and was inspired from [START_REF] Lebeau | Non linear optic and supercritical wave equation[END_REF]. The proof proposed in [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] revisits the small dispersion analysis from [START_REF] Christ | Ill-posedness for nonlinear Schrodinger and wave equations[END_REF] with an additional convolution. Indeed, since their motivation is to make precise the probabilistic well-posedness result, the smooth initial data need to be obtained by convolution with an approximate identity. Then, they use finite propagation speed to indicate that the different bubbles do not interact in short time. Moreover, in order to generate a dense set in H s (R 3 ) and not only around zero, [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] implement the idea from [START_REF] Xia | Generic Ill-posedness for wave equation of power type on three-dimensional torus[END_REF] that consists in performing the standard small dispersion analysis around an arbitrary function in C ∞ (R 3 ) with compact support.

In the case of Schrödinger type equations, we do not have such a finite propagation speed principle. To overcome this, we play with the regularizing convolutions. The observation is that given a fixed scale of regularization ε > 0, the convolution by ρ ε breaks the bubbles that concentrate at scale λ -1 ≪ ε. Hence, at a certain suitable time t ε ≪ 1, the bubble concentrated at scale λ -1 ∼ ε is the only one to inflate. This observation indicates that there is no need to use the finite propagation speed in the proof of Theorem 1.1.4, even in the case of the nonlinear wave equation analyzed in [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF]. Therefore, the same approach probably extends to any equation of Schrödinger type, provided there exists a reasonable Cauchy theory for smooth initial data. Indeed, we do not directly use the particular dispersion of NLS in the analysis. Let us mention some features of the approach developed in this work.

Finite propagation speed and localization of bubbles Since we do not use any finite propagation speed, we have access to Schrödinger type equations. Another consequence is that we can superpose the bubbles in the construction. Indeed, we do not exploit the absence of interaction between the different bubbles, that was crucial in the short time analysis from [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] about the wave equation. Hence, we can take the center of the bubbles to be x = 0, so that the pathological set is left invariant under the action of the rotations and the construction persists in the radial case. Note, however, that we are not able to prove that the solution inflates along the whole sequence as ε goes to zero, but only along the subsequence ε n (see remark 1.1 in [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF], and remark 1.2.5 below).

Focusing and defocusing equations and range of nonlinearities We show that the a priori control on the growth of Sobolev norms is sufficient to establish a lower bound on the time of existence for the smooth solutions obtained by convolution. Hence, we can bypass the use of the coercive energy, and we treat the focusing and defocusing case without distinction. Also, we have access to the energy-supercritical case p > 5. In such a case, we point out that the norm inflation of the H 1 (R 3 )-norm does not contradict the conservation of the energy, since when p > 5 the potential energy of an initial data in H 1 (R 3 ) is infinite. However, to prove that P contains a dense G δ set we need to assume that the smooth solutions are global, say in H Lower bound on s Sun and Tzvetkov observed in [ST20a; ST20c] that a lower bound on s might be required. However, in the statement of Theorem 1.1.4, we do not need such a lower bound, and we propose an explanation of this difference in remark 1.2.9. Yet, we do not know if the limitation in the case of the wave equation is whether technical or not. Note also that we do not have s = 0 because of mass conservation. When s < 0 other mechanisms operate, and the possible norm inflation is due to a high to low frequency cascade.

Other geometries and dimensions Since the construction of the pathological set is local, up to working in local coordinates, our proof works as well in the periodic case, or in the case of a compact Riemannian manifold. Nevertheless, we decided to state the result in the Euclidean case, since there is no general probabilistic Cauchy theory in the spirit of [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF] for NLS on a compact manifold of dimension 3. Our proof can also be adapted to higher dimension, but we write it in the context of [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] for simplicity, in the sense that we only need to compute two derivatives to control the L ∞ -norm of a function in R 3 . In higher dimensions, we would have to use a semiclassical energy with higher order derivatives.

Conclusion

In the scaling-supercritical regime, and at least in the cubic case, we have two different behaviors for the smooth solutions to NLS initiated from regularized initial data obtained by convolution with an approximate identity. On the one hand, Theorem 1.1.4 states that the pathological set of initial data where norm inflation occurs contains a dense G δ set. On the other hand, the statistical approach from Theorem 1.1.3 provides of dense full-measure set of initial data for which the regularized solutions converge to a limiting object that is a strong solution to NLS. This is the counterpart for Schrödinger-type equations to the result from Sun and Tzvetkov [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] for wave equations, and the same comments apply. Namely, there are two different notions of genericity. While the norm inflation set from Theorem 1.1.4 is generic for the topology, it is negligible for the measure. Meanwhile, we know from the Baire category Theorem that the well-posedness set from Theorem 1.1.3, which has full measure, cannot be a dense G δ set.

Outline of the paper We first establish the generic ill-posedness Theorem 1.1.4 in section 1.2. To achieve such a goal, we set up in part 1.2.1 the fundamental bubbles v n that encode the oscillations, and we construct the pathological set in part 1.2.2 by superposing these bubbles. In paragraph 1.2.3 we fix a given scale ε n , its corresponding inflation time t n , and we compare in H s (R 3 ) the solution to NLS initiated from an initial data in the pathological set, with the bubble concentrated at scale ε n , supposed to inflate after the time t n . In a second section 1.3, we prove the generic well-posedness result. After introducing preliminaries for the probabilistic method in part 4.2 and reviewing relevant probabilistic Cauchy theory results in part 1.3.2, we prove in part 1.3.3 the convergence in H s (R 3 ) of the regularized solution by convolution to the solution generated from a randomized initial data. blow-up profile concentrated at scale n -1 , and supposed to inflate at time t n , actually gets smoother and remains bounded at time t n , as soon as n -1 ≪ ε. We draw the attention to Lemmas 1.2.1 and 1.2.4 that indicate such a mechanism. Owing to this observation, we show in Section 1.2.3 how to generalize the small-dispersion analysis from [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] without using finite propagation speed.

Unstable profile

We construct unstable profiles by following the approach in [START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF]. The construction of the bubbles, that have growing H s -norm, is based on the solution V (t) = e iσt to the dispersionless ODE

iV ′ + σ|V | p-1 V = 0 V (0) = 1 .
Note that V is a bounded periodic function. Set a smooth and radial profile φ ∈ C ∞ c ({∥x∥ ≤ 1}), such that 0 ≤ φ ≤ 1, from which we define a bubble concentrated at scale n

v n (0, x) := λ 2 p-1 n φ(nx) = κ n n 3 2 -s φ(nx) .
Given γ > 0 to be determined later in the analysis, the parameters of the above bubble are

κ n := (log(n)) -γ , λ n := κ p-1 2 n n ( 3 2 -s)• p-1 2 .
We recall that we have fixed an approximate identity defined in (1.1.3), and consider the bubble after convolution by the approximate identity

v ε n (0) := ρ ε * v n (0).
Given a time t, we consider the profile

v ε n (t, x) := v ε n (0, x)V (t|v ε n (0, x)| p-1 ) , solution to the dispersionless ODE i∂ t v ε n + σ|v ε n | p-1 v ε n = 0 . v ε n (0) = ρ ε * v n (0) ,
We now fix the parameters,

ε n := 1 100n , t n := log(n) β(p-1) n (s-3 2 )(p-1) = λ -2 n log(n) (β-γ)(p-1) , (1.2.1)
for suitable 0 < γ < β < 1 adjusted in the analysis, and we follow [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] to adapt the bounds from [START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF] in the presence of the convolution parameter.

Lemma 1.2.1 (Growth of the profile). Let 0 ≤ s < 3 2 .6

1. There exists c > 0 independent of n such that for all ε ≤ ε n , we have the lower bound

∥v ε n (t n )∥ H s ≥ cκ n (λ 2 n t n ) s .
2. For all m ∈ N, there exists C > 0 such that for all n ∈ N, t ∈ R and ε > 0,

∥|∇| m v ε n (t)∥ L 2 ≤ Cκ n n m-s (1 + (tλ 2 n ) m ) min 1, 1 εn m+ 3 2 .
In addition,

∥|∇| m v ε n (t)∥ L ∞ ≤ Cκ n n m+ 3 2 -s (1 + (tλ 2 n ) m ) min 1, 1 εn m+ 3 2 .
The factor min 1,

1 εn m+ 3
2 obtained on top of Lemma 2.1 in [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] comes from the fact that we get better estimates when estimating the derivatives and the L ∞

x norm of ρ ε instead of those of v n when ε is large compared to 1 n .

Proof. We first establish the upper bounds.

Upper bounds Let T n (f ) := f (n•) be the scaling operator. We have

v ε n (0, x) = κ n n 3 2 -s φ(n(x -x ′ )) 1 ε 3 ρ x ′ ε dx ′ = κ n n 3 2 -s ε 3 (T n φ * T 1 ε ρ)(0, x), so that when α = (α 1 , α 2 , α 3 ) ∈ N 3 , denoting m = |α| = α 1 + α 2 + α 3 , ∂ α v ε n (0, x) = κ n n 3 2 -s n m T n (∂ α φ)(x -x ′ ) 1 ε 3 ρ x ′ ε dx ′ = κ n n 3 2 -s ε 3 n m (T n (∂ α φ) * T 1 ε ρ)(0, x).
The Young's convolution inequality implies

∥∂ α v ε n (0)∥ L ∞ ≲ κ n n 3 2 -s n m ∥T n (∂ α φ)∥ L ∞ ∥ρ ε ∥ L 1 ≲ λ 2 p-1 n n m , ∥∂ α v ε n (0)∥ L 2 ≲ κ n n 3 2 -s n m ∥T n (∂ α φ)∥ L 2 x ∥ρ ε ∥ L 1 ≲ κ n n m-s . But one can also derivate the ρ part ∂ α v ε n (0, x) = κ n n 3 2 -s 1 ε m T n (φ)(x ′ ) 1 ε 3 (∂ α ρ) x -x ′ ε dx ′ = κ n n 3 2 -s ε 3 1 ε m (T n (φ) * T 1 ε (∂ α ρ)(0, x),
so that there also holds

∥∂ α v ε n (0)∥ L ∞ ≲ κ n n 3 2 -s 1 ε m ∥T n (φ)∥ L 1 1 ε 3 ∥T 1 ε (∂ α ρ)∥ L ∞ ≲ λ 2 p-1 n 1 ε m 1 (εn) 3 , ∥∂ α v ε n (0)∥ L 2 ≲ κ n n 3 2 -s 1 ε m ∥T n (φ)∥ L 1 1 ε 3 ∥T 1 ε (∂ α ρ)∥ L 2 ≲ κ n n 3 2 -s 1 ε m-3 2 1 (εn) 3 = κ n n m-s 1 (εn) m+ 3 2 .
Note that when m ≤ 1, we can still improve this estimate using that

∥∂ α v ε n (0)∥ L 2 ≲ κ n n 3 2 -s 1 ε m ∥T n (φ)∥ L 2 1 ε 3 ∥T 1 ε (∂ α ρ)∥ L 1 ≲ κ n n m-s 1 (εn) 3 .
These two observations sum up as

∥∂ α v ε n (0)∥ L ∞ ≲ λ 2 p-1 n n m min 1, 1 εn m+3 , ∥∂ α v ε n (0)∥ L 2 ≲ κ n n m-s min 1, 1 εn m+ 3 2 .
Next, we deduce the estimates at time t. Using V being bounded by 1, we have

∥v ε n (t)∥ L ∞ ≲ λ 2 p-1 n min 1, 1 εn 3 , ∥v ε n (t)∥ L 2 ≲ κ n n -s min 1, 1 εn 3 2
.

Finally, we compute

|∇v ε n (t)| ≲ t||v ε n (0)| p-1 ∇v ε n (0)V ′ (t|v ε n (0)| p-1 )| + |∇v ε n (0)V (t|v ε n (0)| p-1 )|, (1.2.2)
and

|∆v ε n (t)| ≲t 2 ||v ε n (0)| 2p-3 (∇v ε n (0)) 2 V ′′ (t|v ε n (0)| p-1 )| + 2t||v ε n (0)| p-2 |∇v ε n (0)| 2 V ′ (t|v ε n (0)| p-1 )| + t||v ε n (0)| p-1 ∆v ε n (0)V ′ (t|v ε n (0)| p-1 )| + |∆v ε n (0)V (t|v ε n (0)| p-1 )| .
Iterating, we obtain a similar formula when α ∈ N m is arbitrary, the leading term being the one where all the derivatives fall on the ODE profile V

|∂ α v ε n (t)| ≲ ||v ε n (0)| (p-1)m ∇v ε n (0)| m |v ε n (0)|t m |V (m) t|v ε n (0)| p-1 | .
Since |V | and its derivatives are bounded by 1, we deduce that

∥∇v ε n (t)∥ L ∞ ≲ λ 2 p-1 n n(1 + tλ 2 n ) min 1, 1 εn 1+3 , ∥∆v ε n (t)∥ L ∞ ≲ λ 2 p-1 n n 2 (1 + tλ 2 n + (tλ 2 n ) 2 ) min 1, 1 εn 2+3 .
On the L 2 -based spaces, we also get

∥∇v εn n (t)∥ L 2 ≲ κ n n 1-s (1 + tλ 2 n ) min 1, 1 εn 1+ 3 2 , ∥∆v ε n (t)∥ L 2 ≲ κ n n 2-s (1 + (tλ 2 n ) 2 ) min 1, 1 εn 2+ 3 2 .
The other estimates follow similarly.

Lower bound We now establish the lower bound

∥v ε n (t n )∥ H s ≥ cκ n (t n λ 2 n ) s for every ε ≤ ε n , where we recall that t n λ 2 n ≥ 1 when n ≫ 1. Fix ε ≤ ε n . When 0 ≤ s < 1, we have from interpolation that ∥v ε n (t)∥ 2-s H 1 ≲ ∥v ε n (t)∥ H s ∥v ε n (t)∥ 1-s H 2 .
Using the upper bound

∥v ε n (t)∥ H 2 ≲ κ n n 2-s (1 + (tλ 2 n ) 2
), it is enough to establish the lower bound

∥v ε n (t n )∥ H 1 ≳ κ n (t n λ 2 n )n 1-s .
Since the second term on the right-hand side in (1.2.2) has the upper bound

∥∇v ε n (0)V (t n |v ε n (0)| p-1 )∥ L 2 ≲ κ n n 1-s ,
it is sufficient to establish the lower bound for the leading term

∥t n |v ε n (0)| p-1 ∇v ε n (0)V ′ (t n |v ε n (0)| p-1 )∥ L 2 .
Since V ′ is of modulus one,

∥t n |v ε n (0)| p-1 ∇v ε n (0)V ′ (t n |v ε n (0)| p-1 )∥ L 2 = t n ∥|v ε n (0)| p-1 ∇v ε n (0)∥ L 2 = t n λ 2+ 2 p-1 n n∥|T n (φ) * ρ ε | p-1 (T n (∇φ) * ρ ε ) ∥ L 2 = t n λ 2+ 2 p-1 n n 1-3 2 ∥|φ * ρ nε | p-1 (∇φ * ρ nε ) ∥ L 2 ,
the latter equality coming from the change of variable

T n f * ρ ε (x) = T n (f * ρ nε ).
Thanks to the choice of ε such that nε ≤ nε n = 1 100 , the L 2 norm of φ * ρ nε admits a lower bound that does not depend on n. Moreover, it is bounded from below by c 0 > 0 since φ * ρ nε and its derivative tends to φ and its derivative in the L p spaces when nε goes to 0. To conclude, it only remains to observe that t n λ

2+ 2 p-1 n n 1-3 2 = κ n (t n λ 2 n )n 1-s . When 1 ≤ s < 3
2 , the idea is the same except that we need to interpolate between higher order Sobolev spaces

∥v ε n (t)∥ 3-s H 2 ≲ ∥v ε n (t)∥ H s ∥v ε n (t)∥ 2-s H 3 . The leading term in the expression for ∥v ε n (t)∥ H 2 is ∥t 2 n |v ε n (0)| 2p-3 |∇v ε n (0)| 2 V ′′ (t n |v ε n (0)| p-1 )∥ L 2 = t 2 n ∥|v ε n (0)| 2p-3 (∇v ε n (0)) 2 ∥ L 2 = t 2 n λ 2(2p-1) p-1 n n 2 ∥|T n (φ) * ρ ε | 2p-3 (T n (∇φ) * ρ ε ) 2 ∥ L 2 = t 2 n λ 4+ 2 p-1 n n 2-3 2 ∥|φ * ρ nε | 2p-3 (∇φ * ρ nε ) 2 ∥ L 2 ,
where t 2 n λ

4+ 2 p-1 n n 2-3 2 = κ n (t n λ 2 n ) 2 n 2-s .
To conclude, it only remains to use the upper bound

∥v ε n (t n )∥ H 3 ≲ κ n n 3-s (t n λ 2 n ) 3 .

Pathological set

The pathological set is defined as

P := {f ∈ H s (R 3 ) | lim sup ε,t→0 ∥Φ(t)(ρ ε * f )∥ H s → ∞} . (1.2.3)
In this part, we prove that the pathological contains a dense G δ set, that stems from the "tanghuru" construction [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] consisting in the superposition of inflating bubbles. We fix a ≫ 1. For arbitrary k ∈ N, x k ∈ R 3 , we set n k := e a k , and define the k-th bubble centered at x k ∈ R 3 , and concentrated at scale

n -1 k v 0,k , writes v 0,k (x) := v n k (0, x -x k ) = log(n k ) -γ n 3 2 -s k φ(n k (x -x k )) .
Remark 1.2.2 (Position of the bubbles). In the construction from [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF], the k-th bubble concentrates in the point x k = 1 k , so that the distance between two consecutive bubbles is much larger than the scale of the bubble. Namely,

|x k -x k-1 | ∼ 1 k 2 ≫ e -a k .
Hence, the different bubbles are far enough and do not interact in short time, thanks to the finite propagation speed. Since we do not make use of this, we take arbitrary x k in the construction. Note that when x k = 0 for all k, the construction is radial and provides a pathological set even in the radial case.

Definition 1.2.3 (Dense subset of the pathological set). We denote by S the set of initial data f 0 that can be decomposed under the form

f 0 = u 0 + ∞ k=k 0 v 0,k , k 0 ≥ 1 , u 0 ∈ C ∞ c (R 3 ) .
Our aim is to show that the pathological set contains S, which is dense in H s (R 3 ), and a G δ subset as soon as every smooth solution to NLS is global. For this, we first establish upper bounds for the series in the definition of S, after regularization by convolution. Recall that we have set up the parameters in (1.2.1), and, given k ∈ N, we may denote for simplicity

n = n k = e a k , ε k = ε n k , t k = t n k ,
with a ≫ 1 to be chosen later in the analysis. At time t n k and at scale ε n k we expect the k-th bubble to be responsible for the norm inflation, while the other bubbles concentrated at the larger scales l -1 > k -1 and at smaller scales l -1 < k -1 do not inflate.

Lemma 1.2.4 (H m -bounds for initial data in the pathological set). Let m ∈ N, 1 ≤ k 0 ≤ k, and denote ε k = (100n k ) -1 , with n k = e a k . If m < s, there holds k-1 l=k 0 ∥ρ ε k * v 0,l ∥ H m (R 3 ) ≲ 1 , (1.2.4) ∞ l=k+1 ∥ρ ε k * v 0,l ∥ H m (R 3 ) ≲ n m-s k+1 n k n k+1 3 2
.

(1.2.5)

Moreover, for every m > s, we have

k-1 l=k 0 ∥ρ ε k * v 0,l ∥ H m (R 3 ) ≲ n m-s k-1 , (1.2.6) ∞ l=k+1 ∥ρ ε k * v 0,l ∥ H m (R 3 ) ≲ n m k n -s k+1 n k n k+1 3 2
.

(1.2.7)

Remark 1.2.5. Note that the estimate for the sum of the profiles concentrated at scale smaller than n k is only valid when ε ≥ ε k . As a consequence, we cannot evidence the norm inflation along the whole sequence of regularized initial data (ρ ε * f 0 ) ε>0 , but only along the subsequence (ρ ε k * f 0 ) k≥k 0 , with ε k = (100n k ) -1 . For this reason, we only prove (1.1.5) instead of lim

T,ε→0 ∥u ε ∥ L ∞ t ([0,T ];H s (R 3 )) = ∞ . (1.2.8)
The norm inflation mechanism was also exhibited along a subsequence in [BT08a; BGT05b; Xia21]. However, thanks to the finite propagation speed, [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] proved that the norm inflation for energy-subcritical nonlinear wave equations occurs along the whole sequence of regularized data. Nevertheless, defining the pathological set with the convergence type (1.1.5) is enough for our purpose. Indeed, in the context of probabilistic well-posedness, the convergence of the regularized strong solutions to the local solution from a randomized initial data holds for any sequence ε → 0. Therefore, the pathological set and the good set of randomized initial data are disjoint.

Proof. When m < s, the large bubbles (i.e. with small index l compared to k) contribute the most to the mass of the initial data.

k-1 l=k 0 ∥ρ ε k * v 0,l ∥ H m ≲ k-1 l=k 0 κ n l n m-s l ≲ n m-s k 0 , whereas ∞ l=k+1 ∥ρ ε k * v 0,l ∥ H m ≲ ∞ l=k+1 κ n l n m-s l n k n l 3 2 ≲ n m-s k+1 n k n k+1 3 2
.

This proves (1.2.4) and (1.2.5), respectively. Next, we assume that m > s and we establish (1.2.6) and (1.2.7). It follows from Lemma 1.2.1 that

k-1 l=k 0 ∥ρ ε k * v 0,l ∥ H m ≲ k-1 l=k 0 κ n l n m-s k l min 1, 1 ε k n l m+ 3 2 ≲ n m-s k , Similarly, ∞ l=k+1 κ n l n m-s l min 1, 1 ε k n l m+ 3 2 ≲ 1 ε k m ∞ l=k+1 κ n l n -s l (ε k n l ) -3 2 ≲ n m k n -s k+1 n k n k+1 3 2 .

Perturbative analysis

In this part, we fix f 0 ∈ S and consider the solution u ε k to NLS with initial data f 0 * ρ ε k . We prove that in the scaling-supercritical regime s < s c , and up to the time t n k , the nonlinear part of u ε k dominates the dispersion. Specifically, we prove that u ε k behaves like the ODE profile v ε k k , which is at scale n k . Therefore, u ε k experiences a norm inflation at time t n k as well. To do so, we perform a perturbative analysis by proving some a priori energy estimates on the difference between the ODE profile and u ε k . As a byproduct of the analysis, we can deduce from the a priori energy estimates combined with a standard continuity argument that the regularized solution

u ε k exists in C([0 , t n k ]; H 2 (R 3 )). Proposition 1.2.6 (Perturbative analysis). Let k ≥ 1, n k = e a k , ε k := (100n k ) -1 . When s < s c , the local solution u ε k to NLS with initial data f 0 * ρ ε k can be extended in H 2 (R 3 ) up to time t n k .
Moreover, there exists C > 0 such that for k large enough,

∥u ε k (t n k ) -v ε k n (t n k )∥ H s (R 3 ) ≤ C .
(1.2.9)

Corollary 1.2.7 (Norm inflation). For all 0 < s < s c there exist 0 < c ≪ 1 and a suitable choice of the parameters β, γ, such that for any k large enough we have

∥u ε k (t n k )∥ H s ≥ c log(n k ) c .
Corollary 1.2.7, from which we deduce Theorem 1.1.4, follows from a priori estimate (1.2.9) in the small-dispersion analysis, combined with the lower bound on

v ε k n from Lemma 1.2.1 ∥u ε k (t n k )∥ H s ≥ c log(n) s(β-γ)(p-1)-γ -C .
To prove Proposition 1.2.6, we perform a standard perturbative analysis inspired from [START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF] and [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF], using semi-classical energy estimates. The main difference is the presence of a linear correction,

u ε k L := e it∆ u ε k 0 + k-1 l=k 0 e it∆ v ε k 0,l .
This correction ensures the smallness of the L 2 -norm of w(0). The correction is quite harmless up to time t n k , since u ε k L concentrates at scale at most (n k-1 ) -1 , which is much larger than the scale (n k ) -1 of the inflating bubble provided a is chosen large enough in the construction.

Remark 1.2.8. We remove the linear evolution of the large bubbles in u ε k L , in order to have an initial upper bound for the mass of order at most n -s k initially. Indeed, as a consequence of Lemma 1.2.4 we may have

∥ k-1 l=k 0 v ε k 0,l ∥ L 2 ∼ 1 .
Removing this contribution would deteriorate a bit the estimates on the Sobolev norms H m of u ε k L , that will be of size n m k-1 instead of C. Nevertheless, since we run the argument up to time

t n k ≪ n m
k-1 we shall be able to close the estimates as explained above. Remark 1.2.9. In the case of the wave equation [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF], the norm inflation occurs much later, at time

t n k (N LW ) ∼ λ -1 n k , since it occurs at time t n k (N LS) ∼ λ -2
n k for NLS, respectively. Consequently, a lower bound on s is needed in [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] to control the growth of some source terms (containing

u ε k L ) up to time t n k (N LW ) ≫ t n k (N LS).
Nevertheless, it should be possible to remove this condition by using a refined nonlinear correction instead of u ε k L in the perturbative analysis.

Proof of Proposition 1.2.6. By using the local Cauchy theory for NLS in H σ for σ > 3 2 , we see that a lower bound on the time of existence of u ε k in say H 2 follows from an a priori bound on its H 2 -norm. We deduce such an a priori bound from energy estimates. To do so, we write

w := u ε k -u ε k L -v ε k n .
Using the definition of the linear correction u ε k L , we see that initially,

w(0) = ∞ l=k+1 v 0,l * ρ ε k . (1.2.10)
We shorten the notation by denoting n := n k , and we consider the following semi-classical energy

E n (t) = n 2s ∥w∥ 2 L 2 + n 2(s-2) ∥w∥ 2 H 2 1 2
.

From the Gagliardo-Nirenberg inequality and the interpolation, we have that for σ ∈ [0 , 2],

∥w(t)∥ L ∞ ≲ n 3 2 -s E n (t) , ∥w(t)∥ H σ ≲ n σ-s E n (t) . (1.2.11)
In particular, ∥w(t)∥ H s ≲ E n (t) . Moreover, the H s -norm of the linear correction is bounded uniformly in n since

∥u ε k L ∥ L ∞ t H s x ≤ ∥u ε k 0 ∥ H s + k-1 l=k 0 ∥v ε k 0,l ∥ 1-s L 2 ∥v ε k 0,l ∥ s H 1 ≲ ∥u 0 ∥ H s + k-1 k=k 0 κ n l ≲ 1 . (1.2.12)
For these reasons, inequality (1.2.9) can be reduced to prove the a priori energy bound

sup 0≤t≤tn E n (t) ≤ 1 .
Initially, we have from (1.2.5) and (1.2.7) that

∥w(0)∥ L 2 ≤ Cn -s k+1 n n k+1 3 2 , ∥w(0)∥ H 2 ≤ Cn 2 n -s k+1 n n k+1 3 2 , so that E n (0) 2 = n 2s ∥w(0)∥ 2 L 2 + n -2(s-2) ∥∆w(0)∥ 2 L 2 ≤ 2C n n k+1 3+2s = Cn -c 0 , (1.2.13)
with c 0 := (a -1)(3 + 2s). Now, we run a bootstrap argument to prove that E n (t) < 1 for all t ≤ t n .

For shortness of notation, we drop the complex conjugation sign. By construction, w is solution to the perturbed Schrödinger equation

(i∂ t + ∆)w = |w + u ε k L + v ε k n | p-1 (w + u ε k L + v ε k n ) -|v ε k n | p-1 v ε k n + ∆v ε k n =: F + ∆v ε k n ,
with initial condition w(0) given by (1.2.10), and where the forcing term is

F = w + u ε k L O |w| p-1 + |u ε k L | p-1 + |v ε k n | p-1 .
Denoting Λ := ∆F , we have

|Λ| ≲(|∆w| + |∆u ε k L |) |w| p-1 + |u ε k L | p-1 + |v ε k n | p-1 + (|∇w| + |∇u ε k L |) 2 |w| p-2 + |u ε k L | p-2 + |v ε k n | p-2 + (|∇w| + |∇u ε k L |)|∇v ε k n | |w| p-2 + |u ε k L | p-2 + |v ε k n | p-2 + (|w| + |u ε k L |)|∇ 2 v ε k n | |w| p-2 + |u ε k L | p-2 + |v ε k n | p-2 + (|w| + |u ε k L |)|∇v ε k n | 2 |w| p-3 + |u ε k L | p-3 + |v ε k n | p-3 .
First, we control the source term coming from derivatives of ∆v ε k n . We observe that 2 < ( 3 2 -s)(p -1) when s < s c . 7 Hence, we deduce from Lemma 1.2.1 that

n s ∥∆v ε k n ∥ L ∞ tn L 2 x + n s-2 ∥∆ 2 v ε k n ∥ L ∞ tn L 2 x ≤ C(tλ 2 n ) 4 n 2 ≤ C log(n) 4(β-γ)(p-1) n 2 ≤ Cn ( 3 2 -s)(p-1)-δ 0 , (1.2.14)
with, say,

δ 0 := 1 2 ( 3 2 -s)(p -1) -2 > 0 .
Now, we control the source terms comings from u ε k L and v ε k n in the expression of F and ∆F . First, notice from (1.2.6) and from the H s -bound (1.2.12) that for any m ≥ s, there exists

C m > 0 such that ∥u ε k L ∥ L ∞ t H m x ≤ C m n m-s k-1 , (1.2.15)
whereas when m < s we can only use that thanks to (1.2.12), we have ∥u

ε k L ∥ L ∞ t H m x ≤ C.
Using the Gagliardo-Nirenberg inequality with s < 3 2 , we deduce that

∥u ε k L ∥ L ∞ t L ∞ x ≲ (∥u ε k L ∥ L ∞ t H s x ) 1 2(2-s) (∥u ε k L ∥ L ∞ t H 2 x ) 3-2s 2(2-s) ≲ n 3 2 -s k-1 .
7 This seems to be the only place where we see the condition s < s c .

In order to prevent another unnecessary loss later on, we also observe from the Sobolev embedding

Ḣ 3 4 → L 4 that ∥|∇|u ε k L ∥ L ∞ t L 4 x ≲ ∥u ε k L ∥ L ∞ t H 1+ 3 4 x
, and since 1 + 3 4 > 3 2 > s c , we deduce

∥|∇|u ε k L ∥ L ∞ t L 4 x ≲ n 1+ 3 4 -s k-1 .
Using the estimates on the profile v from Lemma 1.2.1, we can therefore control the source terms u ε k L and v up to time t n ∼ λ -2 n , uniformly in n and estimate

n s ∥F (t)∥ L 2 ≲ n s ∥w∥ L 2 ∥w∥ p-1 L ∞ + ∥u ε k L ∥ p-1 L ∞ + ∥v ε k n ∥ p-1 L ∞ + n s ∥u ε k L ∥ L ∞ ∥w∥ L 2 ∥w∥ p-2 L ∞ + ∥u ε k L ∥ L 2 ∥u ε k L ∥ p-2 L ∞ + ∥v ε k n ∥ L 2 ∥v ε k n ∥ p-2 L ∞ .
As a consequence,

n s ∥F (t)∥ L 2 ≲ E n (t) E n (t) p-1 n ( 3 2 -s)(p-1) + n ( 3 2 -s)(p-1) k-1 + n ( 3 2 -s)(p-1) + n s n 3 2 -s k-1 n -s E n (t)n ( 3 2 -s)(p-2) + n ( 3 2 -s)(p-2) k-1 + n -s n ( 3 2 -s)(p-2) .
Supposing that E n (t) ≪ 1, which is the case at time t = 0, this leads to the estimate

n s ∥F (t)∥ L 2 ≲ E n (t)n ( 3 2 -s)(p-1) + n ( 3 2 -s)(p-1) n k-1 n 3 2 -s + n s n k-1 n ( 3 2 -s)(p-1) .
We recall that we choose n = n k = e a k for some a ≫ 1, so that

n k-1 n = n -θ with θ = 1 -1 a .
Therefore, there holds

n s ∥F (t)∥ L 2 ≲ E n (t)n ( 3 2 -s)(p-1) + n ( 3 2 -s)(p-1) n -θ( 3 2 -s) + n s-θ( 3 2 -s)(p-1) .
We want to ensure that

s < θ 3 2 -s (p -1) , (1.2.16)
so that there exist C > 0 and c > 0 such that

n s ∥F (t)∥ L 2 ≤ CE n (t)n ( 3 2 -s)(p-1) + n ( 3 2 -s)(p-1)-c . (1.2.17) Since s < s c = 3 2 -2 p-1 , the condition (1.2.16) reduces to s c < θ 3 2 -s c (p -1) ,
which is satisfied as soon as θ is sufficiently close to 1, that is when a is chosen sufficiently large so that 3 4 < 1 -1 a .

Similarly, we estimate the increment of the H 2 -norm as in [START_REF] Burq | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations[END_REF], controlling the terms involving

v ε k n , ∇v ε k n , ∆v ε k n or u ε k L , ∇u ε k L , ∆u ε k L in L ∞ : ∥Λ(t)∥ L 2 ≲ ∥∆w∥ L 2 ∥u ε k L ∥ p-1 L ∞ + ∥w∥ p-1 L ∞ + ∥v ε k n ∥ p-1 L ∞ + ∥∆u ε k L ∥ L ∞ ∥w∥ L 2 ∥w∥ p-2 L ∞ + ∥v ε k n ∥ L 2 ∥v ε k n ∥ p-2 L ∞ + ∥∆u ε k L ∥ L 2 ∥u ε k L ∥ p-1 L ∞ + ∥|∇|w∥ 2 L 4 ∥u ε k L ∥ p-2 L ∞ + ∥w∥ p-2 L ∞ + ∥v ε k n ∥ p-2 L ∞ + ∥|∇|u ε k L ∥ 2 L ∞ ∥w∥ L 2 ∥w∥ p-3 L ∞ + ∥v ε k n ∥ L 2 ∥v ε k n ∥ p-3 L ∞ + ∥|∇|u ε k L ∥ 2 L 4 ∥u ε k L ∥ p-2 L ∞ + ∥|∇|w∥ L 2 ∥|∇|v ε k n ∥ L ∞ + ∥|∇|u ε k L ∥ L ∞ ∥|∇|v ε k n ∥ L 2 ∥u ε k L ∥ p-2 L ∞ + ∥w∥ p-2 L ∞ + ∥v ε k n ∥ p-2 L ∞ + ∥w∥ L 2 ∥∆v ε k n ∥ L ∞ + ∥u ε k L ∥ L ∞ ∥∆v ε k n ∥ L 2 ∥u ε k L ∥ p-2 L ∞ + ∥w∥ p-2 L ∞ + ∥v ε k n ∥ p-2 L ∞ + ∥w∥ L 2 ∥|∇|v ε k n ∥ 2 L ∞ + ∥u ε k L ∥ L ∞ ∥|∇|v ε k n ∥ L 2 ∥|∇|v ε k n ∥ L ∞ ∥u ε k L ∥ p-3 L ∞ + ∥w∥ p-3 L ∞ + ∥v ε k n ∥ p-3 L ∞ .
In addition to inequality (1.2.11), we shall use the Sobolev embedding Ḣ 3 4 → L 4 and interpolation to get

∥|∇|w(t)∥ L 4 ≤ n 1+ 3 4 -s E n (t) .
Using the bounds on v ε k n and on u ε k L , we obtain

n s-2 ∥Λ(t)∥ L 2 ≲ E n (t) n ( 3 2 -s)(p-1) k-1 + n ( 3 2 -s)(p-1) (E n (t) p-1 + κ p-1 n ) + n 2+ 3 2 -s k-1 n -2+( 3 2 -s)(p-2) E n (t) p-1 + κ p-1 n + n s-2 n 2-s+( 3 2 -s)(p-1) k-1 + n 3 2 -s E n (t) 2 n ( 3 2 -s)(p-2) k-1 + n ( 3 2 -s)(p-2) (E n (t) p-2 + κ p-2 n ) + n 2+3-2s k-1 n -2+( 3 2 -s)(p-3) E n (t) p-3 + κ p-3 n + n s-2 n 2+ 3 2 -2s+( 3 2 -s)(p-2) k-1 + κ n n 3 2 -s E n (t) + n 1+ 3 2 -s k-1 n -1 n ( 3 2 -s)(p-2) k-1 + n ( 3 2 -s)(p-2) (E n (t) p-2 + κ p-2 n ) + κ n n 3 2 -s E n (t) + n 3 2 -s k-1 n ( 3 2 -s)(p-2) k-1 + n ( 3 2 -s)(p-2) (E n (t) p-2 + κ p-2 n ) + κ 2 n n 2( 3 2 -s) E n (t) + n 3 2 -s k-1 n 3 2 -s n ( 3 2 -s)(p-3) k-1 + n ( 3 2 -s)(p-3) (E n (t) p-3 + κ p-3 n ) .
Once again, we assume that E n (t) ≪ 1. Noting that κ n ≤ 1 and that n k-1 ≤ n, we use the bounds

n ( 3 2 -s)(p-i) k-1 + n ( 3 2 -s)(p-i) (E n (t) p-i + κ p-i n ) ≲ max(1, n ( 3 2 -s)(p-i) ) , i ∈ 1, 2, 3, . . . .
We then simplify the above expression by writing

n s-2 ∥Λ(t)∥ L 2 ≲E n (t)n ( 3 2 -s)(p-1) + n 2+ 3 2 -s k-1 n -2+( 3 2 -s)(p-2) + n s-2 n 2-s+( 3 2 -s)(p-1) k-1 + n 2+3-2s k-1 n -2+( 3 2 -s)(p-3) + n s-2 n 2-s+( 3 2 -s)(p-1) k-1 + n 1+ 3 2 -s k-1 n -1+( 3 2 -s)(p-2) + n 3 2 -s k-1 n ( 3 2 -s)(p-2) .
The cross-terms between n k-1 and n are equal to

n ( 3 2 -s)(p-1) n k-1 n 2+ 3 2 -s + n k-1 n 2-s + n k-1 n 2+3-2s + n k-1 n 1+ 3 2 -s + n k-1 n 3 2 -s . Since n k-1 n = n -θ , there exist C, c > 0 such that n s-2 ∥Λ(t)∥ L 2 ≤ CE n (t)n ( 3 2 -s)(p-1) + n ( 3 2 -s)(p-1)-cθ . (1.2.18)
Combining (1.2.14), (1.2.17) and (1.2.18) with the energy estimate

d dt E n (t) 2 ≤ 2E n (t) n s ∥F (t)∥ L 2 + n s-2 ∥Λ(t)∥ L 2 + n s ∥∆v ε k n ∥ L ∞ tn L 2 x + n s-2 ∥∆ 2 v∥ L ∞ tn L 2 x ,
we get that there exist C, c > 0 such that

d dt E n (t) 2 ≤ CE n (t) 2 n ( 3 2 -s)(p-1) + Cn ( 3 2 -s)(p-1)-c = Cλ 2 n E 2 n + Cλ 2 n n -2c .
It follows from the Gronwall estimate that

E n (t) 2 ≤ E n (0) 2 + Cn -2c t n λ 2 n exp Ct n λ 2 n ≤ E n (0) 2 + Cn -c exp C log(n) (β-γ)(p-1) .
We recall that from (1.2.13), we have E n (0) 2 ≤ Cn -c 0 for some c 0 > 0. Finally, we fix parameters β, γ small enough such that (p -1)β < 1 2 and γ < 1 2 β. Then, we fix n 0 ∈ N such that for all n ≥ n 0 , for some constant C > 0 encountered in the analysis, we have

C log(n) (β-γ)(p-1) ≤ δ 2 log(n) + C(δ) , δ := min(c 0 , c) > 0 , so that E n (t) 2 ≲ δ n -δ n δ 2 ≲ δ n -δ 2 ≪ 1 .
We conclude by a standard bootstrap argument.

We proved that S ⊂ P. Finally, we show that S is dense in H s . Then, by using this inclusion, we deduce that P contains a dense G δ set as soon as NLS is globally well-posed in H 2 . Proposition 1.2.10 (Density). For g ∈ H 2 (R 3 ), we denote Φ(t)g the maximal solution to NLS in

C([0 , T * ]; H 2 (R 3 )) with initial data g. The pathological set P defined in (1.2.3) is dense in H s (R 3 ).
Moreover, in the defocusing energy-subcritical and critical cases p ∈ 3, 5 , where the Cauchy problem associated with NLS is globally well-posed in H 2 (R 3 ), P contains a dense G δ subset of H s (R 3 ).

Proof. By construction, we know that

∞ k=k 0 ∥v 0,k ∥ H s ≲ ∞ k=k 0 κ n k -→ k 0 →∞ 0, therefore S is dense in H s (R 3 ).
We now assume that global well-posedness in H 2 (R 3 ) holds, and we follow the proof of Corollary 1.2 in [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF]. Specifically, we prove that P contains a countable intersection of nonempty open sets. Let us consider the sequences (ε k ) and (t k ) going to zero as in (1.2.1), and define

O N,k := {f ∈ H s (R 3 ) | ∥Φ(t n k )(ρ ε k * f )∥ H s > N } .
By definition of the lim sup, we have

∞ N =1 ∞ k=k 1 ∞ l=k 1 O N,k ⊂ S.
We already know that ∞ k=k 1 O N,k is nonempty since it contains for instance the lollipop ∞ l=k 1 v 0,l . Therefore, it remains to show that for every N and k, O N,k is an open set.

Let f 0 ∈ O N,k , t ≤ t n k and δ > 0 to be chosen later. Using the Duhamel formula and the Cauchy theory in H 2 , we know that there exists C 0 > 0 such that for every f ∈ H s ,

∥Φ(t)(ρ ε k * f ) -Φ(t)(ρ ε k * f 0 )∥ H 2 ≤ C 0 (∥ρ ε k * f ∥ p-1 H 2 + ∥ρ ε k * f 0 ∥ p-1 H 2 )∥ρ ε k * (f -f 0 )∥ H 2 .
Using the inequality

∥ρ ε * f ∥ H 2 ≲ ε -(2-s) ∥f ∥ H s ,
we deduce that for any f ∈ H s ,

∥Φ(t)(ρ ε k * f ) -Φ(t)(ρ ε k * f 0 )∥ H 2 ≤ C 0 ε -(2-s)p k (∥f ∥ p-1 H s + ∥f 0 ∥ p-1 H s )∥f -f 0 ∥ H s .
Since ∥Φ(t)(ρ ε k * f 0 )∥ H s > N , we deduce that when f is at distance less than δ from f 0 in H s , then ∥Φ(t)(ρ ε k * f )∥ H s > N for some suitable δ > 0, so that O N,k is open.

Generic well-posedness

This section is devoted to the proof of Theorem 1.1.3 for the cubic Schrödinger equation, at scalingsupercritical regularity H s (R 3 ), i.e. with 1 4 < s < 1 2 .8 The technics shall be more specific to the Schrödinger semigroup and its dispersive properties. Namely, we make use of Strichartz and bilinear estimates in the framework of Fourier restriction spaces, that we recall in parts 4.2 and 1.3.2. We then adapt the proof of the almost sure local well-posedness result from [START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF] to establish the convergence of the regularized solutions to the solution obtained from the random initial data regularized by convolution, as claimed in Theorem 1.1. 4.

In what follows, we fix s a scaling-supercritical exponent such that 1 4 < s < 1 2 , and a scalingsubcritical (or critical) Sobolev space H σ , with σ satisfying

s c = 1 2 < σ < 2s .

Preliminaries

We first recall the basic dispersive estimate for the free Schrödinger evolution.

Lemma 1.3.1 (Strichartz estimates). Given 2 ≤ p, q ≤ ∞ satisfying the admissibility condition

2 p + 3 q = 3 2 , (1.3.1)
there exists C > 0 such that for all u 0 ∈ L 2 (R 3 ) , we have

∥ e it∆ u 0 ∥ L p t (R;L q x (R 3 ) ≤ C∥u 0 ∥ L 2 x (R 3
) . We emphasize that the admissibility condition is imposed by the scaling invariance. A key feature of the randomization procedure is to rule out the scaling invariance, and to relax the admissibility condition, as written in Lemma 1.3.4. The following bilinear estimate due to Bourgain indicates a nonlinear smoothing effect when high and low frequencies interact. Combined with the probabilistic Strichartz estimates, it is the main mechanism responsible for the smoothing effect observed in the Picard iteration from a randomized initial data.

Lemma 1.3.2 (Bilinear estimate [START_REF] Bourgain | Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity[END_REF]). Given N ≪ M two dyadic integers and u

0 , v 0 ∈ L 2 (R 3 ) such that supp F(u 0 ) ⊂ ξ ∈ R 3 | |ξ| ≲ N , supp F(v 0 ) ⊂ ξ ∈ R 3 | 1 2 M ≤ |ξ| ≤ 2M .
We have

∥ e it∆ u 0 e it∆ v 0 ∥ L 2 t,x (R×R 3 ) ≲ N M -1 2 ∥u 0 ∥ L 2 x (R 3 ) ∥v 0 ∥ L 2 x (R 3
) . Let us now set up the relevant randomization procedure in the Euclidean space, and recall the key probabilistic Strichartz estimates obtained in this context.

Wiener randomization

The Wiener decomposition of a function in f 0 ∈ L 2 is defined from a unit-scale partition of unity of the frequency space given by a function

ψ ∈ C ∞ c (R 3 ) supported in [-2 , 2] 3 , such that k∈Z 3 ψ(ξ -k) = 1 .
Defining the corresponding unit-scale Fourier multiplier P 1,k = F -1 (ψ(• -k)F) on the unit cube centered around k, we decompose any function in L 2 (R 3 ) into a sum of unit-blocks

f 0 = k∈Z 3 P 1,k f 0 .
The randomization procedure consists in decoupling these individual blocks by some independent Gaussian variables. Specifically, given a function 9 f 0 ∈ H s (R 3 ), a probability space (Ω, A, P) and some independent complex Gaussian variables (g k ) k∈Z 3 , we define a probability measure µ supported on H s (R 3 ) induced by the random variable

ω ∈ Ω -→ f ω 0 := k∈Z 3 g k (ω)P 1,k f 0 ∈ H s (R 3 ) a.s.
This randomization procedure is called the Wiener randomization.

Strichartz estimates

The main idea is to combine the unit scale Bernstein estimate with probabilistic decoupling in order to improve the integrability bounds and to relax the Strichartz admissibility condition, for a large measure set of initial data.

Definition 1.3.3. Let M ≫ 1 be a parameter. We say that an event A = A(M ) occurs M -certainly when there exist some positive constants C, c, θ > 0 such that

P(A(M )) ≥ 1 -C exp -cM θ .
Lemma 1.3.4 (Probabilistic Strichartz estimates [START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF], Proposition 1.4). Fix a Strichartz admissible pair (q, r), with 2 ≤ q, r < ∞, satisfying the Strichartz admissibility condition (1.3.1).

For any r such that r ≤ r < ∞, there exist C, c > 0 such that for all λ > 0,

P ω | ∥⟨∇⟩ s e it∆ f ω 0 ∥ L q t (R;L r x (R 3 )) > λ ≤ C exp -cλ 2 ∥f 0 ∥ -2 H s . (1.3.2)
As a consequence, for any λ > 0, the following Strichartz estimate holds λ-certainly

∥⟨∇⟩ s e it∆ f ω 0 ∥ L q t (R;L r x (R 3 )) ≤ λ∥f 0 ∥ H s , (1.3.3) so that almost surely, ∥⟨∇⟩ s e it∆ f ω 0 ∥ L q t (R;L r x (R 3 )) < ∞ .

Probabilistic local well-posedness

With the above refined probabilistic Strichartz estimates at hand, one can observe some nonlinear smoothing effect. Specifically, starting from an initial data f ω 0 ∈ H s that satisfies (1.3.3), and by performing a linear-nonlinear decomposition through the ansatz

u(t) = e it∆ f ω 0 + v ,
we run a contraction mapping argument for v in H σ with 1 2 < σ < 2s, where v is solution to a cubic Schrödinger equation with stochastic forcing terms depending on f (t) = e it∆ f ω 0 , and zero initial condition:

(i∂ t + ∆)v = |v + f | 2 (v + f ) , v(0) = 0 . (NLS f )
We use the following refined Strichartz norm for the linear evolution of the initial data

∥f ∥ S s := ∥⟨∇⟩ s f ∥ L 4 t,x (R×R 3 ) + ∥⟨∇⟩ s f ∥ L 5 t,x (R×R 3 ) .
It follows from Lemma 1.3.4 that for all λ > 0, we have λ-certainly that

∥ e it∆ f ω 0 ∥ S s ≤ λ∥f 0 ∥ H s ,
As a consequence, the set

Σ := f ω 0 | ∥ e it∆ f ω 0 ∥ S s < ∞
has full µ-measure. To control the source term in (NLS f ) we set

Y s := L ∞ t (R; H s ) ∩ S s .
The local existence result due to Bényi, Oh and Pocovnicu for NLS with initial condition f ω 0 ∈ Σ, that we now recall, states as follows.

Theorem 1.3.5 (Almost sure local well-posedness, see [START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF][START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]). For all f ω 0 ∈ Σ, there exists T = T (∥ e it∆ f ω 0 ∥ Y s ) > 0 such that the Cauchy problem NLS with initial data

f ω 0 has a unique solution u ∈ e it∆ f ω 0 + X σ,b T → L ∞ t ([0 , T ]; H s ) , for some b = 1 2 + . Moreover, u ∈ C t ([0 , T ]; H s ) .
Before sketching the proof, let us comment on the Bourgain recentering method used to obtain such a result. The idea is to reduce the Cauchy problem NLS with initial data f ω 0 to the Cauchy problem (NLS f ), with a forcing term f = e it∆ f ω 0 and with zero initial condition, at scalingsubcritical regularity H σ (R 3 ). This is the reason why we have uniqueness for u in an affine subspace of L ∞ t H s x (R 3 ). This strategy generates strong solutions for initial data in Σ, in the sense that the flow map is continuous from the time interval of existence to a functional space embedded in H s . Then, the contraction mapping argument follows from trilinear estimates in Fourier restriction spaces, required to handle the bilinear estimates from Lemma 1.3.2. They are associated to the norm

∥f ∥ X σ,b = ∥⟨ξ⟩ σ ⟨τ + |ξ| 2 ⟩ b f (τ, ξ)∥ L 2 τ,ξ (R×R 3 ) , ∥f ∥ X σ,b T = inf ∥F ∥ X σ,b | F ∈ X σ,b , F | [0,T ] = f .
Combining the bilinear estimate from Lemma 1.3.2 with the probabilistic Strichartz estimate from lemma 1.3.4, we obtain trilinear estimates based on a Littlewood-Paley analysis. Before proceeding further, let us simplify the notation. We write the cubic nonlinearity as a trilinear operator, acting on functions u i 1≤i≤3 of type either e it∆ f ω 0 or v. In this purpose, we define

N (u 1 , u 2 , u 3 ) := u 1 u 2 u 3 , N (u) := N (u, u, u) = |u| 2 u .
Once again, we drop the complex conjugation sign since it plays no role in our short time perturbative analysis, so that the above notation is invariant under permutation of the terms u 1 , u 2 , u 3 . For convenience, we write the Duhamel integral term as (i∂ t -∆) -1 (F ) := such that for all 0 < T ≲ 1, for all

f 1 , f 2 , f 3 ∈ Y s and v, v 1 , v 2 , v 3 ∈ X σ,b ([0 , T ]), we have ∥ i∂ t -∆ -1 N (v 1 , v 2 , v 3 )∥ X σ,b T ≤ CT θ ∥v 1 ∥ X σ,b T ∥v 2 ∥ X σ,b T ∥v 3 ∥ X σ,b T , ∥ i∂ t -∆ -1 N (f 1 , f 2 , f 3 )∥ X σ,b T ≤ CT θ ∥f 1 ∥ Y s ∥f 2 ∥ Y s ∥f 3 ∥ Y s , ∥ i∂ t -∆ -1 N (v 1 , v 2 , f 3 )∥ X σ,b T ≤ CT θ ∥v 1 ∥ X σ,b T ∥v 2 ∥ X σ,b T ∥f 3 ∥ Y s , ∥ i∂ t -∆ -1 N (v 1 , f 2 , f 3 )∥ X σ,b T ≤ CT θ ∥v 1 ∥ X σ,b T ∥f 2 ∥ Y s ∥f 3 ∥ Y s .
In addition, we have the Lipschitz estimate

∥ i∂ t -∆ -1 N v 1 + f -N v 2 + f ∥ X σ,b T ≤ CT θ ∥v 1 -v 2 ∥ X σ,b T ∥v∥ 2 X σ,b T + ∥f ∥ 2 Y .
As a consequence of the above trilinear estimates, we have a good Cauchy theory for (NLS f ).

Lemma 1.3.7 (Local well-posedness for the perturbed equation). Let R > 0 and assume that

∥f ∥ Y s ≤ R .
For any 1 2 < σ < 2s and b = 1 2 + , there exist T = T (R) > 0 and a unique solution v to (NLS f ) in

X σ,b T . Moreover, v ∈ C([0 , T ]; H σ ) , and ∥v∥ X σ,b T ≤ R .
Theorem 1.3.5 follows from Lemma 1.3.7 applied with f := e it∆ f ω 0 , for some initial data f ω 0 ∈ Σ.

Proof of Lemma 1.3.7. We show that for T small enough with respect to R, the map

Γ : v ∈ X σ,b T -→ (i∂ t -∆) -1 |v + f | 2 (v + f ) ∈ X σ,b T is a contraction on the ball B R = v ∈ X σ,b T | ∥v∥ X σ,b T ≤ R .
It follows from the trilinear estimates of Lemma 1.3.6 that for all

v, v 1 , v 2 ∈ X σ,b T , ∥Γv∥ X σ,b T ≤ CT θ ∥v∥ 3 X σ,b + ∥f ∥ 3 Y s , ∥Γ(v 2 ) -Γ(v 1 )∥ X σ,b T ≤ CT θ ∥v 2 -v 1 ∥ X σ,b T ∥v 1 ∥ 2 X σ,b T + ∥v 2 ∥ 2 X σ,b T + ∥f ∥ 2 Y s , so that ∥Γv∥ X σ,b T ≤ 2CT θ R 3 , ∥Γ(v 2 ) -Γ(v 1 )∥ X σ,b T ≤ 2CT θ R 2 ∥v 2 -v 1 ∥ X σ,b T .
We conclude by choosing

T = T (R) > 0 such that 2CT θ R 2 ≤ 1 2 .

Approximate solutions by convolution and convergence

Let f ω 0 ∈ Σ, and u = e it∆ f ω 0 + v be the local solution to NLS obtained in Theorem 1.3.5, with v ∈ X σ,b ([0 , T ]) ⊂ C([0 , T ], H σ ). Let (ε k ) k≥0 be a sequence of positive numbers that go to zero. Using the approximate identity (1.1.3), we define a sequence of regularized initial data

f ω 0,k := f ω 0 * ρ ε k ∈ H ∞ .
Then, we denote u k the maximal solution in H 1 to NLS. Using the conservation of the coercive energy (in the defocusing case), u k is global. The purpose of this section is to prove the convergence claimed in Theorem 1.1.3. We set R > 0 be such that

∥ e it∆ f ω 0 ∥ Y s ≤ R ,
and T = T (R) is as in the proof of Lemma 1.3.7, applied with f := e it∆ f ω 0 .

Proposition 1.3.8. We have the convergence

lim k→∞ ∥u k -u∥ L ∞ ([0,T ],H s ) = 0 .
Proof. We do not have a good Cauchy theory in H s (R 3 ). However, Theorem 1.3.5 provides a Cauchy theory for the perturbed equation (NLS f ), with a forcing term f = e it∆ f ω 0 in Y s . Hence, it is relevant to run a linear-nonlinear decomposition of u k as well

u k = e it∆ f ω 0,k + v k =: f k + v k , f k = e it∆ f ω 0,k ,
We define

δ k := f k -f = e it∆ (f ω 0 * ρ ε k -f ω 0 )
, and note that the convergence

lim k→∞ ∥δ k ∥ L ∞ t H s x = lim k→∞ ∥f 0 -f 0 * ρ ε k ∥ H s x = 0 .
follows from the unitary property of the linear evolution together with the fact that the convolution commutes with the linear evolution. Then, we use the embedding

X σ,b ([0 , T ]) → L ∞ ([0 , T ], H σ )
to reduce the proof of Theorem 1.1.3 to the convergence of the nonlinear term v k in the Fourier restriction space. lim

k→∞ ∥v -v k ∥ X σ,b T = 0 . (1.3.4)
To achieve such a goal, we start by proving that lim k→∞ ∥δ k ∥ S s = 0 .

(1.3.5)

We fix 2 ≤ q, r < ∞ and ω ∈ Ω such that

∥e it∆ f ω 0 ∥ L q t L r
x < ∞ . In particular, we observe that

∥e it∆ f ω 0 ∥ L r x < ∞ , Lebesgue a.e. t ∈ R . (1.3.6)
In addition, when t is such that (1.3.6) holds we

∥δ k (t)∥ L r x ≤ (1 + ∥ρ∥ L 1 x )∥ e it∆ f ω 0 ∥ L r
x , so that the sequence of measurable functions t → ∥δ k (t)∥ L r

x is uniformly bounded in k. From the assumption on f ω 0 , we also have the uniform bound

∥δ k ∥ L q t L r x ≤ (1 + ∥ρ∥ L 1 x )∥ e it∆ f ω 0 ∥ L q t L r
x .

(1.3.7)

Next, we use the fact that the approximate identity converge in L r x , and commutes with the linear evolution e it∆ , to obtain

lim k→∞ ∥δ k (t)∥ L r x = lim k→∞ ∥e it∆ f ω 0 -f ω 0 * ρ ε k ∥ L r x = lim k→∞ ∥e it∆ f ω 0 -e it∆ f ω 0 * ρ ε k ∥ L r x = 0 .
This convergence, combined with the uniform bound (1.3.7), allows us to apply the dominated convergence theorem in L q t and to conclude that the convergence (1.3.5) holds.10 Next, we fix η > 0 arbitrarily small, and k 0 ∈ N such that for all k ≥ k 0 , ∥δ k ∥ Y s ≤ η. We define

w k := v k -v ∈ X σ,b ([0 , T ]) .
By using the equations for v and v k , we see that w k is solution to the following forced Schrödinger equation with zero initial condition

(i∂ t + ∆)w k = N (f k + v k ) -N (f + v) = N (w k + δ k + f + v) -N (f + v) =: F . w k (0) = 0 ,
We perform a bootstrap argument and propagate the smallness of the X σ,b -norm of w k to prove that for all k ≥ k 0 ,

∥w k ∥ X σ,b ([0,T ]) ≤ 2η .
Up to some multiplicative universal constants, the forcing term writes

F ∼ N (w k + δ k ) + N (w k + δ k , w k + δ k , f + v) + N (w k + δ k , f + v, f + v) .
Using the trilinear estimates from Lemma 1.3.6, we get the following a priori estimate

∥w k ∥ X σ,b T ≤ CT θ ∥w k ∥ X σ,b T ∥w k ∥ 2 X σ,b T + ∥δ k ∥ 2 Y s + ∥v∥ 2 X σ,b T + ∥f ∥ 2 Y s + CT θ ∥δ k ∥ Y s ∥w k ∥ 2 X σ,b T + ∥δ k ∥ 2 Y s + ∥v∥ 2 X σ,b T + ∥f ∥ 2 Y s .
We finally exploit the smallness of the Y s -norm of δ k when k ≥ k 0 , the estimates on ∥v∥ X σ,b T from Theorem 1.3.5 and the a priori estimate on ∥w k ∥ X σ,b T from the bootstrap hypothesis to obtain

∥w k ∥ X σ,b T ≤ 2CT θ ∥w k ∥ X σ,b T η 2 + R 2 + 2CT θ η η 2 + R 2 .
Since we chose η arbitrarily small, and T is chosen such that 2CT θ (1 + R 2 ) ≤ 1 2 we conclude that,

∥w k ∥ X σ,b T ≤ 1 2 ∥w k ∥ X σ,b T + 1 2 η .
This finishes the proof of the bootstrap. Abstract. We obtain almost-sure scattering for the cubic defocusing Schrödinger equation in the Euclidean space R 3 , with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness assumption, our result generalizes the work of Bényi, Oh and Pocovnicu [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]. It also extends the results of Dodson, Lührmann and Mendelson [START_REF] Dodson | Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation[END_REF] on the energy-critical equation in R 4 , to the energy-subcritical equation in R 3 . In this latter setting, even if the nonlinear Duhamel term enjoys a stochastic smoothing effect that makes it subcritical, it still has infinite energy. In the present work, we first develop a stability theory from the deterministic scattering results below the energy space, due to Colliander, Keel, Staffilani, Takaoka and Tao.

Then, we propose a globalization argument in which we set up the I-method with a Morawetz bootstrap in a stochastic setting. To our knowledge, this is the first almost-sure scattering result for an energy-subcritical Schrödinger equation outside the small data regime. 99

Introduction

We consider the initial value problem for the cubic defocusing Schrödinger equation

i∂ t u(t, x) + ∆u(t, x) = u(t, x) 2 u(t, x), (t, x) ∈ R × R 3 . u(0, x) = u 0 (x) , (NLS)
The equation has a Hamiltonian structure associated with the energy

E(u) = 1 2 R 3 |∇u(•, x)| 2 dx + 1 4 R 3 |u(•, x)| 4 dx . (2.1.1)
For smooth solutions, the above energy is conserved as time evolves. The mass

M (u) = R 3 |u(•, x)| 2 dx
is also formally conserved. In addition, the equation enjoys the scaling symmetry. Namely, given λ > 0 and u a solution to the equation,

u λ (t, x) = λ -1 u(λ -2 t, λ -1 x) , (t, x) ∈ R × R 3 ,
is also a solution. Since the scaling symmetry leaves invariant the homogeneous Sobolev space Ḣ 1 2 (R 3 ), the scale s c = 1 2 is critical. In the scaling-subcritical regime s > 1 2 , it follows from a contraction mapping argument that the initial value problem is locally well-posed in the following sense :

• (Uniqueness) Given two solutions u 1 , u 2 in H s (R 3 ) and a time t, if u 1 (t) = u 2 (t) then u 1 = u 2 .
• (Existence) For all u 0 ∈ H s (R 3 ), there exists T (∥u 0 ∥ H s ) > 0 and a solution to (NLS)

u(t) = e it∆ u 0 -i t 0 e i(t-τ )∆ |u(τ )| 2 u(τ )dτ , u ∈ C([-T , T ]; H s (R 3 )) . • (Continuity) The solution map u 0 ∈ H s (R 3 ) → u ∈ C([-T , T ], H s (R 3 )) is continuous.

Known results

Note that in the scaling-subcritical regime, the maximal lifespan of the solution is bounded from below by a quantity that only depends on the H s (R 3 ) norm of the initial data. Therefore, one can use a continuity argument and deduce global existence from a priori uniform bound on the H s (R 3 ) norm of the solution.

In the present case, s c < 1 and the equation is energy-subcritical. Note that an initial data in H 1 (R 3 ) has a finite energy. Besides, it follows from the defocusing nature of the equation that the energy is coercive, and yields a uniform control on the H 1 (R 3 ) norm of the solution so that (NLS) is globally well-posed in the energy space H 1 (R 3 ). In addition, the solution display linear scattering in large time. Namely, for all u 0 ∈ H 1 (R 3 ), there exist some unique final states

u ± ∈ H 1 (R 3 ) such that lim t→±∞ ∥ e -it∆ u(t) -u ± ∥ H 1 = 0 .
It is conjectured that (NLS) is globally well-posed in H s (R 3 ) for 1 2 ≤ s < 1, and that the solution scatters at infinity. Still, we do not have a coercive conservation law at these regularities and, in general, we cannot always preclude the H s (R 3 ) norm from blowing up in finite time. In the scaling-critical case, the time of existence of the solution does not only depend on its Ḣ 1 2 (R 3 ) norm, but also on its profile. Nevertheless, by using a concentration-compactness method, Kenig and Merle proved in [KM10] some conditional scattering.

It is well known from [START_REF] Christ | Ill-posedness for nonlinear Schrödinger and wave equations[END_REF] that at scaling-supercritical scales s < 1 2 , the equation is ill-posed because of norm inflation. Furthermore, [ST20a; CG22] have recently evidenced that there exists in H s (R 3 ) a pathological set containing a dense G δ -set of initial data for which the regularized solutions inflate in arbitrarily short time. Nevertheless, following the pioneering work of Bourgain, a probabilistic Cauchy theory has emerged. It displays the existence of a local flow defined on generic sets of initial data, in some regimes where a deterministic Cauchy theory does not hold. In the present work, we address the globalization of such a probabilistic flow for (NLS) in the Euclidean space, with randomized initial data at supercritical regularities 3 7 < s.

Global well-posedness & Scattering below the energy space Let us present the general framework used to study (NLS) at scaling-subcritical regularity, below the energy space, where we still have a local flow. At regularities 1 2 ≤ σ < 1, there is no coercive conservation law that prevents the Sobolev norm to grow up. Hence, to obtain global a priori estimates for a general initial data 1 , one has to make use of some modified energies. The idea is to consider a frequency truncated 1 Note that one can easily prove scattering under smallness assumption on the initial data at regularity version of the solution, and to control the increments of its energy along the evolution. Bourgain implemented this idea in [START_REF] Bourgain | Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity[END_REF] to prove global existence for the cubic Schrödinger equation in H σ (R 2 ), with 2 3 < σ, by using the high-low argument together with some bilinear estimates. The observation is that the nonlinear evolution of the high frequencies is well approximated by the linear evolution. Namely, if we evolve on a short time step the low-frequency modes by the nonlinear flow, and the high-frequency modes by the linear flow, the error we make turns out to be in the energy space H 1 (R 3 ). Furthermore, the methods displays a nonlinear smoothing effect : for all t ∈ R,

u(t) -e it∆ u 0 ∈ H 1 (R 3 ) . (2.1.2)
This globalization method has become standard, and was used in many contexts (see, e.g., [START_REF] Kenig | Global well-posedness for semi-linear wave equations[END_REF] for the wave equation). Bourgain extended the method to the 3D case in [START_REF] Bourgain | Scattering in the energy space and below for 3D NLS[END_REF], where he proved global existence when 11 13 < σ. Then, Colliander, Keel, Staffilani, Takaoka and Tao refined this frequency-cutoff in [START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF] and introduced the I-operator, which goes from H σ (R 3 ) to H 1 (R 3 ). By exploiting the structure of the nonlinearity, the authors evidenced frequency cancellations that come with a nonlinear smoothing effect, loosening therefore the regularity threshold at which the local flow extends globally. Definition 2.1.1 (I-operator). Given 1 2 < σ < 1 and N ≫ 1, the I-operator is the Fourier multiplier

I = F -1 (mF), m(ξ) = 1 if |ξ| ≤ N, N |ξ| 1-σ if |ξ| ≥ 2N . (2.1.3) It holds that ∥∇Iu∥ L 2 ≤ ∥u∥ H σ , ∥u∥ H σ ≤ N 1-σ ∥Iu∥ H 1 . (2.1.4)
As N becomes larger, the energy is supposed to grow to infinity, while we expect it's time derivative to decrease. Specifically, if we assume that J × R 3 is a spacetime slab where we have a smallness assumption on the L 4 t,x norm of u, then

J | d dt E(Iu)(τ )|dτ ≲ 1 N + O 1 N 2 . (2.1.5)
The method usually comes with a rescaling argument, that allows one to assume that the energy of Iu is less than 1 initially. In addition, the interaction Morawetz inequality gives a bound on the L 4 t,x norm, depending on the energy of Iu. Hence, we have a control on the number of spacetime slabs where the increment of E(v) satisfies (2.1.5). By finely tuning the parameters N and s, we would be able to sum (2.1.5) over finitely many spacetime slabs, and to obtain a uniform bound on both the energy of Iu and on the L 4 t,x norm of u, from which scattering follows. The interaction Morawetz estimate reads

J R 3 |u(t, x)| 4 dxdt ≲ ∥u 0 ∥ 2 L 2 sup t∈J ∥u(t)∥ Ḣ 1 2 2 .
It is due to [START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF], and its proof consists in averaging over y the recentered Morawetz action against the mass density, and in tracking some monotonicity. The I-method with a Morawetz bootstrap has been refined, and adapted in different contexts. Let us cite some works that are relevant to our problem. By the use of linear-nonlinear decomposition of the initial data, [START_REF] Dodson | Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when n = 3 via a linear-nonlinear decomposition[END_REF] reached the exponent 5 7 < s, further improved to2 3 < s in [START_REF] Su | Global well-posedness and scattering for defocusing, cubic NLS in R 3[END_REF] by combining the arguments of [START_REF] Dodson | Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when n = 3 via a linear-nonlinear decomposition[END_REF] with a second generation I-method. In the present work, we implement the following key idea. To improve the results of [START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF], Dodson provided a refined version of the energy increment estimate (2.1.5), by keeping some sub-additive quantities on the right-hand-side : for some constant c independent of N ,

J | d dt E(Iu)(τ )|dτ ≲ 1 N 1-∥∇P >cN Iu∥ 2 L 2 t (J;L 6 x (R 3 )) + O 1 N 2 . (2.1.6)
For radial initial data, this comes together with the so-called long-time Strichartz estimate, that yields a uniform a priori estimate on ∥∇P >cN Iu∥ 2

L 2 t L 6
x . This strategy avoids loosing too many powers of N when brutally multiplying the energy increments by the number of spacetime slabs where we have a control. Dodson implemented this idea in [START_REF] Dodson | Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when n = 3 via a linear-nonlinear decomposition[END_REF][START_REF] Dodson | Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when d = 2, 3 and u 0 is radial[END_REF] to prove scattering for s > 5 7 , and for s > 1 2 in the radial case. However, the conjecture is still open for general data. Before ending this paragraph, we formulate the optimal result known at this date. We will use it throughout the analysis. 2 Theorem 2.1.2 (Global well-posedness below the energy space, [Col+04; Dod13; Su12]). Let

2 3 < σ < 1 and u 0 ∈ H σ x (R 3
). The Cauchy problem (NLS) with u(t 0 ) = u 0 is globally well-posed and the solution scatters at infinity in H σ x (R 3 ). Moreover, there exists a constant

C(∥u 0 ∥ H σ x ) > 0 such that ∥u∥ L 5 t,x (R×R 3 ) ≤ C(∥u 0 ∥ H σ x ) . (2.1.7)
Probabilistic Cauchy theory Given a randomized initial data f ω 0 , which is basically the superposition of modes decoupled by multiplying each of them with independent Gaussian variables, the probabilistic approach consists in performing a linear-nonlinear decomposition in the Duhamel integral representation, and in studying the nonlinear term

v(t) = u(t) -e it∆ f ω 0 = t 0 e i(t-t ′ )∆ | e it ′ ∆ f ω 0 | 2 e it ′ ∆ f ω 0 dt ′ + • • • .
The remaining terms correspond to higher order Picard iterates. It turns out that the probabilistic decoupling enhances the nonlinear smoothing effect (2.1.2), even in scaling-supercritical regimes and, in the favorable cases, we expect v to be at critical, or subcritical regularity, where we have some local existence results. Besides, v is solution to (NLS) with a stochastic forcing term.

i∂ t v + ∆v = |v| 2 v + |v + f | 2 (v + f ) -|v 2 |v , (t, x) ∈ R × R 3 . v(t 0 ) = v 0 ∈ H σ (R 3 ) , (NLS f )
At time t = 0, we have v(0) = 0. The strategy is to extend the deterministic Cauchy theory at scaling-subcritical regularities to the perturbed equation (NLS f ), at least locally in time. Specifically, we expect to prove that for almost every ω, (NLS f ) is locally well-posed by some perturbative arguments, with f ω = e it∆ f ω 0 . This strategy corresponds to the Da Prato Debussche trick [START_REF] Da Prato | Two-dimensional Navier-Stokes equations driven by a space-time white noise[END_REF]. In the context of dispersive equations, it was first implemented by Bourgain for the 2D Schrödinger equation on the torus [START_REF] Bourgain | Periodic nonlinear Schrödinger equation and invariant measures[END_REF], and then by Burq and Tzvetkov for the wave equation on compact manifolds without boundaries [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF][START_REF] Burq | Random data Cauchy theory for supercritical wave equations II: a global existence result[END_REF]. In such contexts, the randomization procedure is based on the spectral resolution of the Laplacian. However, on the Euclidean space, we do not have such natural decompositions for the initial data. It turns out that the so-called Wiener randomization proved to be suitable, in the sense that the nonlinear Duhamel term benefits from a nonlinear probabilistic smoothing that makes it subcritical, thanks to some improved spacetime integrability for almost every f ω . Definition 2.1.3 (Wiener randomization). Let f 0 ∈ L 2

x (R 3 ) be radially-symmetric. We define the unit-scale Wiener decomposition of f 0 by

f 0 ∼ k∈Z 3 Q k f 0 ,
where Q k is the Fourier projector associated with a smooth multiplier localized on the unit cube centered around k ∈ Z 3 . Next, given a sequence of real valued mean zero and independent complex Gaussian variables {g k } k∈Z 3 , we define a random variable

ω ∈ Ω → f ω 0 = k∈Z 3 g k (ω)Q k f 0 . (2.1.8)
Let us emphasize that the procedure does not gain regularity in the sense that if

f 0 ∈ H s (R 3 ) \ δ>s H δ (R 3 ), then f ω 0 ∈ H s (R 3 ) \ δ>s H δ (R 3
) for almost every ω. We refer to the works [LM14a; BOP15b; BOP15a; BOP19b; BOP19a; DLM20; DLM19] and the references therein for further details on the Wiener randomization.

Globalization arguments in the stochastic setting Besides the construction of local solutions in supercritical regimes, one may ask about the global existence and asymptotic behavior of these solutions. This question is out of reach in general, and the probabilistic method is mainly concerned with perturbations of the zero solution. Still, there are different globalization procedures in such a stochastic context outside the small data regime that make use, in a mild sense, of the conservation laws. In the compact setting, we can use invariant or quasi-invariant measures as a substitute for the conservation of energy. This approach in the context of dispersive PDE's was initiated by Bourgain, who settled in [START_REF] Bourgain | Periodic nonlinear Schrödinger equation and invariant measures[END_REF][START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] the invariance property of the Gibbs measure for (NLS) on the torus, in dimension 1 and 2. In [START_REF] Burq | Random data Cauchy theory for supercritical wave equations II: a global existence result[END_REF], Burq and Tzvetkov proved almost sure global well-posedness by using the energy increment for v and a Gronwall's argument.

In the Euclidean space, we do not have invariant measures and the globalization arguments are mostly based on deterministic methods. Sill, some works concern the harmonic oscillator. Thanks to the so-called lens transform that intertwine the harmonic oscillator and (NLS), a randomization procedure emerges from the decomposition on the orthonormal basis of the Hermite functions, that are the eigenfunctions of the harmonic oscillator. On the support of the induced measure, that lies in a space just below L 2 , we have global existence and scattering under smallness assumptions (see [START_REF] Poiret | Solutions globales pour l'équation de Schrödinger cubique en dimension 3[END_REF][START_REF] Poiret | Probabilistic well-posedness for the supercritical nonlinear harmonic oscillator[END_REF] in dimension d ≥ 2). Recently, Burq and Thomann evidenced in [START_REF] Burq | Almost sure scattering for the one dimensional nonlinear Schrödinger equation[END_REF] that this randomization procedure provides quasi-invariant measures for Schrödinger equations in 1D, and proved almost-sure scattering. However, in higher dimensions d ≥ 2, the randomization on the Hermite functions induces a gain of regularity in L p for p ̸ = 2, as well as a gain of decay in weighted L 2 -spaces. To address energy-critical equations, for which the conservation law is not enough, the natural strategy in the random data setting consists in using a stability theory for (NLS f ) in the energy space, and to infer a priori spacetime bounds from which we deduce scattering. This method was used by [START_REF] Dodson | Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data[END_REF] to prove almost sure scattering for the 4D energy critical wave equation. Note that the case of the wave equation is more favorable, since the energy controls ∂ t v. Nevertheless, Killip, Murphy and Visan proved the counterpart of [START_REF] Dodson | Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data[END_REF] for the energy critical Schrödinger equation in 4D via a double bootstrap argument, that combines a modified energy and a Morawetz like estimate. In addition, the result of [START_REF] Killip | Almost sure scattering for the energycritical NLS with radial data below H 1 (R 4 )[END_REF] was improved in [START_REF] Dodson | Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation[END_REF], where the authors finely exploited the local smoothing effect. Namely, they refined the probabilistic local well-posedness theory in a new functional framework, inspired by the lateral spaces used to study Schrödinger maps, together with an improved Sobolev embedding for randomized radial initial data. We stress out that the stability-perturbation strategy is more likely to work under some radial assumption on the data, in order to use local energy decay estimate and to gain regularity when estimating the modified energy increments, with rough data. Specifically, one needs to use a radial Sobolev embedding at some point in the analysis, in order to absorb a weight that comes from the need to use local energy decay, or from the Lin-Strauss estimate. However, Oh, Okamoto and Pocovnicu [START_REF] Oh | On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities[END_REF] proved almost sure global existence for the energy critical Schrödinger in higher dimension d ≥ 5, where the potential energy controls more terms. To our knowledge, stability-perturbation arguments for NLS outside the small data regime were only used in the energy-critical case. In the energy-subcritical case, one may be led to consider modified energy arguments as well. Colliander and Oh already used the high-low method in a stochastic context, to prove almost-sure global well-posedness for the cubic periodic Schrödinger equation below L 2 (T) in [START_REF] Colliander | Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L 2 (T)[END_REF], using the mass as a conservation law. Then, Lührmann and Mendelson [START_REF] Lührmann | Random data Cauchy theory for nonlinear wave equations of power-type on R 3[END_REF] applied the high-low method to prove global well-posedness for the NLW. Poiret Robert and Thomann also used this method to prove scattering for supercritical small initial data for the harmonic oscillator in [START_REF] Poiret | Probabilistic well-posedness for the supercritical nonlinear harmonic oscillator[END_REF]. More recently, the authors of [START_REF] Gubinelli | Global Dynamics for the Two-dimensional Stochastic Nonlinear Wave Equations[END_REF] used the I-method in the context of the stochastic nonlinear wave equation. In [START_REF] Fan | Construction of L 2 log-log blowup solutions for the mass critical nonlinear Schrödinger equation[END_REF], the I-method was used to address the log-log blow up profile for the mass critical NLS in L 2 (R 2 ). Let us also mention the scattering result from [START_REF] Spitz | Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data[END_REF] in R 4 , where the author uses an extra randomization of the angular variable as a substitute for the radial assumption. Note, however, that this randomization yields better estimates on the linear evolution than we use under the radial assumption.

In the present work, we consider the 3D case, where the equation is energy-subcritical. Hence, there is a gap to bridge between the scaling-critical space Ḣ 1 2 (R 3 ) and the energy space Ḣ1 (R 3 ). The known results for this equation with supercritical initial data are the followings. On the one hand, we have the almost-sure scattering result for small-data that comes from [START_REF] Poiret | Solutions globales pour l'équation de Schrödinger cubique en dimension 3[END_REF] by the use of the lens transform. On the other hand, the works of [START_REF] Bényi | Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on R 3[END_REF] based on the wiener randomization address scattering for small data, and conditional global existence for general data. Let us also mention the work [START_REF] Bényi | Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on R 3[END_REF], where the authors used higher order expansion in the Duhamel formula. The idea is to refine Da Prato Debussche's trick and include higher order terms in the Ansatz. By doing so, they prove that the nonlinear term lies in H 1 (R 3 ) almost surely, and has finite energy. Yet, it is not clear how to deduce global well-posedness from this observation.

Main result and outline of the proof

Our approach is a bit different. Since v is not in the energy space but just below, we use the scattering theory in H σ (R3 ) for 1 2 < σ < 1 detailed in paragraph 2.1.1, and we establish a stability theory for the perturbed equation (NLS f ) to obtain a conditional scattering result. Then, we obtain global a priori estimates and we prove the following result.

Theorem 2.1.4. Let s, σ be such that 3 7 < s ≤ 1, and 6 7 < σ < 2s. Given a radially-symmetric function f 0 in H s (R 3 ), we define the randomized initial data f ω 0 as in (2.1.8). Then, for almost every ω, there exists a global solution u to the defocusing cubic Schrödinger equation (NLS), with

u -e it∆ f ω 0 ∈ C R; H σ x (R 3 ) .
Moreover, there exist

u ± ∈ H σ x (R 3 ) such that lim t→±∞ ∥u(t) -e it∆ f ω 0 + u ± ∥ H σ x (R 3 ) = 0 . Uniqueness holds for v = u -e it∆ f ω 0 ∈ X σ (R) ⊂ C(R; H σ x (R 3
)), which is a strong solution to the nonlinear Schrödinger equation with some stochastic forcing terms (NLS f ), in the critical space X σ defined in (2.2.2) and with zero initial data. Note that we have scattering at regularity close to H 2s , which has to be compared to Theorem 2.1.2, where we do not directly have scattering in H 1 (R 3 ). The threshold 3 7 < s is certainly not optimal. Even if the local probabilistic flow exists up to 1 4 < s, it would be more realistic to reach 1 3 < s, since the deterministic theory is known for 2 3 < σ. One way of loosening the threshold on s would be to have a local-smoothing estimate as (2.2.20) independent of N , and then to perform a scaling argument to have an energy smaller than 1. In addition, one can also hope to prove a long-time Strichartz estimate in a probabilistic setting, and to refine the local existence theory by the use of lateral spaces as in [START_REF] Dodson | Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation[END_REF].

We stress out that randomized initial data f ω 0 are not radial. 3 Nevertheless, we follow [DLM20; DLM19] and we use the radial assumption on each frequency localized pieces Q k f 0 , for k ∈ Z 3 . Then, we deduce from probabilistic decoupling an estimate on the linear Schrödinger evolution of the whole function f ω 0 ∼ g n (ω)Q k f 0 , which is not radially-symmetric. Hence, the radial-symmetry assumption is concealed by the randomization procedure, but it still gives access to the energy decay estimates. The radial-symmetry assumption might be loosened, but probably not entirely removed. Indeed, when we estimate the modified energy increment, a lot of derivatives fall on the rough term f , and we need at least to gain 1 2 derivative when taking the L 2 -in-time spacetime norms of f . For this purpose, we use the local-energy decay. Subsequently, since we avoid using some decay assumptions, we need to use a radial Sobolev embedding to absorb the weights that comes from the local smoothing estimates. This is also the reason why Dodson settled in [START_REF] Dodson | Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when d = 2, 3 and u 0 is radial[END_REF] the deterministic subcritical theory for radial data only, since the long time Strichartz estimates used in [START_REF] Dodson | Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when d = 2, 3 and u 0 is radial[END_REF] exploit the local energy decay effect.

Remark 2.1.5. A few days after the submission of this paper on arXiv, a preprint [SSW21] also appeared on arXiv, proving a similar result independently, with the same threshold s > 3 7 and radial assumption. Their proof is based on the high-low method.

Outline of the proof First, we fix α > 0, and N 0 = N 0 (α, s, σ, ∥f 0 ∥ H s ) ≫ 1, to be chosen at the end of the analysis. From the large deviation bounds presented in section 2.2, we can find a set Ω α , with P(Ω \ Ω α ) ≤ 2α, such that for every ω, the linear evolution of the random initial data f ω 0 satisfies refined global spacetime estimates. Specifically, for every ω ∈ Ω α and every N ≥ N 0 , we have the finiteness of the quantities F ω (R), F ω ∞ (R) and F ω 2 (R) defined in (2.1.11). The large deviation estimate (2.2.21) for this latter quantity F ω 2 (R) follows from an improved radial Sobolev embedding estimate, which is a weakened version in dimension 3 of the estimate (1.10) from [START_REF] Dodson | Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data[END_REF].

In some sense, it reflects the local energy decay for the linear Schrödinger evolution. Then, we first prove in section 2.3.3 a conditional scattering result, written in Proposition 2.3.8. It states that the nonlinear Duhamel term v scatters in H σ (R 3 ) provided that we have a uniform bound for its H s (R 3 ) norm, on its maximal lifespan. This follows from the stability theory for the perturbed equation (NLS f ), at the subcritical energy scales H s (R 3 ) for 2 3 < s < 1 where one can use the global spacetime bounds provided by the recent progresses collected in Theorem 2.1.2. We avoid the use of (2.2.21) and of the radial assumption in this section, and we restrict the analysis to some ω ∈ Ω α , where we only assume (2.2.12).

The rest of the paper is devoted to the proof of such an a priori bound for v. For this purpose, we fix 6 7 < σ < 1 and v ∈ C(J * ; H σ (R 3 )), the maximal lifespan solution of (NLS f ). Then, we perform a double bootstrap argument that involves a modified energy and a modified interaction Morawetz inequality, together with large deviation estimates (2.2.21) on F ω 2 , that comes from the local energy decay. In section 2.4, we set up the I-method in a stochastic setting. First, we define the modified energy, and compute its increment on some intervals where we assume a smallness condition for the L 4 t,x norm of v. In addition, we establish a modified interaction Morawetz estimate for the frequency truncated solution Iu, where I : H σ (R 3 ) → H 1 (R 3 ) is the I-operator at regularity H σ (R 3 ). Indeed, the usual Morawetz cannot hold for the whole solution, since it requires H 1 2 (R 3 ) regularity. Yet, the general strategy of the globalization argument is to consider Iu instead, solution to the cubic Schrödinger equation with the forcing term H = IN (u) -N (Iu). Namely, H is the commutator between the nonlinearity and I. By doing so, we can exploit the nonlinear smoothing effect yielded by the frequency cancellations in the commutator H. Basically, we can extract a power N -1 from spacetime quantities where H appears, and we write a general key estimate in Lemma 2.4.3 that encapsulate this smoothing effect. Then, we specify this estimate to handle the modified energy increment and the modified Morawetz interaction. Let us now comment a bit on the modified energy we use. It contains two terms, and writes

E(v)(t) := 1 2 R 3 |∇Iv(t, x)| 2 dx + 1 4 R 3 |Iv + If | 4 dx .
The kinetic energy only depends on v, truncated by the I-operator. However, the potential energy depends on f , as in [START_REF] Sun | Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three[END_REF] and [START_REF] Oh | On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities[END_REF]. The motivation for this is to preserve the structure of the unperturbed equation (NLS), and to benefit from the key frequency cancellations. Formally, we have

d dt E(v) = Re R 3 ∂ t (Iv) N (Iu) -IN (u) dx -Re i R 3 ∇(If )N (∇Iu, Iu, Iu)dx .
To control the first term, we exploit the frequency cancellations in the commutator. However, these cancellations do not appear in the second term, since this latter term is purely perturbative. We handle it thanks to local smoothing (2.2.27). Similarly, we write the interaction Morawetz estimate for Iu, and not only Iv, in order to have some perturbation terms that contain H. Note that in the modified energy increment (2.4.11), we use the aforementioned idea from Dodson [Dod13; Dod19b], and we keep track of some sub-additive quantities on the right-hand side as in (2.1.6), not to lose too many powers of N when summing over the spacetime slabs. Finally, we perform the double bootstrap argument in section 2.4 as follows. From the almost conservation of energy, we gain one negative power of N , multiplied by some polynomial powers of N on intervals where the L 4 t,x norm of Iv is small. However, the interaction Morawetz estimate yields a control on such a spacetime norm, that depends on the energy of Iv. Hence, by doing a continuity argument, and after tuning up the parameter σ, we are able to sum the energy increments over a partition of J * made of spacetime slabs where the L 4 t,x norm of Iv is small, and to prove that the energy cannot exceed N 2(1-σ) on the lifespan of v. At the difference of the standard framework, we cannot use the scaling argument in this perturbed setting. Indeed, some perturbation terms coming from

∥∇If ω ∥ L 2 t L 6 x of size N 1-σ 2
appear on the right-hand side of the energy increment estimate, so that the energy cannot remain less than 1 as time evolves. Furthermore, we use the almost-conserved mass, which is harmless, although it is supercritical regarding the scaling.

Organization of the paper The analysis is divided into two parts. In the first one, we prove the conditional scattering result for v, solution to the perturbed equation (NLS f ). In section 2.2, we recall standard large deviation estimates, as well as an improved bound for radially-symmetric randomized initial data. Then, we address the stability theory in section 2.3.3. In the second part of the analysis, we prove the global a priori estimate for v. Namely, we obtain almost conservation laws in section 2.4, used in section 2.5.1 to perform the double bootstrap argument.

Notations

σ is a scaling-subcritical exponent, and s is a scaling-supercritical one. They are linked together by the relation

s < 1 2 < σ < 2s .
We shall keep in mind that the nonlinear Duhamel term is in H σ (R 3 ), whereas the initial data is in H s (R 3 ). The I-operator is defined in (2.1.3), and N is a large dyadic integer. Throughout the analysis, one can keep in mind that the energy is a priori of order

E(v) ≲ N 2(1-s) .
We denote the Duhamel integral by

I([t 0 , t], F ) := -i t t 0 e i(t-t ′ )∆ F (t ′ )dt ′ . (2.1.9)
The nonlinearity is seen as a trilinear operator, and writes N (u (1) , u (2) , u (3) ) := u (1) u (2) u (3) , N (u) := N (u, u, u) .

(2.1.10) Given a time-interval J ⊆ R, a randomized initial data f ω 0 corresponding to some ω ∈ Ω, and its linear evolution at time t denoted f ω (t) = e it∆ f ω 0 , we define

F ω (J) := ∥⟨∇⟩ s f ω ∥ L 10 t,x (J) + ∥⟨∇⟩ s f ω ∥ L 4 t,x (J) + ∥⟨∇⟩ s f ω ∥ L 4 t L 12 x (J) , F ω ∞ (J) := ∥f ω ∥ L ∞ t L 4 x (J) + ∥f ω ∥ L ∞ t L 6 x (J) , F ω 2 (J) := ∥⟨∇⟩If ω ∥ L 2 t L ∞ x (J) + ∥⟨∇⟩If ω ∥ L 2 t L 6
x (J) .

(2.1.11)

Note that the quantity F ω (J) controls the critical norm L 5 t,x , by interpolation between L 4 t,x and L 10 t,x . We will often drop the dependence on ω from the notations. Some large deviation estimates on these terms are proved in section 2.2. 4 In addition, C is a constant that is irrelevant, and that may change from line to line. We may also write ≲. Finally, we denote by P K the Littlewood Paley multiplier around the dyadic frequency of size K = 2 k :

supp P K f ⊂ ξ ∈ R 3 | 2 k-1 ≤ |ξ| ≤ 2 k+1 .
The counterpart in the physical space is χ j : x (R 3 ). We have

supp χ j f ⊂ x ∈ R 3 | 2 j-1 ≤ |x| ≤ 2 j+1 . The Wiener multiplier around k ∈ Z 3 is denoted by Q k , and supp Q k f ⊂ k + [-1 , 1] 3 .

Preliminaries and probabilistic estimates

∥ e it∆ u 0 ∥ L q t (R;L r x (R 3 )) ≲ ∥u 0 ∥ L 2 x (R 3 ) ,
for any Strichartz admissible pair

(q, r) ∈ [2 , +∞], s(q, r) = 2 q + d 1 r - 1 2 = 0 .
Definition 2.2.2 (Functions of bounded 2-variation V 2 and atomic space U 2 ). Let J ⊆ R be a time-interval, and Z be the collection of every finite partitions of J. The set of functions with bounded 2-variation V 2 (J) is the set of functions v : J → L 2 x (R 3 ), endowed with the norm

∥v∥ V 2 (J) := sup {t k } K-1 k=0 ∈Z K k=1 ∥v(t k ) -v(t k-1 )∥ 2 L 2 x 1 2
. 4 We might also incorporate the L 10 3

t,x norm in the definition of F ω (J), which corresponds to the Strichartz admissible pair ( 10 3 , 10 3 ). Indeed, at some point in the analysis, we will need to consider some spacetime slabs J × R 3 for which this quantity is small.

A function a : J → L 2

x (R 3 ) is an atom if there exists a partition {t k } k=0...K in Z and {ϕ k } k=0,...,K-1 some elements in L 2

x such that

a(t) = K k=1 1 [t k-1 ,t k ) (t)ϕ k-1 , K-1 k=0 ∥ϕ k ∥ 2 L 2
x ≤ 1.

The atomic space U 2 (J) is the set of functions u : J → L 2 x (R 3 ) endowed with the norm

∥u∥ U 2 (J) := inf ∥(λ j )∥ ℓ 1 | u = j≥1
λ j a j for some U 2 -atoms (a j ) .

(2.2.1) Definition 2.2.3 (Function spaces adapted to the free Schrödinger evolution). Spaces adapted to the linear propagator e it∆ are the Banach spaces endowed with the norms

∥u∥ U 2 ∆ (J) := ∥ e it∆ u∥ U 2 (J) , ∥v∥ V 2 ∆ (J) := ∥ e it∆ v∥ V 2 (J) .
Then, given σ ∈ R, the critical spaces X σ (J) and Y σ (J) are the Banach spaces endowed with the norms

∥u∥ 2 X σ (J) = K∈2 N K 2σ ∥P K u∥ 2 U 2 ∆ (J) , ∥v∥ 2 Y σ (J) = K∈2 N K 2σ ∥P K v∥ 2 V 2 ∆ (J) . (2.2.2)
Proposition 2.2.4 (Embeddings, [HHK09] Proposition 2.2 and Corollary 2.6 5 ).

U 2 ∆ H σ x → X σ (J) → Y σ (J) → V 2 ∆ H σ x → L ∞ t (J, H σ x ) . (2.2.3) Proposition 2.

(Duality (see [HHK09] and [HTT11])

). There exists a unique bilinear map B , :

U 2 × V 2 → C such that v ∈ V 2 → B(•, v) ∈ (U 2 ) *
is a surjective isometry, and

∥v∥ V 2 = sup ∥u∥ U 2 ≤1 | B(u, v)|, ∥u∥ U 2 = sup ∥v∥ V 2 ≤1 | B(u, v)|. If ∂ t u ∈ L 1 t (J; L 2 x (R 3 )) we have the explicit formula B(u, v) = I (∂ t u | v) L 2 x (R 3 ) dt.
In particular, for the Duhamel term (2.1.9) and when F ∈ L 1 t (J; L 2 x (R 3 )), we have

∥I(J, F )∥ U 2 ∆ ≲ sup ∥v∥ V 2 ∆ ≤1 J×R F vdxdt .
(2.2.4) 5 The spaces U 2 ∆ H σ and V 2 ∆ H σ are generalizations of the spaces U 2 ∆ and V 2 ∆ H σ , with functions u, v : J → H σ (R 3 ) instead of L 2

x (R 3 ).

Proposition 2.2.6 (Transferred linear and bilinear estimates, Lemma 3.3 in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]). Let (q, r) be an admissible pair as in Proposition 2.2.1. We have

∥u∥ L q t (R;L r x (R 3 )) ≲ ∥u∥ U 2 ∆ .
(2.2.5)

Let K, M be two dyadic integers. The bilinear estimate from Bourgain

∥ e it∆ P K u 0 e it∆ P M u 0 ∥ L 2 t,x (R×R 3 ) ≲ KM -1 2 ∥P ≤K u 0 ∥ L 2 x ∥P M v 0 ∥ L 2 x (2.2.6)
has transferred versions to the context of the spaces U 2 and V 2 , that read

∥P K uP M v∥ L 2 t,x (R×R 3 ) ≲ KM -1 2 ∥P K v∥ U 2 ∆ ∥P M v∥ U 2 ∆ , ∥P K uP M v∥ L 2 t,x (R×R 3 ) ≲ KM -1 2 + ∥P K v∥ V 2 ∆ ∥P M v∥ V 2 ∆ .
(2.2.7) Lemma 2.2.7 (Time continuity and sub-additivity, see Lemma A.4 and A.8 in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]).

Let J = [a , b) and u ∈ X σ (J) ∩ C(J; H σ x (R 3 )). The mapping t ∈ J → ∥u∥ X σ [a,t) is continuous. Moreover, given a partition J = L k=1 J k , we have ∥u∥ X σ (J) ≤ L k=1 ∥u∥ X σ (J k ) .

Large deviation estimates for the free Schrödinger evolution

First, we recall the following probabilistic decoupling estimate. It expresses the gain of integrability that emerges from probabilistic averaging effects, due to the cancelation of interferences.

Lemma 2.2.8 (Lemma 3.1 in [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF]). Let {g n } be a sequence of real valued, zero-mean and independent random variables with distribution {µ n } on a probability space Ω, A, P . Assume that there exists c > 0 such that for any γ ∈ R and any n ≥ 0 we have

R e γx dµ n (x) ≤ e cγ 2 .
Then, there exists C > 0 such that for every 2 ≤ p ≤ ∞ and for every sequence of complex numbers {c n } in ℓ 2 (N; C) it holds that n∈N c n g n (ω)

L p ω (Ω) ≤ C √ p n∈N |c n | 2 1 2 . ( 2

.2.8)

From this, we obtain the general large deviation estimate, on which rely all the estimates stated in this section. Lemma 2.2.9 (Large deviation estimate). Let F be a real valued measurable function on a probability space Ω, A, P . Assume that there exist some constants C 0 > 0, M > 0 and p 0 ≥ 1 such that for all p ≥ p 0 we have

∥F ∥ L p ω (Ω) ≤ C 0 √ pM .
(2.2.9)

Then, there exist C = C(C 0 ) and c = c(C 0 ) > 0 independent of M , such that for all λ > 0,

P ω ∈ Ω | |F (ω)| > λ ≤ C exp -cλ 2 M -2 .
(2.2.10)

In the following lemma, we recall the probabilistic Strichartz estimates, that provide improved global spacetime bounds by loosening the Strichartz admissibility condition. Lemma 2.2.10 (Improved Strichartz estimates). Let (q, r) be a Strichartz admissible pair, i.e s(q, r) = 0. For all r ≥ r, there exist C = C(q, r) > 0 and c = c(q, r) > 0 such that for all f 0 ∈ H s (R 3 ) and λ > 0, we have

P ω ∈ Ω | ∥⟨∇⟩ s e it∆ f ω 0 ∥ L q t (R;L r x (R 3 ) > λ ≤ C exp -cλ 2 ∥f 0 ∥ -2 H s . ( 2

.2.11)

Proof. The proof is standard, and we refer to [BOP15a; BOP15b]. To prove (2.2.9) for p > max(q, r), we apply Minkowski inequality and the decoupling estimate (2.2.8). Then, we apply the unit-scale Bernstein estimate on each frequency bloc, that yields a bound on the Fourier multiplier Q k from L r x to L r x , uniformly in k. The Strichartz estimate applied with (q, r) concludes.

Next, we state a uniform in time large deviation estimate for the linear evolution of randomized initial data. Lemma 2.2.11. Let f 0 in H s (R 3 ) for some s > 0. For all q > 2, there exist C, c > 0 such that for all λ > 0,

P ω ∈ Ω | ∥ e it∆ f ω 0 ∥ L ∞ t (R;L q x (R 3 )) > λ ≤ C exp -cλ 2 ∥f 0 ∥ -2 H s x .
Proof. See Lemma 5.15 in [START_REF] Dodson | Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation[END_REF]. This follows from a Sobolev embedding in time, and from the improved Strichartz estimate stated in the previous Lemma.

We deduce from these large deviation bounds, together with lemma 2.2.9, the following spacetime estimates on f ω , for some ω in a large measure set. Lemma 2.2.12. For every α > 0 there exist a set Ω α and a constant C α depending on ∥f 0 ∥ H s such that P(Ω \ Ω α ) ≤ α, and for every ω ∈ Ω α we have

F ω (R) + F ω ∞ (R) ≤ C α . ( 2

.2.12)

In what follows, ω ∈ Ω α is fixed such that (2.2.12) holds. At this stage, we do not need the radial assumption on f 0 .

Additional estimate for randomized radial initial data

The following improved Sobolev embedding for radial functions was introduced in [DLM20], and used in [START_REF] Dodson | Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation[END_REF] in order to prove almost-sure scattering for the energy critical Schrödinger and wave equation in R 4 . Here, we present an analog of this lemma in dimension 3, whose proof needs some slight modifications. Lemma 2.2.13 (Improved Sobolev embedding for radial functions in R 3 ). For all δ > 0, there exist C δ > 0 such that for all radially-symmetric function f 0 in H δ (R 3 ) we have

⟨x⟩ k∈Z 3 |Q k f 0 | 2 1 2 L ∞ x (R 3 ) ≤ C δ ∥f 0 ∥ H δ (R 3 ) . ( 2

.2.13)

Proof. Up to changing the coordinates we assume that x = |x|e 3 , where e 3 = (0, 0, 1) and |x| ≫ 1. We denote by (ρ, θ, φ) the spherical coordinates in the Fourier space R 3 ξ : The function ψ k and its derivatives provide a localization on the unit cube centered around k. More precisely, this localization reads in spherical coordinates as follows : if ξ(ρ, θ, φ) is in supp(ψ k ), then

ξ = ξ(ρ, θ, φ) = ρ cos(φ) sin(θ), ρ sin(φ) sin(θ), ρ sin(φ) cos(θ) , with ρ > 0, θ ∈ (0 , π) and φ ∈ [0 , 2π]. Given k ∈ Z 3 such that 1 ≪ |k|, we write Q k f (x) = R 3 e iξ•x ψ k (ξ) f (ξ)dξ = +∞ 0 2π 0 π 0 e i|x|ρ cos θ ψ k (ξ(ρ, θ, φ)) sin θdθ dφρ 2 f 0 (ρ)dρ .
• ρ is in an interval of size ∼ 1 around |k|.

• θ in an interval of size ∼ 1 |k| around the angle θ k , where sin

(θ k ) ∼ 1 -k |k| e 3 2 .
• φ is in an interval J k of size ∼ min(1,

1 |k| sin (θ k )
). To gain decay with respect to x, we integrate by parts in the variable θ. It holds

I(x, ρ, φ) = - 1 i|x|ρ π 0 ∂ θ e iρ cos θ|x| ψ k (ξ(ρ, θ, φ))dθ = - 1 i|x|ρ e iρ cos θ|x| ψ k (ξ(ρ, θ, φ)) π 0 + 1 i|x|ρ π 0 e iρ cos θ|x| ∂ θ φ k (ξ(ρ, θ, φ))dθ . (2.2.16) Moreover, we have for all ξ(ρ, θ, φ) ∈ R 3 , all x ∈ R 3 and k ∈ Z 3 that | ∂ ∂ θ ψ k (ξ)| ≤ ρ|∇ ξ ψ k (ξ)| . ( 2

.2.17)

Keeping in mind the localization in the variables ρ, θ yielded by the amplitude ψ k and its derivatives, we deduce from (2.2.16) and (2.2.17) that

|I(x, ρ, φ)| ≲ 1 |x|ρ 1 (|k|-1,|k|+1) (ρ)1 J k (φ) . (2.2.18)
Using that φ is localized on the interval J k of size min(1, 1 |k| sin θ k ), integrating over the domain [|k| -1 , |k| + 1] × J k and using Cauchy-Schwarz in ρ, we get

|Q k (f 0 )(x)| ≲ 1 |k||x| min(1, 1 |k| sin θ k )∥1 [|k|-1,|k|+1] (ρ) f 0 (ρ)ρ 2 ∥ L 1 ρ (R) ≲ min(1, 1 |k| sin θ k )∥1 [|k|-1,|k|+1] (ρ) f 0 (ρ)ρ∥ L 2 ρ (R) .
Summing the above estimate over Z 3 gives

|x| 2 k∈Z 3 |Q k (f 0 )(x)| 2 ≲ k∈Z 3 min(1, 1 |k| sin θ k ) 2 ∥1 [|k|-1,|k|+1] (ρ) f 0 (ρ)ρ∥ 2 L 2 ρ (R) ≲ j∈N j l=0 # k ∈ Z 3 | |k| = j, |j sin(θ k ) -l| ≤ 1 (1 + l) 2 ∥1 [j-1,j+1] (ρ) f 0 (ρ)ρ∥ 2 L 2 ρ (R) ≲ j∈N j l=0 1 1 + l ∥1 [j-1,j+1] (ρ) f 0 (ρ)ρ∥ 2 L 2 ρ (R) ≲ j≥1 log(1 + j)∥1 [j-1,j+1] (ρ) f 0 (ρ)ρ∥ 2 L 2 ρ (R) ≲ δ ∥f 0 ∥ 2 H δ (R 3 ) .
This ends the proof of Lemma 2.2.13.

Corollary 2.2.14. Let 2 ≤ r ≤ ∞, K ∈ 2 N and δ > 0. There exists a constant C δ > 0 such that for any radial function f 0 ∈ L 2 (R 3 ) we have

⟨x⟩ 1-2 r k∈Z 3 |Q k P K f 0 | 2 1 2 L r x (R 3 ) ≤ C δ K δ ∥P K f 0 ∥ L 2 x (R 3 ) . ( 2
∥ k∈Z 3 |Q k P K f 0 | 2 1 2 ∥ L 2 x (R 3 ) ≤ ∥P K f 0 ∥ L 2 x (R 3 ) .
It holds

⟨x⟩ 1-2 r k∈Z 3 |Q k P K f 0 | 2 1 2 L r x ≤ ⟨x⟩ k∈Z 3 |Q k P K f 0 | 2 1 2 1-2 r L ∞ x k∈Z 3 |Q k P K f 0 | 2 1 2 2 r L 2 x ≤ C δ ∥P K f 0 ∥ 1-2 r H δ ∥P K f 0 ∥ 2 r L 2 x ≲ C δ K δ ∥P K f 0 ∥ L 2 x .
We will use the local smoothing estimate under the following form.

Lemma 2.2.15 (Local smoothing, [START_REF] Constantin | Local smoothing properties of dispersive equations[END_REF]). For any d ≥ 3 and α > 0, there exists a constant C such that for all

f 0 ∈ L 2 (R d ), sup R>0 ∥ e it∆ f 0 ∥ L 2 t,x (R×{|x|<R}) ≤ CR 1 2 ∥|∇| -1 2 f 0 ∥ L 2 x (R d ) .
(2.2.20)

Recall that the Fourier multiplier I : H σ x → H 1 x was defined in (2.1.3) Proposition 2.2.16. Let s > 1 4 and σ > 1 2 such that σ < 2s. Then, for all r ∈ (4 , +∞] and δ > 0, there exist N 0 , C > 0 and c > 0 depending on r, δ, s, σ such that for all λ > 0 and N ≥ N 0 , we have the large deviation estimates

P ω ∈ Ω | ∥∇I e it∆ f ω 0 ∥ L 2 t (R;L r x (R 3 )) > λ ≤ C exp -cλ 2 (N -δ N 1-σ 2 ∥f 0 ∥ H s x ) -2 .
(2.2.21)

Proof. It suffices to prove that for any δ > 0 small enough (depending on s and σ), there exists a constant C δ such that for any p > r one has

∥∇I e it∆ f ω 0 ∥ L p ω L 2 t L r x ≤ C δ √ pN -δ N 1-σ 2 ∥f 0 ∥ H s x (R 3 ) . ( 2
:= ∥∇I e it∆ f ω 0 ∥ L 2 t (R;L r x (R 3 )) and M = N 0-N 1-σ 2 ∥f 0 ∥ H s x .
To prove (2.2.22), we dyadically decompose f 0 in the frequency space. We reduce the case when r = ∞ to the case where r is finite by applying the Bernstein estimate. For any 3 < r < ∞, we get

∥∇ e it∆ If ω 0 ∥ L p ω L 2 t L r x ≲ ∥∇P ≤1 e it∆ f ω 0 ∥ L p ω L 2 t L r x + K∈2 N * K 3 r ∥∇P K I e it∆ f ω 0 ∥ L p ω L 2 t L r x . (2.2.23)
Estimating the low-frequency term is standard, and we refer to [START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF]. However, to handle the high-frequency terms, we used the radial Sobolev embedding (2.2.19) and the local smoothing estimate. Let us fix K ≥ 2 a dyadic integer. Then, we dyadically decompose the term on right-hand side of (2.2.23) in the physical space, with some cutoff χ j introduced in the notations paragraph. By the triangle inequality, we have

∥∇P K I e it∆ f ω 0 ∥ L p ω L 2 t L r x ≤ j≥0 ∥χ j ∇P K I e it∆ f ω 0 ∥ L p ω L 2 t L r x .
Next, for each j ≥ 0, and p ≥ r, we use Minkowski's inequality and the decoupling estimate (2.2.8) to get

∥χ j ∇P K I e it∆ f ω 0 ∥ L p ω L 2 t L r x ≲ √ p k∈Z 3 |k|∼K |χ j Q k ∇P K I e it∆ f 0 | 2 1 2 L 2 t L r x .
We split the sum into a localized term, and a remainder.

∥χ j ∇P K I e it∆ f ω 0 ∥ L p ω L 2 t L r x ≲ √ p k∈Z 3 |k|∼K |χ j Q k χ ≤j+5 ∇P K I e it∆ f 0 | 2 1 2 L 2 t L r x + √ p k∈Z 3 |k|∼K |χ j Q k χ >j+5 ∇P K I e it∆ f 0 | 2 1 2 L 2 t L r x .
To estimate the remainder, we refer to [START_REF] Dodson | Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation[END_REF]. The proof is a bit technical but only uses Young inequality the local smoothing estimate (2.2.20) and some operator bounds collected in the following lemma that holds for dimension 3 without changing the proof.

Lemma 2.2.17 (Lemma 5.10 and Lemma 5.11 from [START_REF] Dodson | Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation[END_REF]). Let 2 ≤ r ≤ ∞. For any k ∈ Z 3 , any dyadic J, L ∈ 2 N with 2 5 J < L, and any ν > 0, there exists C ν > 0 such that

∥χ j Q k χ l ∥ L 2 x (R 3 )→L r x (R 3 ) ≤ C ν L -ν . (2.2.24)
Similarly, given L ∈ 2 N and k, m ∈ Z 3 such that |k -m| ≥ 100, it holds that

∥Q k χ l Q m ∥ L 2 x (R 3 )→L r x (R 3 ) ≤ C ν L -α |k -m| -ν . (2.2.25)
However, we detail how to handle the main term, since this requires to combine the local smoothing estimate (2.2.20) with the improved radial Sobolev embedding (2.2.13). 6 Given j ≥ 0, we have

k∈Z 3 |k|∼K |χ j Q k χ ≤j+5 ∇P K I e it∆ f 0 | 2 1 2 L 2 t L r x ≲ 2 -j(1-2 r ) ⟨x⟩ 1-2 r k∈Z 3 |k|∼K |Q k χ ≤j+5 ∇P K I e it∆ f 0 | 2 1 2 L 2 t L r x .
By the improved radial Sobolev embedding (2.2.19), we have

⟨x⟩ 1-2 r k∈Z 3 |k|∼K |Q k χ ≤j+5 ∇P K I e it∆ f 0 | 2 1 2 L 2 t L r x ≲ δ K δ ∥χ ≤j+5 |∇|P K I e it∆ f 0 ∥ L 2 t L 2
x .

Next, we apply the local smoothing estimate (2.2.20) to see that the above line is controlled by

k∈Z 3 |k|∼K |χ j Q k χ ≤j+5 ∇P K I e it∆ f 0 | 2 1 2 L 2 t L r x ≲ δ 2 -j(1-2 r -1 2 ) K 1 2 +δ ∥P K If 0 ∥ L 2 x .
The above expression can be summed over j provided that r > 4. In addition, we observe that if K > 2N , there exists γ > 0 such that

K 1 2 +δ ∥P K If 0 ∥ L 2 x ≤ N 1-σ K σ-1 2 -s+δ ∥P K f 0 ∥ H s x .
Indeed, Plancherel estimate and the definition of the I-multiplier (2.1.3) yield

K 1+2δ ∥P K If 0 ∥ 2 L 2 x = K 1+2δ 2 -1 K<|ξ|<2K N |ξ| 2(1-σ) | P K f 0 (ξ)| 2 dξ ≲ N 2(1-σ) K 2(σ-1)+1-2s+2δ ∥P K f 0 ∥ 2 H s x , 6
The case when j = 0 is actually easier and does not require to use the improved radial Sobolev embedding. We refer to the original proof in [START_REF] Dodson | Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data[END_REF] where this term is handled separately.

When K ≤ 2N , we directly get that

∥|∇| 1 2 +δ P K If 0 ∥ L 2 x ≤ N 1 2 -s+δ ∥P K f 0 ∥ H s x .
We now sum over K in estimate (2.2.23) and get that for all r > 4 and δ > 0, 7

∥∇ e it∆ IF ∥ L p ω L 2 t L ∞ x ≲ δ,r √ p∥P ≤ 1f 0 ∥ L 2 x + √ p K∈2 N * K≤2N K 1 2 -s+δ ∥P K f 0 ∥ H s x + √ pN 1-σ K∈2 N * K>2N K σ-1 2 -s+δ ∥P K f 0 ∥ H s x . Define γ 0 (s, σ) = s -σ 2 > 0 such that 1 2 -s = 1-σ 2 -γ 0 .
8 Next, we chose δ and γ 0 such that

σ - 1 2 -s + δ < 0, and 1 2 -s ≤ 1 -σ 2 -γ 0 .
(2.2.26) By using Cauchy Schwarz, we conclude that

∥∇I e it∆ f ω 0 ∥ L p ω L 2 t L ∞ x ≲ δ,r √ pN 1 2 -s+δ ∥P K f 0 ∥ H s x .
This ends the proof of Proposition 2.2.16.

We deduce from the above large deviation estimates the following bounds on the linear evolution of the randomized initial data. Proposition 2.2.18. For all α > 0, there exist a set Ω α ⊂ Ω α , with P(Ω α \ Ω α ) ≤ α, a constant C α > 0 and N 0 (α) such that for all ω ∈ Ω α , and all N ≥ N 0 we have

F ω 2 (R) ≤ C α N 1-σ 2 . ( 2
λ = C α N 0-N 1-σ 2 ∥f 0 ∥ H s x , C 2 α ≥ c -1 ln(Cα -1 ) .

Cauchy theory for the forced NLS equation

The main step of this section is to come with a conditional scattering result for the solutions v to the perturbed Schrödinger equation with a random forcing term (NLS f ). More precisely, we prove that an a priori uniform estimate for the H σ (R 3 ) norm of v on its maximal lifespan yields global existence and scattering, when 2 3 < σ ≤ 1. To prove such a result, we essentially follow the same lines as in the proof of Proposition 3.1 from [START_REF] Killip | Almost sure scattering for the energycritical NLS with radial data below H 1 (R 4 )[END_REF]. However, since v lies below the energy space, 7 In the case when we look at the L 2 t L ∞ x norm, we need to add a K to the power 3 r , for r arbitrarily large, that comes from the Sobolev embedding from W 3 r ,r (R 3 ) to L ∞ (R 3 ). This term is harmless, and can be absorbed in the residual power K δ .

8 Recall the assumption that s ≤ 1 2 and σ < 2s.

we need to develop a stability theory in H σ (R 3 ), and not in the energy space. To achieve such a goal, we appeal to the global Cauchy theory in such subcritical regimes provided by [START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF][START_REF] Dodson | Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when n = 3 via a linear-nonlinear decomposition[END_REF], that provides the spacetime global estimate

∥u∥ L 5 t,x ≤ C(∥u(t 0 )∥ H σ (R 3 )) .
for solutions u to (NLS) with u(t 0 ) ∈ H σ (R 3 ). With this global spacetime bound at hand, we can use sub-additivity and reduce the analysis to intervals J where we have a smallness assumption on the L 5 t,x (J) norm of u, which is a scaling-critical quantity, and hence on v which is expected to stay close to u in H σ (R 3 ). For this reason, we are led to refine the trilinear stochastic estimates from [START_REF] Bényi | Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on R 3[END_REF], and to estimate the Duhamel nonlinear term not only by the critical norm X σ (J), which is not small, but also by some powers of the L 5 t,x (J) norm of v to gain smallness. This is the matter of the refined nonlinear estimates from Propositions 2.3.2 and 2.3.3. Throughout this section, we fix α > 0 and ω ∈ Ω α such that

F ω (R) ≤ C α .
In particular, we do not use the quantity F ω 2 (R), so that we can release the radial assumption on f 0 .

Nonlinear estimates

Before diving into the nonlinear analysis, let us detail how to address the Littlewood-Paley summation of L q norms of dyadic blocs u N , for a given function u that comes with a gain of regularity materialized by the presence of a negative power of N in front of each paraproduct u N .

Observation 2.3.1. Let q > 2 and u in L q t (J; W γ,q x (R 3 )) for some γ ∈ R. We denote by u ∼ N ∈2 N u N the Littlewood-Paley decomposition of u. It holds

N ∈2 N N 0-N γ ∥u N ∥ L q t,x (J×R 3 ) ≲ ∥u∥ L q t (J;W γ,q
x (R 3 )) .

(2.3.1)

Proof. From Hölder, we have

N ∈2 N N 0-N γ ∥u N ∥ L q t,x (J×R 3 ) ≤ N ∈2 N N 0-1 q ′ N ∈2 N N γ ∥u N ∥ L q t,x (J×R 3 ) q 1 q ≲ ∥N γ u N ∥ ℓ q N (2 N ;L q t,x (J×R 3 )) = ∥N γ u N ∥ L q t,x (J×R 3 ;ℓ q N (2 N )) .
Since q ≥ 2, we obtain

∥N γ u N ∥ L q t,x (J×R 3 ;ℓ q N (2 N )) ≤ ∥N γ u N ∥ L q t,x (J×R 3 ;ℓ 2 N (2 N )) .
We conclude the proof of (2.3.1) by applying the Littlewood-Paley square function theorem

∥u N ∥ L q t,x (J×R 3 ;ℓ 2 N (2 N )) = N ∈2 N |u N | 2 1 2 L q t,x
≲ ∥u∥ L q t,x (J×R 3 ) . Now, we establish quadrilinear estimates that involve three types of terms. First, we have the stochastic forcing terms of type f = e it∆ f ω 0 , for which we obtained in Proposition 2.2.10 improved Strichartz estimates. More precisely, we have that F ω (R) < +∞, and using sub-additivity, we will reduce the analysis to a finite number of intervals where F ω (J) is small. Then, we have the term v solution to the forced Schrödinger equation (NLS f ) at the subcritical regularity H σ (R 3 ), and that corresponds to the nonlinear Duhamel term for the solution u = e it∆ f ω 0 + v. We want to obtain a priori estimates for v in the spacetime spaces X σ and L 5 t,x , at least, on some intervals where there holds a smallness assumption on the forcing term. Finally, the terms w ∈ Y 0 that appear in the analysis come from duality.

Proposition 2.3.2 (Trilinear estimates with random terms). Let 1 4 < s ≤ 1 2 and 1 2 < σ < 2s. Denote by f ω = e it∆ f ω 0 the linear evolution of the randomized initial data. There exists a constant C(∥f 0 ∥ H σ ) > 0 such that for all interval J ⊆ R where F ω (J) ≤ 1, and all v ∈ X σ (J) with ∥v∥ L 5 t,x (J) ≤ 1, it holds

∥I •, N (f, f, f ) ∥ X σ (J) ≤ C(∥f 0 ∥ H σ )F ω (J) , (2.3.2) ∥I •, N (f, f, v) ∥ X σ (J) ≤ C(∥f 0 ∥ H σ )F ω (J)∥v∥ X σ (J) , (2.3.3) ∥I •, N (f, v, v) ∥ X σ (J) ≤ C(∥f 0 ∥ H σ )F ω (J) 1 2 ∥v∥ L 5 t,x (J) ∥v∥ X σ (J) . (2.3.4)
In particular, the above trilinear estimates yield the following estimate for the forcing term of (NLS f ) at the regularity H σ

x : there exists

C = C(∥f 0 ∥ H s ) > 0 such that ∥I •, N (v+f )-N (v) ∥ X σ (J) ≲ C(∥f 0 ∥ H s )F ω (J)+C F ω (J)+F ω (J) 1 2 ∥v∥ L 5 t,x (J) ∥v∥ X σ (J) . (2.3.5)
Proof. In the following, we take all the spacetime norms over J × R 3 . Using the duality between U 2 and V 2 (see Proposition 2.2.5), we have 9

∥v∥ 2 X σ (J) ≤ N I 2 N ,
where, denoting w (i) ∈ {v, f } for i ∈ {1, 2, 3}, we have that for all v ∈ X σ (J)

I N := N σ ∥P N v∥ U 2 ∆ (J) ≤ CN σ sup ∥w∥ V 2 ∆ (J) ≤1 J×R 3 P N w (1) w (2) w (3) wdxdt .
Given a fixed function w with ∥w∥ V 2 ∆ (J) ≤ 1, we perform a Littlewood-Paley decomposition of each term 10 and we are left to estimate quadrilinear spacetime integrals of the form

I N ≤ L=(N 1 ,N 2 ,N 3 ) N σ (1) J×R 3 w (1) N 1 w (2)
N 2 w

(3)

N 3 w N dxdt =: L=(N 1 ,N 2 ,N 3 ) N σ (1) I N (L) . ( 2

.3.6)

9 To apply Proposition 2.2.5 and to be in position to apply formula (2.2.4), we need to make sure that N (u) is in L 1 t (J; H σ x ). We omit this short verification which is essentially contained in the analysis, and we refer to [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF] where the authors prove that P ≤N N (u) ∈ L 1 t (J; H σ x ) for any N ∈ 2 N . 10 Note that the sums over N 1 , N 2 , N 3 under the integral are absolutely convergent in C. Hence, we can intertwine the sum and the integral, and then apply the triangle inequality to obtain, (2.3.6).

For dyadic integers N 1 , N 2 , N 3 , we denote by N (1) ≥ N (2) ≥ N (3) the non-increasing ordering among them, and we use the shorthand notation w N i := P N i w (i) , i ∈ {1, 2, 3}. Note that it suffices to consider the nontrivial cases, where N ≲ N (1) . To be able to sum the terms I N over N , we need to gain a negative power of N after summing I L N over L. Hence, we shall bound the above quadrilinear integrals by some negative power of the highest frequency N (1) , and by some appropriate norms of each w i . Then, we proceed as in observation 2.3.1.

Proof of (2.3.2) Here we address the first Duhamel iteration, that contains products with three random terms

I N (N 1 , N 2 , N 3 ) := -i t 0 e i(t-t ′ )∆ N (f N 1 , f N 2 , f N 3 )(t ′ )dt ′ .
By symmetry, we can assume that N 1 ≥ N 2 ≥ N 3 without loss of generality.

• Case 1 : High-high-high N 1 2 1 ≲ N 3 . Applying Hölder with 1 = 3 10 + 7 10 yields

N σ 1 |I N (N 1 , N 2 , N 3 )| = N σ 1 J×R 3 f N 1 f N 2 f N 3 w N dxdt ≲ N σ-s 1 N -s 2 N -s 3 1≤j≤3 ∥⟨∇⟩ s f N j ∥ L 30 7 t,x ∥w N ∥ L 10 3 t,x ≲ N σ-2s 1 1≤j≤3 ∥⟨∇⟩ s f N j ∥ L 10 3 t,x ∥w N ∥ V 2 ∆ .
Recall that σ < 2s. Hence, we have a negative power of the highest frequency, and we can proceed as in observation 2.3.1 to obtain

N 1 ,N 2 ,N 3 ,N N σ 1 |I N (N 1 , N 2 , N 3 )| ≲ N N 0-∥⟨∇⟩ s f ∥ 3 L 10 3 t,x ∥w N ∥ V 2 ∆ .
When we eventually sum over N , we apply Cauchy-Schwarz and use that ∥v∥ Y 0 ≤ 1 to conclude that

N N 0-∥w N ∥ V 2 ∆ ≲ N ∥w N ∥ 2 V 2 ∆ 1 2 ≲ 1 .
We conclude similarly in the other cases.

• Case 2 : High-low-low N 2 ≤ N 1 2
1 and N ∼ N 1 . We apply Cauchy-Schwarz, we use interpolation and the bilinear estimate (2.2.6) in its transferred version (2.2.7) to obtain

N σ 1 |I N (N 1 , N 2 , N 3 )| ≲ N σ 1 ∥f N 1 f N 2 ∥ 1 2 L 2 t,x ∥f N 1 ∥ 1 2 L 4 t,x ∥f N 2 ∥ 1 2 L 4 t,x ∥f N 3 w N ∥ L 2 t,x ≲ N σ-1 4 1 N 1 2 2 N -1 2 +0 N 3 ∥f N 1 (0)∥ 1 2 L 2 x ∥f N 2 (0)∥ 1 2 L 2 x ∥f N 1 ∥ 1 2 L 4 t,x ∥f N 2 ∥ 1 2 L 4 t,x ∥f N 3 (0)∥ L 2 x ∥w N ∥ V 2 ∆ ≲ N σ-1 4 -s 1 N 1 2 -s 2 N -1 2 +0 N 1-s 3 ∥f N 1 (0)∥ 1 2 H s x ∥f N 2 (0)∥ 1 2 H s x ∥⟨∇⟩ s f N 1 ∥ 1 2 L 4 t,x ∥⟨∇⟩ s f N 2 ∥ 1 2 L 4 t,x ∥f N 3 (0)∥ H s x ∥w N ∥ V 2 ∆ . Using that s < 1 2 , N 2 ≤ N 1 2 1 and N 3 ≤ N 1 2 1 ∼ N 1 2 , we have N σ-1 4 -s 1 N 1 2 -s 2 N -1 2 +0 N 1-s 3 ≲ N σ-2s+0 1 .
Then, as in the first case, we then use that σ < 2s and we sum over the N j 's to get

N 1 ,N 2 ,N 3 ,N N σ 1 |I N (N 1 , N 2 , N 3 )| ≲ ∥⟨∇⟩ s f ∥ L 4 t,x ∥f 0 ∥ 2 H s x .
• Case 3 : High-high-low

N 3 ≤ N 1 2 1 ≤ N 2 .
Similarly, it follows from Hölder's inequality, from the bilinear estimate (2.2.6) and from the Strichartz embedding (2.2.5) that

N σ 1 |I N (N 1 , N 2 , N 3 )| ≲ N σ 1 ∥f N 1 f N 3 ∥ L 2 t,x ∥f N 2 ∥ L 5 t,x ∥w N ∥ L 10 3 t,x ≲ N σ-1 4 -s 1 N 1 2 -s 3 ∥f N 1 (0)∥ 1 2 H s x ∥f N 3 (0)∥ 1 2 H s x ∥⟨∇⟩ s f N 1 ∥ 1 2 L 4 t,x ∥⟨∇⟩ s f N 3 ∥ 1 2 L 4 t,x N -s 2 ∥⟨∇⟩ s f N 2 ∥ L 5 t,x ∥w N ∥ V 2 ∆ ≲ N σ-2s 1 ∥f N 1 (0)∥ 1 2 H s x ∥f N 3 (0)∥ 1 2 H s x ∥⟨∇⟩ s f N 1 ∥ 1 2 L 4 t,x ∥⟨∇⟩ s f N 3 ∥ 1 2 L 4 t,x N -s 2 ∥⟨∇⟩ s f N 2 ∥ L 5 t,x ∥w N ∥ V 2 ∆ .
Summing over the different dyadic integers yields

N 1 ,N 2 ,N 3 ,N N σ 1 |I N (N 1 , N 2 , N 3 )| ≲ ∥⟨∇⟩ s f ∥ L 4 t,x ∥⟨∇⟩ s f ∥ L 5 t,x ∥f 0 ∥ H s x .
Proof of estimates (2.3.3), (2.3.4) with mixed terms Here we consider the case when at least one term is random, say w (1) = f , and at least another one is deterministic, say w (2) = v. As for the last term w (3) ∈ {f, v}, we always place it in L 5 t,x (J × R 3 ). • Case 1 : One deterministic term comes with the highest frequency Without loss of generality, we assume that N (1) = N 2 . By applying Hölder's inequality, we get

N σ 2 |I N (N 1 , N 2 , N 3 )| = N σ 2 J×R 3 f N 1 v N 2 w
(3)

N 3 w N dxdt ≲ N σ 2 ∥f N 1 v N 2 ∥ L 2 t,x ∥w (3)
N 3 ∥ L 5 t,x ∥w N ∥ L 10 3 t,x
.

The transferred bilinear estimate and Strichartz embedding yield

N σ 2 |I N (N 1 , N 2 , N 3 )| ≲ N σ 2 ∥f N 1 v N 2 ∥ 1 2 L 2 t,x ∥f N 1 ∥ 1 2 L 5 t,x ∥v N 2 ∥ 1 2 L 10 3 t,x ∥w (3)
N 3 ∥ L 5 t,x ∥w N ∥ V 2 ∆ ≲ N -1 4 2 N 1 2 -s 1 N σ 2 ∥v N 2 ∥ U 2 ∆ ∥f N 1 (0)∥ 1 2 H s x ∥⟨∇⟩ s f N 1 ∥ 1 2 L 5 t,x ∥w (3)
N 3 ∥ L 5 t,x ∥w N ∥ V 2 ∆ .
Finally, we sum over the N i 's to obtain that the contribution of this term to (2.3.6) is less than

N 1 ,N 2 ,N 3 ,N N -1 4 2 N 1 2 -s 1 N σ 2 ∥v N 2 ∥ U 2 ∆ ∥P N 1 f 0 ∥ 1 2 H s x ∥f N 1 ∥ 1 2 L 5 t,x ∥w (3) N 3 ∥ L 5 t,x ∥w N ∥ V 2 ∆ ≲ N 1 ,N 2 ,N 3 ,N N 1 4 -s 2 ∥P N 1 f 0 ∥ 1 2 H s x ∥f N 1 ∥ 1 2 L 5 t,x ∥w (3) N 3 ∥ L 5 t,x ∥w N ∥ V 2 ∆ ≲N 0-∥f 0 ∥ 1 2 H 1 2 x ∥⟨∇⟩ s f ∥ 1 2 L 5 t,x ∥w (3) ∥ L 5 t,x ∥v∥ X σ ∥w N ∥ V 2 ∆ .
Note that we used the assumption that 1 4 < s.

• Case 2 : One random term comes with the highest frequency: N (1) = N 1 . We distinguish between two cases.

-Case 2a : High-high interactions N 1 2

1 ≤ N 2 . We apply Hölder's inequality and the Strichartz embedding (2.2.5), to get

N σ 1 |I N (N 1 , N 2 , N 3 )| ≲ N σ-s 1 N -σ 2 ∥⟨∇⟩ s f N 1 ∥ L 5 t,x ∥⟨∇⟩ σ v N 2 ∥ L 10 3 t,x ∥w (3)
N 3 ∥ L 5 t,x ∥w N ∥ L 10 3 t,x ≲ N σ 2 -s 1 ∥⟨∇⟩ s f N 1 ∥ L 5 t,x ∥⟨∇⟩ σ v N 2 ∥ U 2 ∆ ∥w (3) N 3 ∥ L 5 t,x ∥w N ∥ V 2 ∆ .
Since σ < 2s, we can sum over the dyadic frequencies and obtain

N 1 ,N 2 ,N 3 ,N N σ 1 |I N (N 1 , N 2 , N 3 )| ≲ ∥⟨∇⟩ s f ∥ L 5 t,x ∥v∥ L 5 t,x ∥v∥ X σ ∥w N ∥ V 2 ∆ .
-Case 2b : High-low-low interactions N 2 , N 3 ≪ N 1 2

1 , and N 1 ∼ N . We make a different analysis depending on the type of w (3) . * Case 2b(i) : w (3) = f . The idea is to use Cauchy Schwarz in order to apply the bilinear estimate twice to gain derivatives, and to use Hölder's inequality in order to access the L 5 t,x (J) norm of f , which is small.

N σ 1 |I N (N 1 , N 2 , N 3 )| ≤ N σ 1 ∥f N 1 v N 2 ∥ L 2 t,x ∥f N 3 w N ∥ L 2 t,x ≲ N σ 1 ∥f N 1 v N 2 ∥ 1 2 L 2 t,x ∥f N 1 ∥ 1 2 L 5 t,x ∥v N 2 ∥ 1 2 L 10 3 t,x ∥f N 3 w N ∥ 1 2 L 2 t,x ∥f N 3 ∥ 1 2 L 5 t,x ∥w N ∥ 1 2 L 10 3 t,x
.

Applying bilinear Strichartz estimate yields

N σ 1 |I N (N 1 , N 2 , N 3 )| ≲ N σ-s-1 4 1 N 1 2 -σ 2 N 1 2 -s 2 3 N -1 4 +0 ∥⟨∇⟩ s f N 1 ∥ 1 2 L 5 t,x ∥f N 1 (0)∥ 1 2 H s x ∥⟨∇⟩ σ v N 2 ∥ 1 2 U 2 ∆ ∥f N 3 (0)∥ 1 2 H s x ∥⟨∇⟩ σ v N 2 ∥ 1 2 L 10 3 t,x ∥⟨∇⟩ s f N 3 ∥ 1 2 L 5 t,x ∥w N ∥ V 2 ∆ ≲ N σ-s-1 2 +0 1 N 1 2 -σ 2 N 1 2 -s 2 3 ∥⟨∇⟩ σ v N 2 ∥ U 2 ∆ ∥w N ∥ V 2 ∆ ∥f N 1 (0)∥ H s x ∥f N 3 (0)∥ H s x 1 2 ∥⟨∇⟩ s f N 1 ∥ L 5 t,x ∥⟨∇⟩ s f N 3 ∥ L 5 t,x 1 2 .
Finally, we observe that

N σ-s-1 2 +0 1 N 1 2 -σ 2 N 1 2 -s 2 3 ≤ N σ-s-1 2 +0 1 N 1-σ-s 2 3 ≤ N σ 2 -s 1
, and we can sum over the N i 's to see that the contribution for this term is less than

N i N σ 1 J×R 3 f N 1 v N 2 f N 3 w N dxdt ≲ N 0-∥f 0 ∥ H s x ∥⟨∇⟩ s f ∥ L 5 t,x ∥v∥ X σ . * Case 2b(ii) : w (3) = v.
We use Hölder's inequality and interpolation to get

N σ 1 |I N (N 1 , N 2 , N 3 )| ≲ N σ 1 ∥f N 1 ∥ L 5 t,x ∥v N 2 ∥ L 10 3 t,x ∥f N 1 v N 2 ∥ L 2 t,x 1 2 ∥v N 3 ∥ L 5 t,x ∥w N ∥ L 10 3 t,x ∥v N 3 w N ∥ L 2 t,x 1 2 .
As a consequence of the bilinear estimate (2.2.7) and Strichartz estimates,

N σ 1 |I N (N 1 , N 2 , N 3 )| ≲ N σ-1 4 -s 1 N 1 2 -σ 2 N -1 4 +0 N 3 4 -σ 3 ∥f N 1 (0)∥ H s ∥f N 1 ∥ L 5 t,x 1 2 ∥⟨∇⟩ σ v N 2 ∥ U 2 ∆ ∥⟨∇⟩ σ v N 3 ∥ U 2 ∆ ∥w N ∥ V 2 ∆ .
We get a negative power of N 1 ∼ N as soon as σ > 3 4 . If σ < 3 4 , it follows from

N 1 ∼ N and N 3 ≤ N 1 2 1 that N σ-1 4 -s 1 N 1 2 -σ 2 N -1 4 +0 N 3 4 -σ 3 ≲ N σ-1 2 -s+0 1 N 1 2 -σ 2 N 3 8 -σ 2 1 ≲ N σ 2 -s-1 2 + 3 8 +0 1 .
Next, we use the assumption that σ < 2s to conclude that

N i N σ 1 J×R 3 f N 1 v N 2 v N 3 w N dxdt ≲ N 0-∥f 0 ∥ 1 2 H s x ∥⟨∇⟩ s f ∥ 1 2 L 10 3 t,x ∥v∥ L 5 t,x ∥v∥ X σ ∥w N ∥ V 2 ∆ .
This finishes the proofs of estimates (2.3.3) and (2.3.4), and of Proposition 2.3.2.

In order to perform a double bootstrap argument with ∥v∥ L 5 t,x (J) and ∥v∥ X σ (J) to prove local well-posedness, we need to estimate the spacetime critical L 5 t,x norm of v.

Proposition 2.3.3 (Additional trilinear estimates).

∥I •, N (f, f, v) ∥ L 5 t,x (J) ≲ ∥f 0 ∥ 1 2 H s x F ω (J) 3 2 ∥v∥ X 1 2 (J) , (2.3.7) ∥I •, N (f, v, v) ∥ L 5 t,x (J) ≲ ∥f 0 ∥ 1 2 H s x F ω (J) 1 2 ∥v∥ L 5 t,x (J) ∥v∥ X 1 2 (J)
.

(2.3.8)

Proof. By the TT * Strichartz estimate and the Sobolev embedding, we have

∥I •, N ∥ L 5 t,x (J) ≲ ∥|∇| 1 2 I •, N ∥ L 5 t L 30 11 x ≲ ∥|∇| 1 2 N ∥ L 10 7 t,x
.

Once again we perform a Littlewood-Paley decomposition of each term and conduct a case by case analysis. We write N (1) , N (2) , N (3) the non-increasing ordering among them.

Proof of (2.3.7) We need to estimate

N 1 2 (1) ∥f N 1 f N 2 v N 3 ∥ L 10 7 t,x
.

Without loss of generality, we may assume that N 1 ≥ N 2 .

• Case 1 : N (1) = N 1 . We use Hölder's inequality with 7 10 = 1 5 + 1 2 , interpolation, the bilinear estimate and the Strichartz embedding of L

10 3 t,x → U 2 ∆ to get N 1 2 1 ∥f N 1 f N 2 v N 3 ∥ L 10 7 t,x ≤ N 1 2 1 ∥f N 1 v N 3 ∥ 1 2 L 2 t,x ∥f N 1 ∥ 1 2 L 5 t,x ∥v N 3 ∥ 1 2 L 10 3 t,x ∥f N 2 ∥ L 5 t,x ≲ N 1 2 -s-1 4 1 N 1 2 -σ 3 ∥f N 1 (0)∥ 1 2 H s ∥⟨∇⟩ s f N 1 ∥ 1 2 L 5 t,x N σ 3 ∥v N 3 ∥ U 2 ∆ ∥f N 2 ∥ L 5 t,x .
Under the assumption that 1 4 < s and 1 2 < σ, we can use Observation 2.3.1 and see that in this case,

N i N 1 2 1 ∥f N 1 f N 2 v N 3 ∥ L 10 7 t,x ≲ ∥f 0 ∥ 1 2 H s ∥⟨∇⟩ s f ∥ 3 2 L 5 t,x ∥v∥ X 0 . • Case 2 : N (1) = N 3 . -Case 2a : N 1 < N 1 2
3 . In this case we shall apply the bilinear estimate once.

N 1 2 3 ∥f N 1 f N 2 v N 3 ∥ L 10 7 t,x ≤ N 1 2 3 ∥f N 2 v N 3 ∥ L 2 t,x ∥f N 1 ∥ L 5 t,x ≲ N -1 4 3 N 1 2 -s 2 N -s 1 ∥f N 2 (0)∥ 1 2 H s x ∥⟨∇⟩ s f N 2 ∥ 1 2 L 5 t,x N 3 ∥v N 3 ∥ U 2 ∆ ∥v N 3 ∥ L 10 3 t,x 1 2 ∥⟨∇⟩ s f N 1 ∥ L 5 t,x ≲ N -s 3 ∥f N 2 (0)∥ 1 2 H s x ∥⟨∇⟩ s f N 2 ∥ 1 2 L 5 t,x N 1 2 3 ∥v N 3 ∥ U 2 ∆ ∥⟨∇⟩ s f N 1 ∥ L 5 t,x .
Summing over the N i 's yields

N i N 1 2 3 ∥f N 1 f N 2 v N 3 ∥ L 10 7 t,x ≲ ∥f 0 ∥ 1 2 H s x ∥⟨∇⟩ s f ∥ 3 2 L 5 t,x ∥v∥ X 1 2 .
-Case 2b : N 1 2

3 ≤ N 1 . In this case there is no need to apply the bilinear estimate and we only use Hölder. We have

N 1 2 3 ∥f N 1 f N 2 v N 3 ∥ L 10 7 t,x ≤ N 1 2 3 N -s 1 ∥⟨∇⟩ s f N 1 ∥ L 5 t,x ∥f N 2 ∥ L 5 t,x ∥v N 3 ∥ L 10 3 t,x ≤ N -s 2 3 ∥⟨∇⟩ s f N 1 ∥ L 5 t,x ∥f N 2 ∥ L 5 t,x N 1 2 3 ∥v N 3 ∥ U 2 ∆ .
Then we sum over the N i 's and get

N i N 1 2 3 ∥f N 1 f N 2 v N 3 ∥ L 10 7 t,x ≲ ∥⟨∇⟩ s f ∥ L 5 t,x ∥f ∥ L 5 t,x ∥v∥ X 1 2 .
This concludes the proof of (2.3.7).

Proof of (2.3.8) We need to estimate

N 1 2 (1) ∥f N 1 v N 2 v N 3 ∥ L 10 7 t,x
.

Without loss of generality, we assume that N 2 ≥ N 3 .

• Case 1 : N (1) = N 1 . We use Hölder with 7 10 = 1 2 + 1 5 to apply the bilinear estimate once.

N 1 2 1 ∥f N 1 v N 2 v N 3 ∥ L 10 7 t,x ≤ ∥f N 1 v N 3 ∥ L 2 t,x ∥v N 2 ∥ L 5 t,x ≲ N 1 4 -s 1 N 1 2 3 ∥⟨∇⟩ s f N 1 ∥ 1 2 L 5 t,x ∥f N 1 (0)∥ 1 2 H s x ∥v N 3 ∥ 1 2 U 2 ∆ ∥v N 3 ∥ 1 2 L 10 3 t,x ∥v N 2 ∥ L 5 t,x ≲ N 1 4 -s 1 ∥⟨∇⟩ s f N 1 ∥ 1 2 L 5 t,x ∥f N 1 (0)∥ 1 2 H s x ∥v N 3 ∥ X 1 2 ∥v N 2 ∥ L 5 t,x .
Under the assumption that 1 4 < s we can use Observation 2.3.1 and see that in this case,

N i N 1 2 1 ∥f N 1 v N 2 v N 3 ∥ L 10 7 t,x ≲ ∥f 0 ∥ 1 2 H s x ∥⟨∇⟩ s f ∥ 1 2 L 5 t,x ∥v∥ L 5 t,x ∥v∥ X 1 2 .
• Case 2 :

N (1) = N 2 .
-Case 2a :

N 1 < N 1 2
2 . We proceed similarly, and get

N 1 2 2 ∥f N 1 v N 2 v N 3 ∥ L 10 7 t,x ≤ ∥f N 1 v N 2 ∥ L 2 t,x ∥v N 3 ∥ L 5 t,x ≲ N 1 4 2 N 1 2 -s 1 ∥⟨∇⟩ s f N 1 ∥ 1 2 L 5 t,x ∥f N 1 (0)∥ 1 2 H s x ∥v N 2 ∥ 1 2 U 2 ∆ ∥v N 2 ∥ 1 2 L 10 3 t,x ∥v N 3 ∥ L 5 t,x ≲ N -s 2 2 ∥⟨∇⟩ s f N 1 ∥ 1 2 L 5 t,x ∥f N 1 (0)∥ 1 2 H s x ∥v N 2 ∥ X 1 2 ∥v N 3 ∥ L 5 t,x . Hence, N i N 1 2 2 ∥f N 1 v N 2 v N 3 ∥ L 10 7 t,x ≲ ∥f 0 ∥ 1 2 H s x ∥⟨∇⟩ s f ∥ 1 2 L 5 t,x ∥v∥ L 5 t,x ∥v∥ X 1 2 .
-Case 2b : N 1 2

2 ≤ N 1 . We apply Hölder and get

N 1 2 2 ∥f N 1 v N 2 v N 3 ∥ L 10 7 t,x ≤ N 1 2 2 ∥f N 1 ∥ L 5 t,x ∥v N 2 ∥ L 10 3 t,x ∥v N 3 ∥ L 5 t,x ≤ N -s 2 2 ∥⟨∇⟩ s f N 1 ∥ L 5 t,x N 1 2 2 ∥v N 2 ∥ U 2 ∆ ∥v N 3 ∥ L 5 t,
x . We deduce from this and from Observation 2.3.1 that in this case,

N i N 1 2 2 ∥f N 1 v N 2 v N 3 ∥ L 10 7 t,x ≲ ∥⟨∇⟩ s f ∥ L 5 t,x ∥v∥ L 5 t,x ∥v∥ X 1 2 .

Local well-posedness

In the following local well-posedness statement, the smallness assumption comes from the L 5 t,x norm of the free evolution of the initial data. Since the problem is at a subcritical regularity scale 1 2 < σ, we could avoid using such a critical norm. Nevertheless, the L 5 t,x norm provides a blow-up and scattering criterion. Then, we prove in the next section that this blow-up criterion can be exploited as soon as we are able to obtain an a priori uniform bound on the H σ norm of the solution.

Proposition 2.3.4 (Local well-posedness). Let 1 4 < s ≤ 1 2 and ω ∈ Ω α . Then, for every 1 2 < σ < 2s, the Cauchy problem (NLS f ) with data v(t 0 ) ∈ H σ is locally well-posed. More precisely, for every E > 0, there exists ε 0 (E) such that for every ε ≤ ε 0 , every interval J that contains t 0 on which

F ω (J) ≤ ε, ∥ e it∆ v 0 ∥ L 5 t,x (J) ≤ ε , (2.3.9)
and for every v 0 ∈ H σ with ∥v 0 ∥ H σ ≤ E, there exists a unique solution v to (NLS f ) in X σ (J), with data v(t 0 ) = v 0 . Moreover, there exists C = C(∥f 0 ∥ H s ) > 0 such that ∥v∥ X σ (J) ≤ 2E, ∥v∥ L 5 t,x (J) ≤ Cε , and v ∈ C(J; H σ ) .

Proof. We perform a contraction mapping argument in the Banach space,

B E,ε = v ∈ X σ (J) | ∥v∥ X σ (J) ≤ 2E, ∥v∥ L 5 t,x (J) ≤ Cε ,
where C = C(∥f 0 ∥ H σ ) > 0 is an irrelevant constant that comes from the multilinear estimates of Proposition 2.3.3. We endow this space with the norm X σ (J), and we define the mapping

Φ : v ∈ B E,ε → e it∆ v 0 -i t t 0 e i(t-t ′ )∆ N (v + f )(t ′ )dt ′ .
From the Duhamel integral representation formula, we see that v is solution to

(NLS f ) with v(t 0 ) = v 0 if an only if Φ(v) = v. Let us prove that Φ(B E,ε ) ⊆ B E,ε
when ε ≪ E -1 . We get from Strichartz and from the trilinear estimate (2.3.5) that for all v ∈ B E,ε ,

∥Φ(v)∥ X σ (J) ≤ ∥v 0 ∥ H σ x + ∥I •, N (v) ∥ X σ (J) + ∥I •, N (v + f ) -N (v) ∥ X σ (J) ≤ E + C∥⟨∇⟩ σ N (v) ∥ L 10 7 t,x + C(∥f 0 ∥ H s )F ω (J) + C(∥f 0 ∥ H s ) F ω (J) + ∥v∥ L 5 t,x (J) ∥v∥ X σ (J) .
Applying the Fractional Leibniz rule with 7 10 = 2 5 + 3 10 yields

∥⟨∇⟩ σ N (v) ∥ L 10 7 t,x ≤ ∥v∥ 2 L 5 t,x (J) ∥⟨∇⟩ σ v∥ L 10 3 t,x ≤ ∥v∥ 2 L 5 t,x (J) ∥v∥ X σ (J) ≤ C 2 ε 2 E .
Hence, for all v ∈ B E,ε and under the smallness assumption (2.3.9) we get

∥Φ(v)∥ X σ (J) ≤ E + Cε 2 E + C(1 + C)εE .
(2.3.10)

Similarly, we have

∥Φ(v)∥ L 5 t,x ≤ ∥ e it∆ v 0 ∥ L 5 t,x (J) + ∥I •, N (v) ∥ L 5 t,x (J) + ∥I •, N (f ) ∥ L 5 t,x (J) + ∥I •, N (f, f, v) + N (f, v, v) ∥ L 5 t,x (J) .
(2.3.11) First, observe from the Sobolev embedding and from (2.3.2) that

∥I •, N (f ) ∥ L 5 t,x (J) ≤ C∥I •, N (f ) ∥ X 1 2 (J) ≤ C(∥f 0 ∥ H σ )F ω (J) .
By Hölder's inequality, Sobolev embedding and Strichartz embedding we have

∥I •, N (v) ∥ L 5 t,x (J) ≤ ∥v∥ X 1 2 (J) ∥v∥ 2 L 5
t,x (J) . Then, using the trilinear estimates presented in Proposition 2.3.3 we get that for any v ∈ B E,ε ,

∥I •, N (f, f, v) + N (f, v, v) ∥ L 5 t,x (J) ≤ C F ω (J) 3 2 + F ω (J) 1 2 ∥v∥ L 5 t,x (J) ∥v∥ X σ (J) .
Collecting the above estimates, we deduce from (2.3.11) that under the smallness condition (2.3.9), we have for v ∈ B E,ε

∥Φ(v)∥ L 5 t,x ≤ ε + Cε + 2C 3 ε 2 E + 2CE ε 3 2 + Cε 3 2 .
(2.3.12) Combining (2.3.10), (2.3.12) and choosing η = 3Cε, R = 2CE, for ε ≤ ε 0 (E) with, say ε 0 (E) ∼ E -1 when E ≫ 1, we obtain ∥Φ(v)∥ X σ (J) ≤ 2CE, ∥Φ(v)∥ L 5 t,x (J) ≤ 3Cε . for any v ∈ B E,ε . Thus , Φ : B E → B E . Similarly, we prove that

∥Φ(v 1 ) -Φ(v 2 )∥ X σ (J) ≤ C ∥v 1 ∥ L 5 t,x (J) ∥v 1 ∥ X σ (J) + ∥v 2 ∥ L 5 t,x (J) ∥v 2 ∥ X σ (J) ∥v 1 -v 2 ∥ X σ (J)
. Choosing ε 0 (E) ≪ E -1 , we see that Φ is a contraction mapping, and it admits a unique fixed point v, solution to (NLS f ). Finally, we deduce the continuity of t → v(t, •) from the embedding X σ (J) -→ C J; H σ x (R 3 ) . Proposition 2.3.5 (Blow-up criterion). Let v be the maximal lifespan solution to (NLS f ) on J * × R 3 given by the local well-posedness theory. If we have

∥v∥ L 5 t,x (J * ) < +∞ , (2.3.13)
then sup J * = +∞ and the solution scatters as t goes to +∞.

Proof. Denote E = ∥v(0)∥ H σ x , and ε > 0 to be chosen later on. We proceed by contradiction and assume that ∥v∥ L 5 t,x < +∞, but T * = sup J * < +∞. We will raise the contradiction by extending the solution up to T * . By the local Cauchy theory from Proposition 4.2.2, it is enough to prove that there exist a constant C(E, f 0 ), t 0 ∈ J * and δ > 0 such that

∥v(t)∥ L ∞ t H σ x (J * ) ≤ C(E, f 0 ) , (2.3.14) ∥ e i(t-t 0 )∆ v(t 0 )∥ L 5 t,x [t 0 -δ,T * +δ] ≤ ε 0 , (2.3.15)
where ε 0 = ε 0 (E) is as in the local well-posedness Proposition 4.2.2. To prove the above estimates, we proceed as follows. First, we use the global assumption (2.3.13) to decompose J * into a finite number L(ε) of intervals {I l } 1≤l≤L such that for j ∈ {1, . . . , L}, ∥v∥ L 5 t,x (I j ) ≤ ε . Using the assumption (2.2.12) that ∥f ∥ L 5 t,x (R) < +∞, and up to an extra decomposition of J * into O(L) intervals, we may assume that for each l ∈ 1, . . . , L ,

F (J l ) ≤ ε .
Next, we take J ⋐ J * a compact sub-interval of J * , and denoting J ∩ J l by J l , we prove by induction on l ≤ L that there exists C(∥f 0 ∥ H s ) such that

∥v∥ X σ ( J l ) ≤ C2 l E .
(2.3.16)

For this purpose, we apply the trilinear estimates (2.3.5) to get

∥v∥ X σ J 1 ) ≤ ∥v(0)∥ H σ + Cε + 2Cε∥v∥ X σ ( J 1 ) .
Hence, choosing ε small enough we see that

∥v∥ X σ ( J 1 ) ≤ 2E .
Iterating this on J 2 , . . . , J L with the same ε, we obtain that for l ∈ 1, . . . , L ,

∥v∥ X σ ( J l ) ≤ 2 l E .
Since ε does not depend on J ⋐ J * , we have from sub-additivity of the norm X σ (see Lemma 3.4.4) that

∥v∥ L ∞ t H σ (J * ) ≤ sup J⋐J * ∥v∥ X σ ( J) ≤ L l=1
2 l E ≤ C(L, E) .

(2.3.17)

Similarly, the multilinear estimates from Proposition 2.3.3 yield for all t 0 ∈ J * ,

∥I [t 0 , •], N (f + v) ∥ L 5 t,x ( J) ≤ C(E, f 0 )
. Therefore, taking the sup over all the possible J we obtain a uniform bound with respect to t 0 on the L 5 t,x (J * ) norm of the Duhamel integral. Consequently, we have from Duhamel's formula that

∥ e i(t-t 0 )∆ v(t 0 )∥ L 5 t,x [t 0 ,T * ] = ∥v -I [t 0 , •], N (f + v) ∥ L 5 t,x [t 0 ,T * ] ≤ ∥v∥ L 5 t,x (J * ) + ∥I [t 0 , •], N (f + v) ∥ L 5 t,
x (t 0 ,T * ) , and we conclude the proof of (2.3.15) by monotone convergence lim t 0 →T * ∥ e i(t-t 0 )∆ v(t 0 )∥ L 5 t,x [t 0 ,T * ] = 0 . As explained above, this finishes the proof of the global existence. At that point, it is standard to deduce scattering from these global bounds. Indeed, when J * = [0 , +∞) and ∥v∥ L 5 t,x (J * ) < +∞, we use the uniform bound (2.3.17), the nonlinear estimates from Proposition 2.3.2 and monotone convergence to deduce from the Duhamel integral formulation that e -itn∆ v n is a Cauchy sequence in H σ x as t n goes to +∞. This ends the proof of Proposition 2.3.5.

Stability theory

In this subsection, we prove that global existence and scattering for v can be deduced from a priori uniform estimate on the H σ norm of v(t). Note that this is true when v is a solution to (NLS) without forcing terms, and when σ > 2 3 (see Theorem 2.1.2). Hence, since v is actually solution to the forced equation (NLS), we will settle a stability theory at regularity H σ . In particular, on some spacetime slabs where the perturbation satisfies a smallness condition, we shall be able to approach v by a solution to (NLS) and to infer local spacetime bound on v. Then, provided we have a global a priori estimate for the H σ norm of v, we perform a bootstrap argument to extend the spacetime bound to the whole maximal lifespan of v, and to deduce conditional scattering. For now on, we fix σ > 2 3 , which corresponds to the lowest regularity where the global Cauchy theory for (NLS) is known.

Lemma 2.3.6 (Short-time stability in H σ x ). Let t 0 ∈ R, E ≥ 0 and v 0 ∈ H σ x with ∥v 0 ∥ H σ x ≤ E.
Let v be the local solution to (NLS f ) as in Proposition 4.2.2 associated with v(t 0 ) = v 0 . Then, take u 0 ∈ H σ x and denote by u the global solution to (NLS) in C ∞ (R; H σ x ), with data u(t 0 ) = u 0 . There exist

ε 1 = ε 1 (E) and C 0 (E) such that if ∥v 0 -u 0 ∥ H σ x ≤ ε , F ω (J) ≤ ε , ∥u∥ L 5 t,x (J) ≤ ε , (2.3.18)
for some ε ≤ ε 1 , then v stays close to u on any compact interval

J ⋐ J * that contains t 0 ∥v -u∥ X σ (J) ≤ C 0 (E)ε . (2.3.19)
Proof. Let us consider the difference w = v -u11 , solution to the equation

i∂ t w + ∆w = N (v + f ) -N (v) + N (v) -N (u) , (t, x) ∈ R × R 3 . w t=0 = v 0 -u 0 , (2.

3.20)

We introduce the function g : t ∈ J → ∥w∥ X σ (t 0 ,t) , and we perform a continuity argument. Note that as a consequence of the smallness condition ∥w(0)∥ H σ x ≤ ε and from Lemma 3.4.4, we have that g is continuous and lim sup t→t 0 g(t) ≲ ε. Moreover, it follows from Duhamel's formulation that

∥w∥ X σ (J) ≤ C∥v 0 -u 0 ∥ H σ x + ∥I •, N (v + f ) -N (v) ∥ X σ (J) + ∥I •, N (u) -N (v) ∥ X σ (J)
. From the trilinear estimate (2.3.5) of Proposition 2.3.2 and the smallness assumptions, we get

∥I •, N (v + f ) -N (v) ∥ X σ [t 0 ,t] ≤ CF ω (J) + C∥v∥ X σ [t 0 ,t] F ω (J) + ∥v∥ L 5 t,x ([t 0 ,t]) F ω (J) 1 2 ≤ Cε + C(∥u∥ X σ [t 0 ,t] + ∥w∥ X σ ([t 0 ,t]) ) ε + ε 1 2 ∥w∥ L 5 t,x [t 0 ,t] + ε 1 2 ∥u∥ L 5 t,x (J) ≤ Cε + Cε 1 2 CE + g(t) 2ε + Cg(t) .
Moreover,

∥I •, N (u) -N (v) ∥ X σ [t 0 ,t] ≤ C∥w∥ X σ ([t 0 ,t]) ∥v∥ 2 L 5 t,x ([t 0 ,t]) + ∥u∥ 2 L 5 t,x (J) ≤ C∥w∥ X σ [t 0 ,t] ∥w∥ 2 L 5 t,x [t 0 ,t] + ∥u∥ 2 L 5 t,x (J) ≤ Cg(t) g(t) 2 + ε 2 ) .
Hence, by combining the above estimates, we prove that for all t ∈ J with t 0 ≤ t, g(t) ≤ 2Cε + 2Cε

3 2 E + g(t) ε 1 2 CE + ε 3 2 + g(t) 2 + ε 2 ,
and the result follows from a continuity argument, by choosing

ε 1 ≪ E -1 2 .
Next, we combine the local well-posedness result with the blow-up criterion to turn the short-time stability result into a long time stability statement.

Lemma 2.3.7 (Long-time stability). Let Λ > 0, t 0 ∈ R, J = [t 0 , T ], v 0 ∈ H σ x with ∥v 0 ∥ H σ x ≤ E. Let u be the global-in-time solution to (NLS) in H σ
x with data u(t 0 ) = v 0 , and assume that

∥u∥ L 5 t,x (J) ≤ Λ .
There exists ε 2 = ε 2 (E, Λ) > 0 such that under the smallness condition

F ω (J) ≤ ε 2 ,
there exist a unique solution v to (NLS f ) and a constant C(E, Λ) > 0 such that for every compact interval J ⋐ J, we have v ∈ X σ ( J), and

∥v∥ L 5 t,x (J) ≤ C(E, Λ) . ( 2

.3.21)

Proof. Let v be the maximal lifespan solution to (NLS f ) with v(t 0 ) = v 0 , on an interval J * × R 3 . The whole statement reduces to the estimate (2.3.21). Indeed, we would be in position to apply the blow-up criterion, and to deduce that J ⊂ J * . The strategy to prove such a global estimate is to break J into a finite number L(E, Λ) of intervals where the L 5 t,x norm of u is small, so that we can apply the short-time stability Lemma 2.3.6. Then, we sum over the different spacetime slabs. Since their number does only depend on E and Λ, we obtain (2.3.21).

We proceed as follows. First, we use the deterministic global well-posedness Theorem 2.1.2 and we get a constant C 0 (E), such that for any J ⋐ J * , we have

∥u∥ X σ ( J) ≤ C 0 (E) .
In addition, we take ε 1 = ε 1 (2C 0 (E)) as in Lemma 2.3.6, and ε 2 < ε 1 , to be determined shortly. Next, we decompose J into L(ε, Λ, E) intervals {J i } 1≤i≤L , with J i = [t i , t i+1 ), such that for i ∈ {0, . . . , L -1} and for some ε < ε 2 ,

∥u∥ L 5 t,x (J i ) ≤ ε .
Now, we write w = v -u, we fix J * ⋐ J * and we denote J i = J * ∩ J i . We would like to apply Lemma 2.3.6 on each J i . To this end, we need to make sure that for all i ∈ 1, . . . , L -1 , we have

∥v(t i )∥ H σ x ≤ 2C 0 (E) , ∥w(t i )∥ H σ ≤ ε 1 (2C 0 (E)) .
Indeed, we already know by assumption and by construction that

F ω ( J i ) ≤ ε 1 , ∥u∥ L 5 t,x ( J i ) ≤ ε 1 .
Hence, we shall prove by induction on i that for some ε 2 < ε 1 , for every ε < ε 2 and for every i ∈ 0, . . . , L 1 , there exists

C i = C i (E, Λ) such that ∥w∥ X σ ( J i ) ≤ C i ε , ( sup 1≤i≤L C i )ε 2 ≤ ε 1 , ∥v(t i )∥ H σ x ≤ 2C 0 (E) . (2.3.22)
In the case when i = 0, we have w(t 0 ) = 0 and it follows from the short-time stability Lemma 2.3.6 applied on J 1 , where v and f satisfy the smallness condition (2.3.18), that there exists C 1 (E) with

∥w∥ X σ ( J 1 ) ≤ C 1 (E)ε .
Hence, we chose ε 2 such that C 1 (E)ε 2 < ε 1 (2C 0 (E)). Next, we assume that (2.3.22) holds up to time t i . By Duhamel's integral formulation, and since w(t 0 ) = 0, we have

∥w(t i )∥ H σ x ≤ ∥I t 0 , t i , N (v + f ) -N (u) ∥ H σ x .
Then, we proceed as in the proof of the short-time stability Lemma 2.3.6, and we use the multilinear estimates from Propositions 2.3.2, 2.3.3, as well as the induction assumption, to see that

∥I t 0 , t i , N (v + f ) -N (u) ∥ H σ x ≤ C i-1 l=0 ∥I •, N (v + f ) -N (u) ∥ X σ ( J l ) ≤ C i-1 l=0 C l (E)ε .
We define

C i = C(E)C 1≤l≤i-1 C l and choose ε 2 such that C i ε 2 ≤ ε 1 (2C 0 (E))
. Subsequently, the smallness condition (2.3.18) is satisfied on the interval J i = [t i , t i+1 ), so that we can apply Lemma 2.3.6 and obtain

∥w∥ X σ ( J i ) ≤ C(E) 0≤l≤i-1 C j ε .
In addition, we get from the triangle inequality that

∥v(t i+1 )∥ H σ x ≤ ∥u∥ L ∞ (J i+1 ;H σ x ) + ∥w∥ L ∞ (J i+1 ;H σ x ) ≤ C 0 (E) + C i+1 ε ≤ 2C 0 (E) .
This finishes the proof of the induction result (2.3.22). Moreover, it follows from the sub-additivity of the X σ norm (see Lemma 3.4.4) that

∥v∥ X σ ( J) ≤ ∥u∥ X σ (J) + L j=1 ∥w∥ X σ (J j ) ≤ C(E, Ξ) .
The proof of (2.3.21) follows by using the embedding X σ ( J) -→ L 5 t,x ( J), and the monotone convergence theorem which implies that ∥v∥ L 5 t,x (J) = sup J⋐J ∥v∥ L 5 t,x ( J) .

Proposition 2.3.8 (Uniform bound in H σ x implies scattering). Let v 0 ∈ H σ x , and let v be the maximal-lifespan solution in H σ x to (NLS f ) from initial data v(0) = v 0 . Suppose that we have the uniform a priori bound sup

t∈J * ∥v(t)∥ H σ x ≤ E .
Then, the solution v is global and scatters as t goes to ∞.

Proof. We get from the blow-up criterion of Lemma 2.3.5 that it suffices to prove the global spacetime bound ∥v∥ L 5 t,x (J * ) < +∞ .

(2.3.23)

Recall that the deterministic Cauchy theory in H σ , for 2 3 < σ ≤ 1, provides such a bound. Indeed, Theorem 2.1.2 claims that there exists Λ(E) such that for any solution u to (NLS) starting from a data u(t 0 ) with ∥u(t 0 )∥ H σ

x ≤ E, we have

∥u∥ L 5 t,x (R) ≤ Λ(E) .
Subsequently, we fix ε 2 (E, Λ(E)) = ε 2 (E) as in Lemma 2.3.7, and we divide J * into a finite number L(E) of intervals J i = [t i , t i+1 ), with t 0 = 0, such that

F ω (J i ) ≤ ε 1 , j ∈ {1, . . . , L} .
Next, for each i ∈ 0, . . . , L -1 , we consider the global solution u to (NLS) with initial data u(t i ) = v(t i ). Since ∥v(t i )∥ ≤ E by assumption, we have

∥v∥ ≤ Λ(E) .
Hence, we can apply Lemma 2.3.7 on J i , and obtain

∥v∥ L 5 t,x (J i ) ≤ C i (E) .
By summing over the spacetime slabs, we conclude that

∥v∥ L 5 t,x (J * ) ≤ C(E, Λ(E)) ≤ C(E) .
This finishes the proof of Proposition 2.3.8.

Almost conservation laws

In this section, we prove some almost conservation laws in order to obtain the uniform bound on the H σ (R 3 ) norm of v. Recall that throughout this section, α > 0 and ω ∈ Ω α are fixed, such that for N large enough,

F ω (R) + F ω ∞ (R) ≤ C α , F ω 2 (R) ≤ C α N 1-σ 2 -γ .
Since there is no coercive conservation law at the level of H σ for (NLS), we need to use a modified energy. More precisely, we consider the energy of Iv, which is finite but not preserved, in order to damp the frequencies of size larger than N . However, we expect the time-derivative of the modified energy to be small when N is large, at least on a spacetime slab where there hold some smallness conditions for the L 4 t,x norm of v and spacetime bounds for the stochastic forcing term.

Setting up the I-method

Let v ∈ H σ x (R 3 ) be the maximal lifespan solution to (NLS f ) on [0 , T * ). There is no difficulty to estimate the low frequencies of v since the L 2 norm is almost-conserved, and the forcing term is uniformly in L 2 . Lemma 2.4.1 (Almost conservation of the mass (see Lemma 7.1 in [START_REF] Oh | On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities[END_REF])). Let v be solution to (NLS f ) on a time interval J. Then,

R 3 |v(t, x)| 2 dx ≲ R 3 |v 0 (x)| 2 dx + R 3 |f 0 (x)| 2 dx .
Proof. The proof is a straightforward consequence of the conservation of mass for (NLS) and the unitarity of the linear Schrödinger propagator e it∆ , from which we deduce that

R 3 |v(t)| 2 dx = -2Re v(t)f ω (t)dx ≤ 1 2 |v(t)| 2 dx + 2 |f ω (t)| 2 dx .
We refer to Lemma 7.1 in [START_REF] Oh | On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities[END_REF] for further details.

To obtain some estimates on the high frequencies of v, which has infinite energy, we need to smooth it by applying the I-operator, and by considering Iv, which has finite energy, or sometimes

Iu = I(v + f ). The aim is to prove the uniform bound sup t∈[0,T * ) ∥Iv(t)∥ H 1 (R 3 ) < +∞ .
(2.4.1) Indeed, we deduce from this that sup

t∈[0,T * ) ∥v(t)∥ H σ x (R 3 ) < +∞ . (2.4.2)
With this uniform estimate (2.4.2) on theH σ (R 3 ) norm of v at hand, we are in position to apply Proposition 2.3.8 to show that the solution v to (NLS f ) is global-in-time and scatters. Now, observe that the truncated solution Iu, as well as Iv, satisfies the perturbed cubic Schrödinger equation

i∂ t Iu + ∆Iu = N (Iu) + H , (t, x) ∈ R × R 3 , (I-NLS)
where we denoted by H the commutator

H := IN (u) -N (Iu) . (2.4.3)
This perturbation H can be seen as a trilinear operator acting on functions u (j) of type v or f . In some appropriate spacetime norms, H enjoys some decay with respect to N thanks to frequency cancellations stemming from the gauge-invariant structure of the equation, and from the definition of the I-operator.

Z I (J) := ∥Iv∥ S 1 (J) = sup s(q,r)=0 ∥⟨∇⟩Iv∥ L q t (J;L r x (R 3 )) ,
where we recall that s(q, r) = 2 q -d 1 2 -1 r encodes the admissibility condition for the Strichartz estimate to hold.

We shall prove that Z I (J) is of size ∼ N 1-σ provided E(Iv(t 0 )) ≤ N 2(1-σ) and ∥Iv∥ 4 L 4 t,x (J) ≪ 1. This is the matter of Lemma 2.4.4. Before this, let us state and prove the main result of this section, where we estimate the spacetime Lebesgue norm of wH for a given function w that lies uniformly in L 2

x .

Lemma 2.4.3. For all δ > 0, there exists a constant C such that for every w ∈ L ∞ t (J; L 2 x (R 3 )) and

N ∈ 2 N , if J ⊂ [0 , T * ) is such that Z I (J) < +∞, then J R 3 w(t, x)H(t, x)dx dt ≤ CN -1+δ sup K∈2 N K -1 ∥P K w∥ L ∞ t L 2 x (J) Z I (J) 2 + F ω 2 (J) 2 Z I (J) + F ω ∞ (J) . (2.4.4)
The idea is to use the frequency cancellations from the I operator in the error term H to gain a negative power of N . As we will see, we shall often come across terms as in the left-hand-side of (2.4.4), especially when we estimate the modified energy and the modified Morawetz interaction. Hence, obtaining estimates like (2.4.4) turns out to be the crucial part of our analysis.

Proof. The strategy follows the sames lines as the one of Proposition 3 from [START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF], but with some differences. At a fixed time t ∈ J, we use Plancherel's formula, and we develop the nonlinearity in Fourier to obtain

R 3 w(t, x)H(t, x)dx = Σξ i =0 1 - m(ξ 2 + ξ 3 + ξ 4 ) m(ξ 2 )m(ξ 3 )m(ξ 4 ) w(t, ξ 1 ) Iu(t, ξ 2 ) Iu(t, ξ 3 ) Iu(t, ξ 4 ) . (2.4.5)
First, we perform a Littlewood-Paley decomposition of each term appearing in the above integral, localized around a dyadic frequency N i . We denote by B the pointwise bound of the Fourier multiplier sup

ξ 1 +ξ 2 +ξ 3 +ξ 4 =0 ξ i ∼N i 1 - m(ξ 2 + ξ 3 + ξ 4 ) m(ξ 2 )m(ξ 3 )m(ξ 4 ) ≤ B(N 1 , N 2 , N 3 , N 4 ) .
As observed in Proposition 3.1 in [START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF], a case by case analysis leads to the following decay estimate for B : for all δ > 0, there exists C > 0 such that for all N 1 , N 2 , N 3 , N 4 , N ∈ 2 N , we have

B(N 1 , N 2 , N 3 , N 4 ) N 1 N 2 N 3 ≤ CN -1+δ max(N 1 , N 2 , N 3 , N 4 ) -δ . (2.4.6)
The above estimate stems from frequency cancellations and indicates an almost conservation of the energy. Next, we factorize (2.4.5) by the pointwise bound B on the Fourier multiplier. We are left to control

B(N 1 , N 2 , N 3 , N 4 ) R 3 Λ(w, Iu, Iu)(ξ) Iu(ξ)dξ ,
where Λ is a multilinear operator obtained from (2.4.5) (see also (3.22) from [START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF]). To estimate such a multilinear integral, we apply Plancherel, Hölder's inequality and the Coifman-Meyer estimate (see Page 179 in [START_REF] Meyer | Commutateurs d'intégrales singulièeres et opérateurs multilinéeaires[END_REF]). We also use Hölder's inequality in time, to see that the left-hand-side of (2.4.4) is bounded from above by

N i B(N 1 , N 2 , N 3 , N 4 )∥P N 1 w∥ L ∞ t L 2 x (J) ∥P N 2 Iu∥ L 2 t L 6 x (J) ∥P N 3 Iu∥ L 2 t L 6 x (J) ∥P N 4 Iu∥ L ∞ t L 6 x (J) .
Using the decay estimate (2.4.6), we can sum over the N i 's and gain one power of Let us now address the a priori estimate for Z I (J) on a spacetime slab, where we assume a bound for the modified energy of v, as well as a smallness condition on spacetime norms of Iv and f . More precisely, provided that the energy of Iv is initially of size N 2(1-σ) , we prove that Z I (J) is of size ∼ N 1-σ on an interval J where there holds (2.4.8) and (2.4.9). Lemma 2.4.4 (Modified local existence theory). Let t 0 ∈ J * be such that

N J R 3 w(t, x)H(t, x)dx dt ≤ CN -1+δ sup N 1 N -1 1 ∥P N 1 w∥ L ∞ t L 2 x ∥∇Iu∥ L 2 t L 6 x (J) ∥∇Iu∥ L 2 t L 6 x (J) ∥Iu∥ L ∞ t L 6 x (J) . ( 2
E(v)(t 0 ) ≤ N 2(1-σ) .
There exist C > 0 and η > 0 that do not depend on N , ε N = N -2(1-σ)-δ for some arbitrarily small δ, and N 0 = N 0 (C α , f 0 ) such that for any interval J = [t 0 , b) ⊂ [0 , T * ) on which the following smallness conditions12 are satisfied

∥Iv∥ 4 L 4 t,x (J) ≤ ε N , (2.4.8) ∥f ∥ L 4 t L 3 x (J) ≤ η ,
(2.4.9)

we have that for all N ≥ N 0 , Z I (J) ≤ CN 1-σ .

(2.4.10)

For now on, we assume that N > N 0 , such that (2.4.10) holds whenever the assumptions of Lemma 2.4.4 are satisfied.

Proof. Recall that Iv is solution to (I-NLS). By the Duhamel's integral formulation for equation (I-NLS), we have that

Iv(t) = e i(t-t 0 )∆ Iv(t 0 ) - t t 0 e i(t-τ )∆ N (Iu) + H dτ = e i(t-t 0 )∆ Iv(t 0 ) + I [t 0 , t]; N 1 + I [t 0 , t]; N 1 + I [t 0 , t]; H , where 13 N 1 := N (v, v, v) + N (f, f, f ) + N (f, v, v) , N 2 = N (f, f, v) .
It follows from the dual Strichartz estimate of Proposition 2.2.1 that

Z I (J) ≤ C ∥Iv(t 0 )∥ H 1 + ∥⟨∇⟩IN 1 ∥ L 6 5 t L 18 11
x (J)

I 1 + ∥⟨∇⟩IN 2 ∥ L 1 t L 2 x (J) I 2 + ∥I([t 0 , t], ⟨∇⟩IH)∥ U 2 ∆ (J) I 3
.

Term I 1 : We deduce from the Leibniz rule that

∥⟨∇⟩I(N (Iv))∥ L 6 5 t L 18 11 x ≤ ∥⟨∇⟩Iv∥ L 2 t L 6 x ∥Iv∥ 2 L 6 t L 9 2 x
.

By interpolation and Sobolev embedding, we have ∥Iv∥

L 6 t L 9 2 x ≤ ∥Iv∥ 2 3 L 4 t,x (J) ∥Iv∥ 1 3 L ∞ t L 6 x (J) ≤ Cε 1 6 N Z I (J) 1 3 .
Consequently, the contribution of this term is less than

∥⟨∇⟩I(N (Iv))∥ L 6 5 t L 18 11 x ≤ Cε 1 3 N Z I (J) 1+ 2 3 .
Similarly, we have

∥⟨∇⟩I(N (Iv, Iv, If ))∥ L 6 5 t L 18 11 x ≤ ∥⟨∇⟩If ∥ L 2 t L 6 x ∥Iv∥ 2 L 6 t L 9 2 x + ∥⟨∇⟩Iv∥ L 2 t L 6 x ∥Iv∥ L 6 t L 9 2 x ∥If ∥ L 6 t L 9 2 x ≤ CZ I (J) F ω 2 (J)ε 1 3 N Z I (J) -1 3 + ε 1 6 N Z I (J) 1 3 ∥If ∥ 2 3 L 4 t,x ∥If ∥ 1 3 L ∞ t L 6 x ≤ CZ I (J) F ω 2 (J)ε 1 3 N Z I (J) -1 3 + ε 1 6 N Z I (J) 1 3 F ω ∞ (J) 1 3 F ω (J) 2 3 .
Additionally,

∥⟨∇⟩(If )(If ) 2 ∥ L 6 5 t L 18 11 x ≤ ∥⟨∇⟩If ∥ L 2 t L 6 x ∥If ∥ 2 L 6 t L 9 2 x ≤ C∥f 0 ∥ 2 L 2 x F ω 2 (J) .
Hence, the contribution of the first term is less than

I 1 ≤ CZ I (J) F ω 2 (J)Z I (J) -1 + ε 1 3 N Z I (J) 2 3 + ε 1 3 N N 1-σ 2 Z I (J) -1 3 C α + ε 1 6 N Z I (J) 1 3 C α .
Term I 2 : Once again, when a derivative falls on the term If , we need to place it inside L 2 t , and use the improved Sobolev embedding for randomized radial initial data. It follows from the Leibniz rule that

∥⟨∇⟩IN 2 ∥ L 1 t L 2 x ≤ C∥(⟨∇⟩If )(If )(Iv)∥ L 1 t L 2 x (J) + ∥(⟨∇⟩Iv)(If ) 2 ∥ L 1 t L 2
x (J) . We use Hölder's inequality, Sobolev embedding and the assumption that s ≥ 3 7 to get

∥(⟨∇⟩If )(If )(Iv)∥ L 1 t L 2 x (J) ≤ C∥⟨∇⟩If ∥ L 2 t L 6 x (J) ∥If ∥ L 4 t L 12 x (J) ∥Iv∥ L 4 t,x (J) ≤ CC 2 α ε 1 4 N N 1-σ 2 .
Finally, ∥⟨∇⟩Iv(If

) 2 ∥ L 1 t L 2 x (J) ≲ ∥⟨∇⟩Iv∥ L 2 t L 6 x (J) ∥If ∥ 2 L 4 t L 3 x (J) ≲ η 2 Z I (J) .
Collecting the above estimates, we see that the contribution of this term is less than

I 2 ≤ CZ I (J) C 3 α N 1-σ 2 Z I (J) -1 + C 2 α ε 1 4 N N -1-σ 2 + η 2 .
Term I 3 : It follows from the duality estimate in U 2 ∆ as stated in Proposition 2.2.5 and from Lemma 2.4.4 that

∥I((t 0 , t), ⟨∇⟩IH)∥ U 2 ∆ ≤ C sup ∥w∥ V 2 ∆ ≤1 J×R 3 w(t, x)⟨∇⟩IHdxdt ≤ CN -1+δ sup ∥w∥ V 2 ∆ ≤1 sup K∈2 N K -1 ∥P K (⟨∇⟩Iw)∥ L ∞ t L 2 x (J) Z I (J) 2 + F ω 2 (J) 2 Z I (J) + F ω ∞ (J) ≤ CN -1+δ Z I (J) 2 + F ω 2 (J) 2 Z I (J) + F ω ∞ (J) .
Next, we take γ > 0 arbitrarily small, and we fix

ε N = N -2(1-σ)-γ .
We have

I 1 ≤ CZ I (J) N -2(1-σ) 3 -γ 3 Z I (J) 2 3 + C α N -2(1-σ) 3 -γ 3 N 1-σ 2 Z I (J) -1 3 + C α N -1-σ 3 -γ 3 Z I (J) 1 3
.

If Z I (J) ∼ N 1-σ
, then the term in the parenthesis on the right-hand-side of the above estimate is smaller than a negative power of N . Moreover, we have lim

|J|→0 Z I (J) = 0 .
Hence, we apply a continuity argument in this case to prove that Z I (J) is indeed of size ∼ N 1-σ . Similarly, we have

I 2 ≤ CZ I (J) C 3 α N 1-σ 2 Z I (J) -1 + C 2 α Z I (J) -1 + η 2 .
Here again, we shall be able use the continuity method. We emphasize that η does not have to depend on N . The last contribution is estimated as follows :

I 3 ≤ CN -1+δ Z I (J) 2 + N 1-σ Z I (J) + C α .
In this case, we observe that when Z I (J) ∼ N 1-σ , the parenthesis in the right-hand-side of the above estimate is bounded by a negative power of N provided 1 2 < σ. Finally, we conclude by collecting the above contributions and by using the continuity method for 1 2 < σ, N > N 0 large enough and η small enough. We stress out that η and N 0 only depend on some universal constants and on C α , which is fixed throughout the analysis. This concludes the proof of Lemma 2.4.4.

Modified energy increment

In this section, we revisit in a probabilistic setting the analysis of modified-energy increment performed when using the I-method (see for instance [START_REF] Dodson | Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when n = 3 via a linear-nonlinear decomposition[END_REF][START_REF] Dodson | Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when d = 2, 3 and u 0 is radial[END_REF][START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF]). A similar strategy was developed in [START_REF] Fan | Construction of L 2 log-log blowup solutions for the mass critical nonlinear Schrödinger equation[END_REF]. Here, we propose a different approach that strongly relies on the structure of the nonlinearity. Subsequently, we follow [START_REF] Oh | On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities[END_REF] and we include the probabilistic forcing term in the potential part of the energy

E(v(t)) := 1 2 R 3 |∇Iv(t, x)| 2 dx + 1 4 R 3 |Iv + If | 4 dx .
The idea is to benefit from the frequency cancellations appearing in H. More precisely, when we handle the increment of the energy, we aim to come up with terms under the form wH, for which we already obtained some spacetime bounds in Lemma 2.4.3. In the next Proposition, we estimate the energy increments on some intervals where the L 4 t,x norm of Iv satisfies a smallness condition.

Proposition 2.4.5. Let t 0 ∈ J * , such that

E(v)(t 0 ) ≤ N 2(1-σ) .
Then, for all interval J = [t 0 , t] ⊂ [0 , T * ) where Iv satisfies the smallness conditions (2.4.8) and (2.4.9), that lead to the modified local well-posedness statement of Lemma 2.4.4, there exists

C(α) such that J d dt E(v) dt ≤ CN 0+ N 6(1-σ)-1 + CN 1-σ F ω 2 (J) 2 + CN 1-σ F ω 2 (J)∥Iv∥ 2 L 4 t,x (J) .
(2.4.11)

Observe that we gain a negative power of N from the perturbative term that contains H. However, as explained in the introduction, we keep track of some sub-additive quantities when we estimate the other terms. This will be useful when we will sum over the intervals where we have the modified local well-posedness result of Lemma 2.4.4. Still, we are forced to lose some powers of N when we estimate the terms where the gradient hits If by F 2 . This indicates that we have no hope to prevent the energy from extending O(1), even if we rescale the initial data at the beginning of the analysis.

Proof. We deduce from (I-NLS) that

d dt E(Iv(τ )) = Re R 3 ∂ t (Iv) N (Iu) -IN (u) dx + Re R 3 ∂ t (If )N (Iu)dx = Re R 3 ∂ t (Iv)Hdx + Re R 3 i∆(If )N (Iu)dx .
Consequently,

J d dt E(Iv(τ )) dτ ≤ C J R 3 ∂ t (Iv)Hdx dt I + ∥∆(If )N (Iu)∥ L 1 t,x (J) II .
(2.4.12)

Estimate of term I In light of Lemma 2.4.3, it suffices to prove that sup

K∈2 N K -1 ∥(∂ t Iv∥ L ∞ t L 2 x ≤ N 3(1-σ) .
(2.4.13) Indeed, we have

Z I (J) ≤ CN 1-σ , F ω ∞ (R) ≤ C α , F ω 2 (J) ≤ C α N 1-σ 2 -γ .
The first estimate on Z I (J) follows from the assumptions (2.4.8), (2.4.9), and Lemma 2.4.4. Subsequently, we use Lemma 2.4.3 to get that

J R 3 ∂ t (Iu)Hdx dt ≤ CN -1+δ sup K∈2 N K -1 ∥P K ∂ t (Iu) ∥ L ∞ t L 2 x (J) Z I (J) 2 + F ω 2 (J) 2 Z I (J) + F ω ∞ (J) ≤ C(C α )N -1+δ sup K∈2 N K -1 ∥P K ∂ t (Iu) ∥ L ∞ t L 2 x (J) N 3(1-σ) .
Let us turn to the proof of (2.4.13), and fix K ∈ 2 N . Recall that Iv solves

∂ t Iv = i∆Iv -iIN (u) .
First, we have from the definition of Z I (J) and from (2.4.10) that

∥P K ∆Iv∥ L ∞ t L 2 x (J) ≤ CKZ I (J) ≤ CKN 1-σ .
It remains to handle the nonlinear term IN (u) ∼ IN (u (1) , u (2) , u (3) ) with u (i) ∈ v, f . For this purpose, we shall lose some derivatives and apply the fractional Leibniz rule with the operator ∇I, especially when all the three terms are of type v since we only have some controls on norms of ∇Iv, and not on ∇v. One way of loosing some derivatives is to use Sobolev embedding. We eventually prove that

∥P K IN (u)∥ L ∞ t L 2 x (J) ≤ CK Z I (J) 3 + F ω ∞ (J) 3 . (2.4.14)
In what follows, all the spacetime norms will be taken over J × R 3 . First, we observe that

∥v∥ L ∞ t L 3 x ≤ C∥⟨∇⟩ 1 2 v∥ L ∞ t L 2 x ≤ C∥⟨∇⟩Iv∥ L ∞ t L 2
x ≤ CZ I (J) .

This follows from the Sobolev embedding and from the definition of the I-operator (see (2.1.3)).

Next, we perform a case by case analysis depending on the type of terms that occur in the trilinear form.

Case 1 : Three random terms In this case, we use Hölder inequality and the probabilistic estimates (2.2.12) to get

∥P K I(f 3 )∥ L ∞ t L 2 x (J) ≤ ∥f ∥ 3 L ∞ t L 6 x (J) ≤ F ω ∞ (J) 3 ≤ C 3 α .
Case 2 : Two random terms In this case, we use Sobolev embedding, Hölder's inequality and the almost conservation of the mass to obtain

∥P K I(f 2 v)∥ L ∞ t L 2 x ≤ C∥⟨∇⟩P K I(f 2 v)∥ L ∞ t L 6 5 x ≤ CK∥f ∥ 2 L ∞ t L 6 x ∥v∥ L ∞ t L 2 x ≤ CKF ω ∞ (J) 2 ∥f 0 ∥ L 2 x .
Case 3 : One random term Similarly,

∥P K I(f v 2 )∥ L ∞ t L 2 x ≤ C∥⟨∇⟩P K I(f v 2 )∥ L ∞ t L 6 5 x ≤ CK∥f ∥ L ∞ t L 6 x ∥v∥ L ∞ t L 3 x ≤ CKF ω ∞ (J)Z I (J) 2 .
Case 4 : Three deterministic terms Here, we need to use the Leibniz rule for I⟨∇⟩. We obtain

∥P K I(v 3 )∥ L ∞ t L 2 x ≤ C∥⟨∇⟩ 5 4 P K I(v 3 )∥ L ∞ t L 12 11 x ≤ CK∥⟨∇⟩ 1 4 P K I(v 3 )∥ L ∞ t L 12 11 x ≤ CK∥⟨∇⟩ 1 4 Iv∥ L ∞ t L 4 x ∥v∥ 2 L ∞ t L 3 x ≤ CK∥⟨∇⟩Iv∥ L ∞ t L 2 x ∥v∥ 2 L ∞ t L 3 x ≤ CKZ I (J) 3 .
This addresses the contribution for the term I in the energy increment.

Estimate of term II After integrating by parts and applying the Leibniz rule, we write

∥∆(If )N (Iu)∥ L 1 t,x (J) ≤ C J×R 3 |∇(If )N ∇Iu (1)
, Iu (2) , Iu (3) |dtdx (2.4.15) with u (i) ∈ f, v for i ∈ 1, 2, 3 . Next, we perform a case by case analysis to estimate the contribution of the different terms on the right-hand side of (2.4.15).

No random term. In this case, we have u (i) = v for i ∈ 1, 2, 3 . We apply Cauchy-Schwarz inequality to obtain

(2.4.15) ≤ C∥∇If ∥ L 2 t L ∞ x (J) ∥∇Iv∥ L ∞ t L 2 x (J) ∥Iv∥ 2 L 4 t,x ≤ CF ω 2 (J)Z I (J)∥Iv∥ 2 L 4 t,x (J) .
To handle the other cases, we apply Cauchy-Schwarz and we use the estimate (2.2.27).

One random term.

• Case 1 :

u (1) = f, u (2) = u (3) = v. It follows from the almost conservation from Lemma (2.4.1) that (2.4.15) ≤ C∥∇If ∥ 2 L 2 t L ∞ x (J) ∥Iv∥ 2 L ∞ t L 2 x ≤ CF ω 2 (J) 2 .
• Case 2 :

u (1) = u (2) = v, u (3) = f . Similarly, (2.4.15) ≤ C∥∇If ∥ L 2 t L ∞ x (J) ∥If ∥ L 2 t L ∞ x (J) ∥∇Iv∥ L ∞ t L 2 x (J) ∥Iv∥ L ∞ t L 2 x (J) ≤ CF ω 2 (J) 2 Z I (J) .
Two random terms.

• Case 1 :

u (1) = u (2) = f, u (3) = v.
We deduce from Cauchy-Schwarz and from the almost conservation of the mass that 2.4.1 that

(2.4.15) ≤ ∥∇If ∥ 2 L 2 t L ∞ x (J) ∥If ∥ L ∞ t L 2 x (J) ∥Iv∥ L ∞ t L 2 x (J) ≤ CF ω 2 (J) 2 .
• Case 2 :

u (1) = v, u (2) = u (3) = f , u (2) = u (3) = f . In this case, we have (2.4.15) ≤ C∥∇If ∥ L 2 t L ∞ x (J) ∥∇Iv∥ L ∞ t L 2 x (J) ∥If ∥ L 2 t L ∞ x (J) ∥Iv∥ L ∞ t L 2 x (J) ≤ CF ω 2 (J)∥If ∥ L ∞ t L 2 x (J) Z I (J) ≤ CF ω 2 (J)Z I (J) .
Three random terms. Here, we have

u (i) = f for i ∈ 1, 2, 3 , so that (2.4.15) ≤ C∥∇If ∥ 2 L 2 t L ∞ x (J) ∥If ∥ 2 L ∞ t L 2 x (J) ≤ CF ω 2 (J) 2 .
Collecting all the contributions, we obtain the desired estimate on the modified energy increment.

Modified interaction Morawetz

We revisit the standard interaction Morawetz inequality in a perturbed setting, for the equation (I-NLS) satisfied by Iu. First, we recall the formal computations that leads to the so-called Lin-Strauss inequality. Given a convex weight a, we define the action14

M 0 (t) = 2Im R 3 ∂ j a(x)(∂ j Iu)(x)Iu(x)dx = 2Im R 3 ∂ j aT 0,j dx .
It corresponds to the contraction of the current density T 0,j = T j,0 = 2Im ∂ j (Iu)Iu against the vector field ∇a. Thanks to the convexity of a, we hope to get some monotonicity for M 0 . The general virial identity reads

d dt M y (t) = R 3 -∆ 2 a|Iu| 2 + 4∂ 2 kj aRe(∂ k (Iu)∂ j (Iu)) + ∆a|Iu| 4 + 2Re 2∂ j a∂ j (Iu)H + ∆a(Iu)H dx .
Given a point y ∈ R 3 , the recentered Morawetz action, denoted M y , corresponds to the particular weight a y (x) = |x -y|, that satisfies

∂ j a = x j -y j |x -y| , ∆a = 2 |x -y| , ∆ 2 a = -2πδ y , ∂ 2 jk a = I - x -y |x -y| ⊗ x -y |x -y| =: P .
Hence,

- d dt M y (t) = π|Iu(t, y)| 2 + 2 R 3 | P ∇Iu(x)| 2 dx + R 3 1 |x -y| |Iu(x)| 4 dx + 4Re x j -y j |x -y| ∂ j (Iu)Hdx + 4Re R 3 1 |x -y| IuHdx . (2.4.16)
Observe the presence of two extra terms, that come from the forcing term H. Next, we average the recentered Morawetz action against the mass density |Iu(y)| 2 to obtain the Morawetz interaction,

R 3 M y (t)|Iu(y)| 2 dy .
Note that this quantity is bounded by the conserved L 2 norm times the critical

Ḣ 1 2 (R 3 ) norm of Iu. | R 3 M y (t)|Iu(y)| 2 dy| ≤ C∥Iu(t)∥ 2 L 2 x ∥Iu(t)∥ 2 Ḣ 1 2 .
(2.4.17)

We can now state and prove the modified Morawetz interaction, that will provide a spacetime control on Iu.

Proposition 2.4.6 (Modified interaction Morawetz). Assume that J ⊂ J * is an interval such that Z I (J k ) < ∞. Then, for any δ > 0, there exists C > 0 such that for any partition J = L-1 k=0 J k , there holds

J×R 3 |Iu| 4 dtdx ≤ C∥Iu∥ 2 L ∞ Ḣ 1 2 (J) ∥Iu∥ 2 L ∞ L 2 (J) + CN -1+δ ∥Iu∥ L ∞ t L 2 x (J) ∥Iu∥ 2 L ∞ t H 1 2 (J) L-1 k=0 Z I (J k ) 2 + F 2 (J k ) 2 Z I (J k ) + F ∞ (J k ) . (2.4.18)
Proof. By using the Leibniz rule, and then integrating over time, we obtain

R 3 M y (t)|Iu(y)| 2 dy t=t 1 t=t 0 = J×R 3 |Iu(t, y)| 2 d dt R 3 x j -y j |x -y| T 0,j (x)dx dydt I + J×R 3 ×R 3 x j -y j |x -y| T 0,j (x)2Re(∂ t Iu(y)Iu(y))dxdydt II .
Next, we plug (2.4.16) into I and II. We have

I = π J×R 3 |Iu(t, y)| 4 dydt + 2 J×R 3 ×R 3 |P x,y ∇u(t, x)| 2 |Iu(y)| 2 dxdydt + J×R 3 ×R 3 2 |x -y| |Iu(x)| 4 |Iu(y)| 2 dxdydt + 4Re R×R 3 ×R 3 |Iu(y)| 2 x j -y j |x -y| ∂ j (Iu)(x)H(x)dxdydt I 1 + 4Re J×R 3 ×R 3 |Iu(y)| 2 |x -y| Iu(x)H(x)dxdydt I 2 . (2.4.19)
To expand the term II we use equation (I-NLS) and we write the time derivative of Iu in terms of the perturbation H.

II = - J×R 3 ×R 3 x j -y j |x -y| T 0,j (x)2Re(i∆Iu(y)Iu(y))dxdydt + J×R 3 ×R 3 x j -y j |x -y| I 0,j (x)2Re(iIu(y)H(y))dxdydt II 1 . (2.4.20)
Next, we observe that the terms which does not come from the perturbation have a sign. Hence, we can use the bound (2.4.17) and see that

π J×R 3 |Iu(t, y)| 4 dydt ≤ 2C∥Iu∥ 2 L ∞ t L 2 x (J) ∥Iu∥ 2 L ∞ t Ḣ 1 2 (J) + | I 1 + I 2 + II 1 | .
We are left to estimate three terms I 1 , I 2 and II 1 , that contain a spacetime norm of the commutator H. Hence, we will use Lemma 2.4.3 to address these three terms. Up to a partition, and using sub-additivity, we may assume that L = 1.

Term I 1 : We use the triangular inequality and Lemma 2.4.3 to get

| I 1 | ≤ ∥Iu∥ 2 L ∞ t L 2 y (J) J sup y R 3 x j -y j |x -y| ∂ j (Iu)(x)H(x)dx dt ≤ CN -1+δ ∥Iu∥ 2 L ∞ t L 2 y (J) sup y∈R 3 K∈2 N K -1 ∥P K x j -y j |x -y| ∂ j (Iu)(x) ∥ L ∞ t L 2 x (J) Z I (J) 2 + F ω 2 (J) 2 Z I (J) + F ω ∞ (J) .
We fix y ∈ R 3 , t ∈ J and K ∈ 2 N , and we proceed as follows.

∥P K x j -y j |x -y| ∂ j (Iu)(x) ∥ L 2 x = sup ∥g∥ L 2 ≤1 R 3 x j -y j |x -y| ∂ j (Iu)(t, x) P K g(x)dx
We deduce from the Leibniz rule that for each

x ∈ R 3 , x -y |x -y| • ∇Iu(x) = div x -y |x -y| Iu(x) - 2 |x -y| Iu(x) .
Next, we apply Hardy's inequality to handle the singular term with |x -y| -1 , and we obtain

∥P K x j -y j |x -y| ∂ j (Iu)(x) ∥ L 2 x ≤ sup ∥g∥ L 2 ≤1 R 3 x -y |x -y| • Iu(t, x)∇P K g(x) + Iu(t, x) 2 |x -y| P K g(x)dx ≤ C∥Iu∥ L 2 x sup ∥g∥ L 2 ≤1 K∥g∥ L 2 x + ∥ 1 |x -y| P K g(x)∥ L 2 x ≤ CK∥Iu∥ L 2 x .
This addresses the estimate for the term I 1 :

| I 1 | ≤ CN -1+δ ∥Iu∥ 3 L ∞ t L 2 x (J) Z I (J) 2 + F ω 2 (J) 2 Z I (J) + F ω ∞ (J) .
Term I 2 : We proceed similarly. In this case, we are left to estimate

∥P K Iu(x) |x -y| ∥ L 2 x (J) = sup ∥g∥ L 2 ≤1 R 3 Iu(t, x) 1 |x -y| P K g(x)dx ≤ ∥Iu∥ L 2 x ∥ 1 |x -y| g∥ L 2 x ≤ CK∥Iu∥ L 2 x ∥g∥ L 2 x ,
where we used Hardy's inequality in the last line. Hence,

| I 2 | ≤ CN -1+δ ∥Iu∥ 3 L ∞ t L 2 x (J) Z I (J) 2 + F ω 2 (J) 2 Z I (J) + F ω ∞ (J) .
Term II 1 :

| II 1 | ≤ sup t,y R 3 x j -y j |x -y| I 0,j (x)dx J R 3 Iu(y)H(y)dy dt ≤ CN -1+δ ∥Iu∥ 2 L ∞ t Ḣ 1 2 (J) ∥Iu∥ L ∞ t L 2 x (J) Z I (J) 2 + F ω 2 (J) 2 Z I (J) + F ω ∞ (J) .
This concludes the proof of Proposition 2.4.6.

Almost-sure global well-posedness and scattering

In this section, we glue together the pieces collected in the previous sections to prove the main theorem. First, given α > 0 we take ω ∈ Ω α such that

F ω (R) + F ω ∞ (R) ≤ C α .
Then, we apply the local probabilistic Cauchy theory developed in Section 2.3.2. For such an ω, we have a local solution u = e it∆ f ω 0 + v to (NLS) with initial data u |t=0 = f ω 0 , where v satisfies (NLS f ) in H σ (R 3 ) with zero initial condition, up to a maximal lifetime T * . In addition, v extends globally in time scatters in H σ (R 3 ) as soon as its H σ (R 3 ) norm is uniformly bounded up to the time T * . To obtain such an a priori uniform bound on v, we perform a double bootstrap argument with a modified interaction Morawetz and the modified energy E(v). Recall that we obtained in this section a priori estimates on the time increments of these two quantities, under the extra assumption that f 0 is radially-symmetric, and that ω ∈ Ω α , yielding a control the L 2 t (R; L ∞ x (R 3 )) norm of ⟨∇⟩I e it∆ f ω 0 .

Double bootstrap argument

Recall that we fixed α > 0, N 0 (α) large enough and Ω α with P(Ω α \ Ω α ) ≤ α, such that for every ω ∈ Ω α and N ≥ N 0 (α), there holds (2.2.27). Furthermore, we can assume from the large deviation estimate on the L p x norm of the randomized initial data (see [START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF], Lemma 2.3), and from the fact that the I-operator commutes with the randomization and that it is bounded on L 2 , that for all ω ∈ Ω α and all N ,

∥If ω 0 ∥ L 4 x ≤ C α . (2.5.1)
Then, we write C the constant that appears in the modified Morawetz inequality (2.4.18), and we define

M 0 := ∥u∥ L ∞ t L 2 x ([0,T * )) , M = 4CM 3 0 . (2.5.2)
It follows from the almost conservation of the mass from Lemma 2.4.1 that the bound on M 0 does only depend on ∥f 0 ∥ L 2 x . Afterwards, we consider the set

Θ N = T ∈ [0 , T * ) | sup t∈[0,T ] E(v(t)) ≤ N 2(1-δ) , ∥Iv∥ 4 L 4 t,x [0,T ] ≤ M N 1-σ .
We claim that for every 6 7 < σ < 2s ≤ 1, there exists N ′ 0 (ε, ∥f 0 ∥ L 2 x (R 3 ) , s, σ) to be defined later such that for all N ≥ max(N 0 , N ′ 0 ), we have

Θ N = [0 , T * ) . (2.5.3)
Before diving into the proof, we note that Θ N is nonempty. Indeed, it follows from (2.5.1) and from σ) , and hence T = 0 ∈ Θ N . Moreover, we see from a continuity argument that θ N is also closed. Hence, it remains to prove that Θ N is an open interval. Let us fix T ∈ Θ N , with T < T * . By the local well-posedness theory for (NLS f ) presented in Proposition 4.2.2, there exists a small T 2 > T such that

v(0) = 0 that E(v)(0) = 1 4 ∥If ω 0 ∥ 4 L 4 x ≤ C α ≪ N 2(1-
∥Iv∥ 4 L 4 t,x [0,T 2 ] ≤ 2M N 1-σ . (2.5.4)
The double bootstrap argument is the following. First, by using the a priori estimate (2.5.4), by tuning the parameter and by summing over the spacetime slabs where

∥Iv∥ 4 L 4 t,x [T 1 ,T ] ≤ ε N ,
we observe from the local-in-time energy increment of Proposition 2.4.5 that if a time T 1 is such that E(v(T 1 )) ≤ N 2(1-σ) , where ε N ∼ N -2(1-σ) is as in Lemma 2.4.4, then the energy remains less than N 2(1-σ) as long as we control the L 4 t,x norm of Iv. Next, we deduce from the modified interaction Morawetz (2.4.18) that we eventually have (2.5.4) with M N 1-σ on the right-hand side, instead of 2M N 1-σ . Proposition 2.5.1 (Double bootstrap argument). Let 6 7 < σ < 2s. There exists

N 0 (ε, σ, s, ∥f 0 ∥ L 2 ) such that for all N ≥ max(N 0 , N 0 ), if T 1 is in Θ N and T 1 < T 2 < T * is such that ∥Iv∥ 4 L 4 t,x (0,T 2 ) ≤ 2M N 1-σ , then T 2 is also in Θ N .
Proof. Given a fixed N ≥ N 0 , we consider ε N = N -2(1-σ)-δ as in Lemma 2.4.4, where δ > 0 is arbitrarily small. As a consequence of the assumption ∥Iv∥ L 4 t,x [0,T 2 ] ≤ 2M N 1-σ , there are at most L = 2M N 1-σ ε -1 N spacetime-slabs on which the smallness condition ∥Iv∥ 4

L 4 t,x ≤ ε N is satisfied. We decompose therefore [0 , T 2 ) into L intervals J k = [t k , t k+1 ) such that for k ∈ 0, . . . , L -1 ∥Iv∥ 4 L 4 t,x (J k ) ≤ ε N , with 0 = t 0 < t 1 < • • • < t L = T 2 , and L = 2M N 1-σ ε N = 2M N 3(1-σ)+δ .
In order to apply Lemma 2.4.4 on each spacetime slab J k , we need the L 4 t L 3

x norm of f on J k to be less than η, which is an universal constant. This is possible up to some extra decomposition of the intervals J k into a finite number of sub-intervals, that only depends on ∥f 0 ∥ H s and on η. Hence, the total number of spacetime-slabs on which we perform the analysis is bounded by

L ≤ C(f 0 , η) 2M N 1-σ ε N ≤ C(f 0 , η)2M N 3(1-σ)+δ .
Now, by iteration on k, we deduce from Proposition 2.4.5 that

sup t∈[0,t k ] E(v(t)) ≤ N 2(1-σ) .
(2.5.5) First, we note from the definition of Θ N that (2.5.5) is already known when t k ≤ T 1 , since we assumed that T 1 ∈ Θ N . Afterwards, we consider k such that (2.5.5) holds for t k . We prove that it still holds for any t ∈ J k . To do so, we fix t ∈ J k and we use sub-additivity :

E(v(t)) ≤ E(v(0))) + t k+1 0 d dt E(Iv(τ )) dτ = E(v(0))) + 0≤i≤k-1 J i d dt E(v(τ )) dτ .
Initially, we know from (2.5.1) that E(v(0)) ≤ C α . Moreover, we have by construction that E(Iv(t i )) ≤ N 2(1-σ) , and ∥Iv∥ 4 L 4 t,x (J i ) ≤ ε N for each i ∈ 0, . . . , k . Hence, we can apply the modified energy increment (2.4.11) on each interval J i , and get

E(v(t)) ≤ C α + C 0 0≤i≤k-1 N 6(1-σ)-1 + N 1-σ F 2 (J i ) 2 + N 1-σ F 2 (J i )∥Iv∥ 2 L 4 t,x (J i ) ,
where C 0 = C 0 (f 0 , s, σ) is a universal constant that arises in (2.4.11). Next, we brutally multiply the first term by the number of spacetime-slabs 15 , and we use sub-additivity for the second term and the third term. When summing over the spacetime slabs, we also apply Cauchy-Schwarz on the third term, and we obtain

E(v(t)) ≲ C α + LN 6(1-σ)-1 + N 1-σ 0≤i≤k-1 F 2 (J j ) 2 + 0≤i≤k-1 F 2 (J i ) 2 1 2 0≤i≤k-1 ∥Iv∥ 4 L 4 t,x (J j ) 1 2
.

We conclude by using the upper bound on L and the estimate (2.2.27) on F 2 that

E(v(t)) ≤ C α + C 0 LN 6(1-σ)-1 + N 1-σ F ω 2 (R) 2 + F ω 2 (R)(2M N 1-σ ) 1 2 ≤ C α + C 0 2M N 9(1-σ)-1+δ + N 1-σ N -γ N 1-σ + N 0-N 1-σ 2 N 1-σ 2 ≤ N 2(1-σ) C α N -2(1-σ) + 2C 0 M N 7(1-σ)-1+δ + N -2γ + N -γ ≤ N 2(1-σ) ,
provided that 6 7 < σ, and

N ≥ N ′ 0 = N ′ 0 (C 0 , γ, M
). This finishes the first part of the proof of Proposition 2.5.1. To show that T 2 ∈ Θ N , it remains to prove that

[0,T 2 )×R 3 |Iv| 4 dtdx ≤ M N 1-σ .
For this purpose, we perform the same analysis as in the first part of the proof to handle the remainders that contain the commutator H. Namely, we decompose [0 , T 2 ] into L intervals J k 0≤k≤L-1 such that (2.4.8) and (2.4.9) hold on each J k . Since we already know from Proposition (2.5.1) that sup

t≤T 2 E(v(t)) ≤ N 2(1-σ) ,
15 If we prove a long time Strichartz estimate as in [START_REF] Dodson | Global well-posedness and scattering for nonlinear Schrödinger equations with algebraic nonlinearity when d = 2, 3 and u 0 is radial[END_REF], we might be able to use also some sub-additivity to estimate the first term, and to loosen therefore our restriction on s > 3 7 .

we can deduce from Lemma 2.4.4 that for each k

Z N,J k (v) ≤ CN 1-σ .
Therefore, it follows from the modified interaction Morawetz that (2.4.18)

[0,T 2 )×R 3 |Iu| 4 dtdx ≤ C ∥Iv∥ 2 L ∞ t ([0,T 2 ]; Ḣ 1 2 ) + ∥If ∥ 2 L ∞ t ([0,T 2 ]; Ḣ 1 2 ) ∥Iu∥ 2 L ∞ t ([0,T 2 ];L 2 x ) + CN -1+δ L∥Iu∥ L ∞ t ([0,T 2 ];L 2 x ) ∥Iu∥ 2 L ∞ t ([0,T 2 ]; Ḣ 1 2 ) N 2(1-σ) + C 2 α N 1-σ N 1-σ + C α .
As a consequence of the global energy estimate on [0 , T 2 ] shown in the first part of the proof, we deduce from interpolation and from the coercivity of E(v) that

∥Iv∥ 2 L ∞ t ([0,T 2 ]; Ḣ 1 2 ) ≤ ∥Iv∥ L ∞ t ([0,T 2 ]; Ḣ1 ) ∥Iv∥ L ∞ t ([0,T 2 ];L 2 x ) ≤ √ 2E(v) 1 2 M 0 ≤ √ 2M 0 N 1-σ . (2.5.6) Moreover, ∥If ∥ 2 L ∞ Ḣ 1 2 ≤ CN 1-2s ∥f 0 ∥ 2 H s ≤ C(f 0 )N 0-N 1-σ , (2.5.7)
for some irrelevant constant C(f 0 ) = C(∥f 0 ∥ H s , s, σ). Consequently, the almost conservation of the mas and estimates (2.5.6), (2.5.7) yield

C ∥Iv∥ 2 L ∞ t ([0,T 2 ]; Ḣ 1 2 ) + ∥If ∥ 2 L ∞ t ([0,T 2 ]; Ḣ 1 2 ) ∥Iu∥ 2 L ∞ t ([0,T 2 ];L 2 x ) ≤ CM 3 0 √ 2N 1-σ + C(f 0 ) 1-σ-δ ≤ 2CM 3 0 N 1-σ ≤ M 2 M 3 0 N 1-σ , (2.5.8)
provided that 6 7 < σ, and that δ is taken sufficiently small. We conclude that

[0,T 2 )×R 3 |Iu| 4 dtdx ≤ M 2 N 1-σ + N 1-σ N 0-C 0 + C ′ 0 .
Moreover, using the bound 16 on F ω written in (2.1.11),

[0,T 2 )×R 3 |Iu| 4 dtdx ≲ [0,T 2 )×R 3 |If | 4 dtdx + [0,T 2 )×R 3 |Iu| 4 dtdx ≤ 4F ω ∞ ([0 , T 2 ]) 4 + M 2 N 1-σ + 4N 0-(C 0 + C ′ 0 )N 1-σ ≤ M N 1-σ .
In the last estimate, we took N ≥ N 0 " = N 0 "(f 0 ) large enough. This concludes the proof of Proposition 2.5.1 with N 0 = max(N ′ 0 , N 0 "). 16 We actually control F ω defined with f instead of If . But we can easily replace f by If in the proof, if we replace f 0 by If 0 using the fact I is a Fourier multiplier and commutes therefore with the randomization. Moreover, the operator norm of I acting on L 2 is less than 1. Hence, we do not lose any power of N .

Proof of Theorem 2.1.4

We proved that for every α > 0 and ω ∈ Ω α as in Lemma 2.2.27, the Cauchy problem (NLS f ) admits a unique local solution v, with a maximal lifespan [0 , T * ). Moreover, we find a set Ω α such that P(Ω α \ Ω α ) ≤ α, where we proved in Proposition 2.5.1 by a double bootstrap argument on E(v) and a modified Morawetz estimate that if ω ∈ Ω α , there exists N 0 = N 0 (ε, ∥f 0 ∥ H s , s, σ) such that for all N ≥ N 0 , we have Θ N = [0 , T * ). In particular, sup

0≤t≤T * E(v(t)) ≤ N 2(1-σ) .
Hence, we deduce from the above estimate on the coercive modified energy, from the operator bound (2.1.4) for the I-operator and from the almost conservation of the mass, that there exists N large enough and a constant C N such that sup

0≤t≤T * ∥v∥ H σ x ≤ ∥v∥ L ∞ ([0,T * ];L 2 ) + CN 1-σ ∥Iv∥ L ∞ ([0,T * ]; Ḣ1 ) ≤ CN 3(1-σ) =: C N .
In what follow we drop the dependence on N . Since we have a global a priori estimate on the H σ norm of v, we can apply the stability theory presented in Section 2.3.2, and deduce from Proposition 2.3.8 that T * = +∞, and that v scatters at infinity. Finally, we take

Σ = α>0 Ω α .
Since P(Ω \ Ω α ) ≤ 2α for every α > 0, we have that P(Σ) = 1. By construction, a local solution to (NLS) initiated from an initial data u(0) = f ω 0 ∈ H s with ω ∈ Σ extends globally in time and scatters.

Chapter 3 Asymptotic stability of small ground states for NLS under random perturbations

From the publication [START_REF] Camps | Asymptotic stability of small ground states for NLS under random perturbations[END_REF].

Résumé. Nous étudions l'équation de Schrödinger cubique dans l'espace euclidien, perturbée par un potentiel de courte portée V . La présence d'une valeur propre négative pour -∆ + V génère une famille à un paramètre d'états fondamentaux non linéaires, qui sont localisés en espace et qui donnent lieu à de petites solutions périodiques, asymptotiquement stables dans l'espace d'énergie. Nous étudions la persistance de ces états cohérents sous l'effet de perturbations irrégulières et aléatoires. Plus précisément, nous construisons un ensemble de mesure pleine constitué de petites solutions sur-critiques, qui ont en outre une énergie infinie, mais qui laissent la famille d'états fondamentaux asymptotiquement stable. La principale difficulté réside dans l'analyse des interactions de termes localisés en espace avec des termes dispersifs qui ont une faible régularité.

Abstract. We consider the cubic Schrödinger equation on the Euclidean space perturbed by a short-range potential V . The presence of a negative simple eigenvalue for -∆ + V gives rise to a curve of small and localized nonlinear ground states that yield some time-periodic solutions known to be asymptotically stable in the energy space. We study the persistence of these coherent states under rough perturbations. We shall construct a large measure set of small scaling-supercritical solutions below the energy space that display some asymptotic stability behavior. The main difficulty is the need to handle the interactions of localized and dispersive terms in the modulation equations. To do so, we use a critical-weighted strategy to combine probabilistic nonlinear estimates in critical spaces based on U p , V q (controlling higher order terms) with some local energy decay estimates (controlling lower order terms). We also revisit in the perturbed setting the analysis of [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF] on the probabilistic global well-posedness and scattering for small supercritical initial data. We use a distorted Fourier transform and semiclassical functional calculus to generalize probabilistic and bilinear Strichartz estimates.

Introduction

In this paper we consider the cubic Schrödinger equation perturbed by a short-range potential in the Euclidean space of dimension d ≥ 3:

i∂ t ψ + ∆ψ = µ|ψ| 2 ψ + V ψ , ψ t=0 = ψ 0 .
(NLS)

The constant µ, whose sign dictates whether the nonlinearity is focusing or defocusing, will play no role since we look at small initial data. Hence, we shall fix µ = 1. The short-range potential V , whose properties are detailed below, is assumed to be in the Schwartz class S(R d ). Our main result concerns the case when d = 3 and σ(H) = e 0 ∪ σ c (H) with no resonance at zero, and where e 0 < 0 is a simple negative eigenvalue with positive and normalized eigenfunction ϕ 0 . Then, the 1-dimensional eigenspace spanned by ϕ 0 bifurcates around zero to a family of small and localized nonlinear ground states. These ground states satisfy the elliptic equation

∆ -V + |Q| 2 Q = EQ , (3.1.1)
and are written under the form

Q(z) = zϕ 0 + q(z), E(z) = e 0 + e(z) , (3.1.2)
where z is the complex modulation parameter. They give rise to periodic solutions to (NLS) of the form u(t, x) = e -itE Q(x). Soffer and Weistein [SW90], followed by Pillet and Wayne [START_REF] Pillet | Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations[END_REF] and Gustafson & al. [GNT04] addressed the problem of asymptotic stability of these small ground states in the energy space. More precisely, any small local solution ψ to (NLS) in H 1 (R 3 ) is global and can be decomposed into

ψ(t) = Q(z(t)) + η(t, z(t)), (3.1.3)
where the radiation term η satisfies a time dependent orthogonality condition (see (3.1.7)). In large time, asymptotic stability holds in the sense that z(t) has a limit as t goes to infinity, and η(t) scatters in H 1 (R 3 ). This result is a particular instance of the so-called soliton resolution conjecture (see [START_REF] Tao | Why are solitons stable?[END_REF] for a general presentation). Our aim is to prove that the asymptotic stability property of these coherent states still holds at very low regularity, in a supercritical regime where we have a local probabilistic flow. Before we present our main result, we recall that the critical exponent s c for the homogeneous (NLS) equation (with V=0) is the regularity exponent for which the homogeneous Sobolev norm is invariant under the scaling

u λ (t, x) = λ -1 u(λ -2 t, λ -1 x) , u 0,λ (x) = λ -1 u 0 (λ -1 x) .
For the cubic Schrödinger equation in R d , we have

s c = d -2 2 .
In light of the conservation laws of mass and energy

M(ψ)(t) = R d |ψ(t, x)| 2 dx, E(ψ)(t) = R d 1 2 |∇ψ(t, x)| 2 + µ 4 |ψ(t, x)| 4 + V |ψ(t, x)| 2 dx ,
we say that the problem is mass-critical when s c = 0 and energy-critical when s c = 1. There exists a vast literature on the Cauchy theory for (NLS) and we refer to the books [BKK07; Caz03; Dod19a; Tao06] or to the survey [GV79a; GV79b] and the references therein for a general presentation. Note that the equation is energy-subcritical for d = 3, and we have a local flow for s ∈ [1/2 , 1], where s c = 1/2. On the other hand, in the scaling-supercritical regime where s < s c , the local Cauchy problem for (NLS) is known to be ill-posed (see [START_REF] Christ | Instability of the periodic nonlinear Schrodinger equation[END_REF]). Nevertheless, the probabilistic Cauchy theory initiated by Bourgain [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] and developed by many authors since then provides some large measure sets made of scaling-supercritical initial data u ω 0 in H s (R d ) for s < s c that give rise a local solutions to (NLS) of the form ψ = u ω + v. Here, u ω = e -itH u ω 0 is the propagation of u ω 0 under the linear flow and v is solution to the cubic Schrödinger equation with a random forcing term, but with v(0) = 0.

Main result

We consider (NLS) in dimension d = 3 with small randomized initial data that lie in H s (R 3 ) ∩ Ran P c (H) for some s ∈ (1/4 , 1/2], where P c (H) is the projection onto the continuous spectral subspace for H, and we assume that σ(H) = e 0 ∪ σ c (H) with no resonance at zero, where e 0 < 0 is a simple negative eigenvalue. We generalize the result of [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF] in the presence of such of potential by first constructing a local probabilistic flow, and extending to a global flow. Then, we prove that at each time of the evolution, the solution decouples into a sum of a ground state and a radiation term that scatters at infinity. More precisely, we fix s ∈ (1/4 , 1/2], a function u 0 in H s and a small parameter ε. Then, we perform the Wiener randomization procedure on u 0 as detailed in section 3.3.1, and we get a large measure set Ω ε made of rough and small initial data u ω 0 ∈ H s (R 3 ) with improved Strichartz estimates. These initial data give rise to global solutions to (NLS) under the form (3.1.3) that display an asymptotic stability dynamic.

Theorem 3.1.1 (Probabilistic asymptotic stability of small ground states). Assume that H has no resonance at zero and that σ(H) = e 0 ∪ σ c (H) with e 0 < 0 a simple eigenvalue. There exist a set Ω ε and δ 0 such that for all ψ 0 with ∥ψ 0 ∥ H 1/2 < δ 0 and all ω ∈ Ω ε , the initial value problem

i∂ t ψ + ∆ψ = |ψ| 2 ψ + V ψ , (t, x) ∈ R × R 3 , ψ |t=0 = εu ω 0 + ψ 0 , (3.1.4)
admits a unique global-in-time solution ψ of the form

ψ(t) = ε e it(∆-V ) u ω 0 + v(t), where v ∈ C(R, H 1/2 (R 3 )) .
Moreover, the solution uniquely resolves into ψ(t) = Q(z(t)) + η(t), and there exist z + ∈ C and a final state

η + ∈ H 1/2 (R 3 ) ∩ Ran(P c ) such that lim t→+∞ z(t) exp i t 0 E(z)dτ = z + , lim t→+∞ ∥η -e -itH εu ω 0 + η + ∥ H 1/2 (R 3 ) = 0 . (3.1.5)
Remark 3.1.2. The measure of the set Ω ε of initial with improved Strichartz estimates is all the more large since ε∥u 0 ∥ H s is small. There exist C, c > 0 such that for all u 0 and ε > 0,

P Ω \ Ω ε ≤ C exp -cε -2 ∥u 0 ∥ -2 H s (R 3 ) .
Remark 3.1.3. We have conditional uniqueness for v in the critical space defined in (3.4.13), and embedded into L ∞ (R, H 1/2 ). Hence, v gains some regularity and lies in a space where there exists a deterministic Cauchy theory.

By taking u 0 = 0 in the statement of Theorem 3.1.1 and using persistence of regularity, we may extend the deterministic result of [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF] to the intercritical regime, where 1/2 ≤ s ≤ 1.

Corollary 3.1.4 (Deterministic asymptotic stability). For 1/2 ≤ s ≤ 1, there exists δ 0 > 0 such that for all ∥ψ 0 ∥ H s ≤ δ 0 , the Cauchy problem (NLS) with initial data ψ 0 has a unique global solution ψ ∈ C(R; H s ) that resolves into ψ(t) = Q(z(t)) + η(t), and asymptotic stability (3.1.5) holds in H 1/2 .

Background

Asymptotic stability for small solitons in the energy space Previous results on the asymptotic stability of small ground states hold for initial data ψ 0 in the energy space H 1 (R 3 ), where (NLS) is known to be well-posed. Soffer and Weinstein proved in [START_REF] Soffer | Multichannel nonlinear scattering for nonintegrable equations[END_REF] that the equation displays some multichannel scattering for small and localized data in the energy space. More precisely, any initial data small in H 1 (R 3 ) ∩ L 1

x (R 3 ) gives rise to a global solution that resolves into a fixed ground state and a radiation term whose L 6

x (R 3 )-norm decays to zero, as well as some of its L 2 x (R 3 )-weighted norm (see Theorem 4.1 in [START_REF] Soffer | Multichannel nonlinear scattering for nonintegrable equations[END_REF]). By the use of the center stable manifold method, Pillet and Wayne then extended this result in [START_REF] Pillet | Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations[END_REF] to the case where the initial data are localized in some L 2

x (R 3 )-weighted spaces rather than in L 1 x (R 3 ). While these works impose some fixed orthogonality conditions to the modulation parameters, Gustafson, Nakanishi and Tsai introduced a time-dependent orthogonality condition that leads to an asymptotic stability result in H 1 (R 3 ) (Theorem 1.7 in [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF]) without any assumption on the decay of the initial data, and the radiation term scatters in H 1 (R 3 ). A natural question that arises as was to whether the asymptotic stability holds true below the energy space. Indeed, we still have a local flow for 1/2 ≤ s ≤ 1, and a local probabilistic flow for 1/4 < s < 1/2. We note that our proof does not use any decay assumption on the initial data. Besides, the randomized initial are not likely to decay for a general u 0 (see the discussion in paragraph 3.3.1).

Probabilistic well-posedness theory Even though the Cauchy problem (NLS) is in general ill-posed for scaling-supercritical initial data below H 1/2 (R 3 ), the probabilistic method can provide some large measure sets of initial data that give rise to global-in-time solutions. Many works were done in this direction for different dispersive PDE's after the pioneer work of Jean Bourgain [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] for the NLS equation on T 2 , followed by the work of Burq and Tzvetkov on the nonlinear wave equation (NLW) on a 3D compact Riemannian manifold without a boundary [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF][START_REF] Burq | Random data Cauchy theory for supercritical wave equations II: a global existence result[END_REF]. The idea is to find a randomization procedure which improves some Strichartz estimates on the free evolution of the randomized data, but does not improve their regularity.

To extend local well-posedness below the scaling-critical Sobolev space, the strategy consists in decomposing the solution into ψ = u ω + v where u ω := e -itH u ω 0 is the linear evolution of the randomized data. The remaining term v is smoother and satisfies (NLS) with a stochastic forcing term:

i∂ t ψ -Hψ = N (ψ), ψ| t=0 = ψ 0 + u ω 0 . ⇐⇒ i∂ t v -Hv = N (v + u ω ), v| t=0 = ψ 0 . (3.1.6)
Besides, we need to solve equation (3.1.6) for v in a well-chosen subspace of C(I, H sc ) by a contraction mapping argument. Thanks to its random structure, the linear evolution of the randomized data u ω displays some enhanced integrability properties that make it possible to gain regularity on the stochastic Duhamel term. After that, we globalize the local solutions by using the Bourgain invariant measure argument in the case of T d , or by using a priori estimates on some critical norm of v.

The general randomization procedure consists in finding a well-chosen decomposition of the initial data and in decoupling the terms of this decomposition by multiplying each of them by some independent random variables g n (ω) n centered around zero. Then, taking averages cancels interference and improves therefore the integrability of the data. There exists many versions of the randomization procedure, and we refer the interested reader to the survey [START_REF] Bényi | On the Probabilistic Cauchy Theory for Nonlinear Dispersive PDEs[END_REF] and the references therein for a detailed presentation. One has to distinguish between two cases. In the confining case where the physical space is a compact manifold, or when the equation contains a confining potential, we consider the decomposition u ω 0 = n g n (ω)u n e n (x) where e n is an orthonormal basis made of eigenfunctions of the Schrödinger operator. In the Euclidean case, the Schrödinger operator does not provide such a natural decomposition, and we use instead unit scale frequency decomposition on Wiener cubes (see [BOP15a; BOP15b; DLM19; DLM20]), or microlocal decomposition (see [START_REF] Bringmann | Almost sure scattering for the energy critical nonlinear wave equation[END_REF][START_REF] Burq | Randomization improved Strichartz estimates and global well-posedness for supercritical data[END_REF][START_REF] Murphy | Random data final-state problem for the mass-subcritical NLS in L 2[END_REF]). In some specific cases, we can also use a compactifying transformation such as the Lens transform for NLS (see [START_REF] Burq | Almost sure scattering for the one dimensional nonlinear Schrödinger equation[END_REF]) or the Penrose transform for NLW (see [De 13]) to apply in the Euclidean setting variants of the Bourgain invariant measure argument. Nevertheless, it must be emphasized that the probabilistic method used in the aforementioned works concerns perturbations of the zero solution and ends up with asymptotic results like global well-posedness and scattering for small data or in the defocusing case.

Still, Kenig and Mendelson recently addressed in [START_REF] Kenig | The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data[END_REF] the problem of asymptotic stability of large solitons for the quintic focusing wave equation with randomized radial initial data. They introduced a randomization procedure based on the distorted Fourier transform adapted to the linearized operator around a soliton, and proved some intricate kernel estimates due to the presence of a resonance at zero for this operator. Then, Bringmann showed the stability of the ODE blow-up for the radial energy critical NLW in 4D under random perturbations below the energy space [START_REF] Bringmann | Stable blowup for the focusing energy critical nonlinear wave equation under random perturbations[END_REF]. However, for nonlinear focusing Schrödinger equations the asymptotic stability around solitons is more subtle. Indeed, the solitons for (NLS) with V = 0 are stable when the nonlinearity is small and the critical exponent is negative. Hence, we cannot run the probabilistic strategy since the smoother term v, which is expected to lie in the critical space C(R; H sc (R d )), cannot serve as a substitute for the conservation laws. In addition, the linearized operator around the soliton is not self-adjoint. We refer to [CM21; Tao08] and the references therein for a general insight on the results and techniques about stability of solitons for NLS.

Schrödinger equation with a short-range potential The study of dispersive PDE's with a potential is a very general problem. For instance, a potential can arise in the modulation equations obtained to address stability problems around soliton solutions by the analytic method (see [START_REF] Tao | Why are solitons stable?[END_REF]). There exists a vast literature about Schrödinger equations perturbed by a localized potential (see for instance [GHW14; GPR18; PS20]). These works rely on the use of a distorted Fourier transform F V , which is the analogue adapted to H of the Fourier transform. When it exists, the distorted Fourier transform defines a partial isometric map onto L 2 ac (R d ) that conjugates H with the operator of multiplication by |ξ| 2 . Additionally, the distorted Fourier transform is related with the wave operators W ± . We refer the interested reader to the seminal work of Agmon [START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF], grounded on the previous works [AS71 ;[START_REF] Ikebe | Eigenfunction expansions associated with the Schrödinger operators and their applications to scattering theory[END_REF]. In the probabilistic context, we shall follow the strategy used by Kenig and Mendelson for the quintic focusing NLW in [START_REF] Kenig | The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data[END_REF] to provide a randomization procedure based on a distorted frequency decomposition. This randomization procedure commutes with the flow e -itH and is therefore suited to the underlying linear dynamic of (NLS). In the present work we consider the Schrödinger equation perturbed by a potential in order to generate a curve of small nonlinear ground states, and our analysis is easier than in [START_REF] Kenig | The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data[END_REF] since we can assume no resonance at zero. Let us now precise the assumptions made on V .

Assumptions on the potential: V is a real-valued potential in the Schwartz class S(R d ; R).

We write

H := -∆ + V, H 0 = -∆.
Note that V is short-range in the sense of Agmon: there exists ε > 0 such that the operator

u ∈ H 2 (R d ) → (1 + |x|) 1+ε V u ∈ L 2 (R d )
is a compact operator. Namely, H is H 0 -compact. Hence, H admits a unique self-adjoint realization on L 2 (R d ) with domain D(H) = H 2 (R d ) and has the same essential spectrum as H 0 , that is the half line [0 , +∞). In addition, Agmon proved in [START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF] that if V is short-range then the spectrum of H is given by σ

(H) = σ p (H) ∪ σ ess (H) = σ p (H) ∪ [0 , +∞),
where σ p (H) (the discrete spectrum of H) is a countable set made of eigenvalues with finite multiplicity. We have the spectral decomposition:

L 2 (R d ) = L 2 ac (R d ) ⊕ L 2 p (R d ),
where L 2 p (R d ) is the space spanned by the associated eigenvectors and L 2 ac (R d ) is the absolute continuous spectral subspace for H. In what follows, we assume that V is generic, in the sense that (i) σ p (H) ⊂ (-∞ , 0), (ii) 0 is not a resonance for H.

We refer to [START_REF] Germain | Nonlinear Resonances with a Potential: Multilinear Estimates and an Application to NLS[END_REF] and the references therein for precise discussions about optimal assumptions on V . In the present work, we assume extra regularity on V to get the global-in-time local smoothing estimates of Proposition 1.33 in [START_REF] Rodnianski | Effective limiting absorption principles, and applications[END_REF]. We also chose to take V smooth in order to derive a bilinear estimate for perturbed linear Schrödinger evolutions based on semiclassical functional calculus, and that does not rely on an explicit structure theory for wave operators (see [START_REF] Beceanu | Structure of wave operators for a scaling-critical class of potentials[END_REF][START_REF] Hong | Global well-posedness of NLS with a rough potential below the energy norm[END_REF]). The assumption that V is in S(R d ) is far from being optimal and is made for simplicity.

Organization of the paper

Spectral theory We present in section 3.2 the basic properties of the distorted Fourier transform, and it's connections with wave operators and functional calculus. Under our assumptions on -∆ + V , the distorted Fourier transform is a unitary operator onto the continuous spectral subspace L 2 c (R d ) that provides very useful frequency decompositions adapted to the perturbed framework. Indeed, the distorted Fourier multipliers involved in these decompositions commute with the perturbed linear flow e -itH . In particular, they preserve the continuous spectral subspace Ran(P c ). We state an analogue of the Fourier multiplier theorem for this transformation that follows from the L p -boundedness of wave operators, and we also stress out that distorted Fourier multipliers by radial functions coincide with spectral multipliers defined by the functional calculus on self-adjoint operators. Then, we shall briefly revisit some nonlinear tools such as the Littlewood-Paley inequality in order to perform standard harmonic analysis techniques for nonlinear dispersive PDE's in the perturbed setting.

Probabilistic and Bilinear Strichartz estimates for the inhomogeneous Schrödinger evolution First, we generalize the refined global-in-time Strichartz estimates for randomized data from [BOP15a], Lemma 2.3. To do so, we introduce a distorted Wiener decomposition by using distorted Fourier multipliers localized on unit cubes (whereas standard Fourier multipliers are used in the flat case). As mentioned above, the reason for this is to come with a randomization that commutes with e -itH and that preserves Ran P c (H). Proposition (3.3.2) yields a large measure set Ω of initial data in Ran P c (H) that display refined Strichartz estimates. In order to revisit the standard proof written in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF], we use the variant of a Bernstein's estimate for distorted Fourier multipliers (3.2.12) presented in section 3.2. Furthermore, we stress out that since u ω 0 is in Ran(P c ) we can apply the local smoothing property (3.2.7) for e -itH . For all these reasons, randomization adapted with a distorted frequency decomposition turns out to be the best suited to the dynamic of (NLS). Nevertheless, the distorted Fourier transform does not change a product into a product of convolution. Therefore, the nonlinear analysis is more intricate in this setting, and we cannot generalize bilinear estimates so easily. Still, we prove an extended version of the bilinear Strichartz estimate from J. Bourgain to the inhomogeneous case with V in the Schwartz class. Our proof does not rely on a structure formula for the wave operator, and we use instead semiclassical functional analysis to intertwine Fourier and distorted Fourier multipliers. More precisely, we quantify the interactions between functions of the form φ(N -1 H) and φ(M -1 H 0 ) in order to intertwine localization with respect to H and H 0 . Then, we decompose the perturbed evolution e -itH into a superposition of free evolution e it∆ of flat-Fourier localized data, and we apply the original Bourgain's estimate for the free evolution e it∆ , as well as local energy decay.

Probabilistic global existence and scattering on the continuous spectral subspace The purpose of section 3.4 is to lay the foundations for the analysis conducted in section 3.5 about the stability of small nonlinear ground states under rough and randomized perturbations. First, we recall the definitions and key properties of critical spaces of functions from an interval I to

H d-2 2 (R d ), written X d-2
2 (I). This space is built upon the space of functions of finite q-variation V q and its predual, the atomic space U p . Next, we generalize Theorem 1.2 in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF] to the perturbed framework in order to understand the dynamic for solutions to (NLS) projected on the continuous spectral subspace for H. To do so, we shall specify and develop a bit on the random nonlinear estimates derived in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]. We obtain the following result.

i∂ t ψ + (∆ -V )ψ = P c |ψ| 2 ψ , ψ t=0 = εu ω 0 + ψ 0 , (NLS c ) admits a unique global-in-time solution ψ in the class ε e -itH u ω 0 + X d-2 2 [0 , ∞) ⊆ C [0 , ∞), H s (R d ) ∩ Ran(P c ) .
In addition, there exists

v + ∈ H d-2 2 (R d ) ∩ Ran(P c ) such that lim t→+∞ ∥ψ -e -itH εu ω 0 + v + ∥ H d-2 2 (R d ) = 0 .
Remark 3.1.6. If the pure point spectrum of -∆ + V is empty then P c = Id and (NLS c ) is (NLS). By time reversibility of (NLS) the same results hold true for negative times, and conditional uniqueness holds in the following sense: let v 1 , v 2 be two solutions in

X d-2 2 (0, ∞). If there exists t ∈ [0 , ∞) such that v 1 (t) = v 2 (t) then v 1 ≡ v 2 .
In section 3.5, we consider perturbed solutions of (NLS) around a ground state under the form Q + η. In particular, the radiation term η satisfies a nonlinear Schrödinger equation with a stochastic forcing term that contains localized and linear terms. The main difficulty is that the higher order terms are handled in critical spaces, while the localized lower order ones can only be controlled in weighted Sobolev spaces by the use of some local smoothing estimates. Therefore, we shall present in Proposition 3.4.15 a critical-weighted strategy that gives a way to perform nonlinear analysis in critical spaces and in weighted spaces simultaneously. In dimension d = 3, our analysis shows that this technique also gives an alternative version of Theorem 3.1.5 where global existence and uniqueness for the nonlinear part of the solution hold in V 2 intersected with the weighted Sobolev space L 2 t (R; H 1,-1/2-(R 3 )). A similar approach for Korteweg-de Vries equation can be found in [START_REF] Koch | Small data scattering and soliton stability in ( Ḣ-1/6 for the quartic KdV equation[END_REF].

Outline of the proof of Theorem 3.1.1 Let us now present the framework and some notations used in [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF]. In what follows, we identify C ≃ R 2 , with z = z 1 + iz 2 = (z 1 , z 2 ). For convenience, we denote the operator D z f (z) := (∂ z 1 f (z), ∂ z 2 f (z)) acting on C seen as a real vector space of dimension 2 endowed with the real scalar product

⟨(z 1 , z 2 ), (z ′ 1 , z ′ 2 )⟩ = z 1 z ′ 1 +z 2 z ′ 2 .
Then, we construct a local probabilistic flow for (NLS), and we decompose the solution at each time under the form (3.1.3) with the following time-dependent orthogonality condition imposed for the radiation term η(z):

η(z) ∈ H c (z) := η ∈ L 2 (R 3 ) | R 3 ⟨iη, ∂ z 1 Q(z)⟩dx = R 3 ⟨iη, ∂ z 2 Q(z)⟩dx = 0 . (3.1.7)
In particular, observe that H c (0) = Ran(P c ) = L 2 c (R 3 ). In broad outline, we ask for η to be orthogonal to the center manifold of ground states at each time, and this has for effect to cancel some linear terms in the modulation equations (3.5.8). Note that these orthogonality conditions are all the more natural since any small function ϕ in L 2 (R 3 ) can actually be decomposed into

ϕ = Q(z) + η , (3.1.8)
where z ∈ C and η ∈ H c (z) satisfies (3.1.7) (see Lemma 2.3 in [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF]). Furthermore, the decomposition (3.1.8) is explicit and P p (η), the discrete part of η, can be expressed as a function of P c (η) and z. Indeed, for each z small enough there exists a bijective operator R(z) : H c (0) → H c (z) written in Lemma 3.5.4 such that η = R(z) P c (η) .

We introduce ν = P c (η) -εu ω . Since u ω is in Ran(P c ), we have that ν ∈ Ran P c , and

P p (η) = (R(z) -I) εu ω + ν .
Therefore, the evolution reduces to a system with two degrees of freedom z and ν

ψ = Q(z) + η(z) , η(z) = R(z) εu ω + ν(z) . (3.1.9)
The aim is to obtain some global a prioi estimates for (ν, m(z)), solution to the coupled system of modulation equations (3.5.8), (3.5.10), where m(z) denotes bijective gauge defined in (3.5.7). By injecting the ansatz (3.1.9) into the equation (NLS), we see that ν satisfies the perturbed Schrödinger equation with a stochastic forcing term (3.5.10). It has the form

i∂ t ν + (∆ -V )ν = F (ν, Q(z), z, εu ω ) ,
and stress out that the forcing term contains some linear terms with respect to η and its complex conjugate η. The main difficulty is to prove scattering with these linear terms. Indeed, they are time dependent, and they are not self-adjoint perturbations of H. Hence, we can not absorb them in the left-hand side in order to get a perturbed operator as studied in our framework. We rather need to consider linear terms as source terms. Fortunately, these terms come with powers of Q or ϕ 0 which are small in H 2 and localized. Hence, we shall use local energy decay to control them. On the other hand, we need the critical spaces based on U 2 and V 2 to control the term N (u ω + ν). Consequently, we shall exploit the aforementioned critical-weighted strategy (see Proposition 3.4.15) to simultaneously cope with the localized and critical terms and to get the desired a priori estimate on ν.

The dynamic of m(z) is governed by an ODE that arises as when differentiating the orthogonality conditions (3.1.7) with respect to time. As explained above, orthogonality conditions (3.1.7) cancel terms which are linear with respect to η. Hence, the obtained ODE contains terms which are at least quadratic in η. Since the local smoothing estimate controls some global-in-time and weighted quadratic norms of η, a global-in-time control of ṁ(z) in L 1 t (R) follows from the analysis of the ODE and yields the convergence result written in (3.1.5).

Eventually, the analysis leads to the global a priori estimates on ν and m(z) stated in Proposition 3.5.9, from which we deduce global existence by the use of a continuity argument. The desired asymptotic dynamic also follows from these a priori estimates.

Notations Let s, σ ∈ R. We denote the Sobolev space by H s (R d ), and we define the weighted Sobolev space by

L 2,σ (R d ) = u | ⟨x⟩ σ u ∈ L 2 (R d ) , H 1,σ (R d ) = u | ⟨x⟩ σ u , ⟨x⟩ σ ∇u ∈ L 2 (R d ) .
The projection onto the continuous spectral subspace L 2 c (R d ) for H is written P c , and P p = I -P c = (• | ϕ 0 )ϕ 0 is the projection onto the pure point subspace for H. Note that

Ran(P c ) = L 2 c (R) = H c (0) ,
where spaces H c (z) encode the orthogonality conditions (3.1.7) to the ground state manifold. We write e -itH = e -itH P c the perturbed Schrödinger evolution projected on the continuous spectral subspace, whereas by e it∆ = e -itH 0 is the free Schrödinger evolution. For I ⊆ R, the space-time Lebesgue space L q t (I; L p x (R d )) is written L q t L r x (I), or L q t L r x when the dependence on I is clear. Given a Borel function f : R + → C, we denote the operators defined by the usual functional calculus by f (H 0 ) and f (H) = f (H) P c (H). In particular, we define the Littlewood-Paley multipliers (see (3.2.16)) around the dyadic frequency N ∈ 2 N by

∆ N u = φ(N -1 H 0 ) , Π N = K≤N ∆ K u , ∆ N = φ(N -1 H) , Π N = K≤N ∆ K u .
Given a function m : R d → C we denote the distorted Fourier multiplier F * V P c (H)mF V by M m (H) (see Definition 3.2.7) where F V is the distorted Fourier transform (see Proposition 3.2.6). Given a function u 0 in L 2 c (R d ), we define its Wiener randomization by u ω 0 in Definition 3.3.1, and u ω = e -itH u ω 0 is the linear propagation of the randomized data by the under perturbed flow.

Spectral theory for generic short-range Schrödinger operators

In this section, we present well known facts about functional calculus on perturbed Schrödinger operators that gives rise to a natural framework to work with -∆ + V and -∆ all together. First, we collect some definitions and basic properties of wave operators, such as the intertwining property as well as their L p -boundedness. As a consequence, we will see how dispersive estimates known for free Schrödinger evolution carry over to the perturbed setting. Additionally, we recall the construction of a distorted Fourier transform based on the limiting absorption principle. We shall see how the wave operators connect this transformation to the usual -or flat -Fourier transform, and lead to a useful generalization of the Fourier multiplier theorem. Some of the results we state here may not hold true in low dimensions d < 3.

Wave operators and the intertwining property

The wave operators are defined by the strong limit

W ± = s -lim t→±∞ e itH e -itH 0 in B(L 2 (R d ))
.

They aim at understanding scattering of Schrödinger operators -∆ + V by comparing the asymptotic behavior of e -itH with the asymptotic behavior of e -itH 0 . We say that the wave operators are asymptotically complete if

(i) W ± is bounded and surjective from L 2 (R d ) onto L 2 ac (R d ) (ii) σ sc (H) = ∅, and hence L 2 ac (R d ) = L 2 c (R d ).
Agmon proved in [START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF] that the wave operators W ± exist and are asymptotically complete under the decaying assumption ⟨x⟩ 1+ε V ∈ L ∞ (R d ). Consequently, wave operators are partial isometries onto L 2 c in this case, that is

W ± * W ± = I, W ± W ± * = P c .
It follows from the definition that e -itH W ± = W ± e -itH 0 .

This leads to the so-called intertwining property: for any Borel function f : R → C

H P c = W ± H 0 W ± * , f (H) P c = W ± f (H 0 ) W ± * , (3.2.1)
where f (H) and f (H 0 ) are defined by the functional calculus on self-adjoint operators. In a seminal paper [START_REF] Yajima | The W k,p -continuity of wave operators for Schrödinger operators[END_REF], Yajima proved the W k,p (R d ) boundedness for wave operators when some extra regularity and decay on V and F(V ) are assumed. We state the result in L p (R d ), since we do not need more in our study.

Proposition 3.2.1 (L p -bound of wave operators, Theorem 1.1 in [START_REF] Yajima | The W k,p -continuity of wave operators for Schrödinger operators[END_REF]). Let V be a generic potential with some regularity and decaying assumptions detailed in [START_REF] Yajima | The W k,p -continuity of wave operators for Schrödinger operators[END_REF], and 1 ≤ p ≤ +∞. The wave operators W ± can be extended to bounded operators in L p (R d ): there exists C p such that for all

f in L 2 (R d ) ∩ L p (R d ), ∥ W ± f ∥ L p (R d ) ≤ C p ∥f ∥ L p (R d ) . (3.2.2)
Given 1 ≤ p, q ≤ +∞, it follows from the intertwining property (3.2.1) and from the L p bound (3.2.2) that f (H) and f (H 0 ) have equivalent norms on B(L p (R d ), L q (R d )). There exists C p,q such that for any Borel function f : R → C,

C -1 p,q ∥f (H)∥ B(L p ,L q ) ≤ ∥f (H 0 )∥ B(L p ,L q ) ≤ C p,q ∥f (H)∥ B(L p ,L q ) .
One striking but straightforward consequence of Proposition 3.2.1 is the generalization of the dispersive estimate to perturbed linear Schrödinger evolutions.

Proposition 3.2.2 (Dispersive estimate for perturbed Schrödinger evolution). Let V be a potential as in Proposition 3.2.1, 2 ≤ p ≤ +∞ and q such that 1/p + 1/q = 1. There exists a constant

C p such that for all f ∈ L 2 (R d ) ∩ L q (R d ) and t ∈ R \ {0} ∥ e -itH P c f ∥ L p (R d ) ≤ C p |t| -d(1/2-1/p) ∥f ∥ L q (R d ) . (3.2.3)
By the T T * argument, we deduce from the above dispersive estimate the global-in-time Strichartz estimate for the perturbed linear evolution: for all Schrödinger admissible pairs 2 ≤ q, r ≤ ∞, with

3 ≤ d and 2 q + d r = d 2 ∥ e -itH P c f ∥ L q t (R;L r x (R d )) ≤ C q,r,d ∥f ∥ L 2 (R d ) . (3.2.4)
This result was originally proved in [START_REF] Journé | Decay estimates for Schrödinger operators[END_REF], under the assumption that H has no eigenvalue nor resonance at zero, and that V satisfies the decay condition

|V (x)| ≲ ⟨x⟩ -2-ε for some 0 < ε .

Distorted Fourier multipliers

In order to refine the perturbed Strichartz estimates of Proposition 3.2.2 for randomized initial data as in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF], we have to use unit-scale decomposition of the frequency space adapted to the operator -∆ + V . More precisely, we need to set out a distorted Fourier transform that conjugates the operator H with the multiplication by |ξ| 2 , and to generalize the Fourier multiplier theorem. To construct such a transformation, the usual strategy is to determine some generalized plane waves e(x, ξ) that are perturbations of the plane waves e ix•ξ and that formally satisfy the Helmholtz equation

-∆ + V e(•, ξ) = |ξ| 2 e(•, ξ) , (3.2.5)
with the asymptotic condition

v(x, ξ) := e(x, ξ) -e ix•ξ = O |x|→∞ |x| -1 ,
as well as the Sommerfeld radiation condition

r ∂ r -i|ξ| v(x, ξ) -→ |x|→0 0 .
Equation (3.2.5) cannot be solved in L 2 (R d ) since |ξ| 2 is in the essential spectrum of H. However, this equation can be solved in weighted spaces by a procedure called the limiting absorption principle. This principle states that the resolvent R(z

) := H -z -1 , which is bounded from L 2 (R d ) to H 2 (R d )
and that defines an analytic operator valued function on C \ [0 , +∞), can be extended to the boundary of its domain of definition in the following sense.

Proposition 3.2.3 (Limiting absorption principle, Theorem 4.2 in [START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF]). Let λ > 0 and δ > 1/2. The sequence

lim z→λ Im(z)>0 R(z) =: R + (λ) converges in B ⟨x⟩ -δ L 2 (R d ), ⟨x⟩ δ H 2 (R d
) endowed with the uniform operator norm topology.

Then, the existence of a family of distorted plane waves follows from the above limiting absorption principle. Indeed, the solutions to equation (3.2.5) satisfy the Lippman-Schwinger equation

e(x, ξ) = e ix•ξ -R + (|ξ| 2 )(V e ix•ξ ) .
(3.2.6) Proposition 3.2.4 (Generalized plane wave, Theorem 5.1 in [START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF]). There exists a measurable function

e in L 2 loc R d × R d \ {0} such that for every fixed ξ in R d \ {0}, the function e(•, ξ) belongs to H 2 loc (R d ) ∩ C(R d
) and is a solution to equation (3.2.5) in H 2 loc (R d ). See [START_REF] Ikebe | Eigenfunction expansions associated with the Schrödinger operators and their applications to scattering theory[END_REF] for further properties of the generalized plane waves when V is regular. Another important consequence of the limiting absorption principle is the local smoothing estimate, or local energy decay, for the perturbed linear Schrödinger evolution. We state a version taken from [RT15] (Proposition 1.33) where some extra regularity and decay assumptions on V are required. This local smoothing effect and its transferred version into the space U 2 is a key ingredient in the proof of Theorem 3.1.1, when we study the perturbations around the ground state. Proposition 3.2.5 (Global-in-time local smoothing estimate for H, Proposition 1.33 in [START_REF] Rodnianski | Effective limiting absorption principles, and applications[END_REF]). Under our assumptions made on V , and for ψ solution to the forced Schrödinger equation

i∂ t ψ + (∆ -V )ψ = F , ψ |t=0 = ψ 0 ,
we have the following global-in-time control of the local energy of ψ

R ∥ P c (ψ)∥ 2 H 1,-1/2-dt ≲ V ∥ψ 0 ∥ 2 H 1/2 (R d ) + R ∥F (t)∥ 2 L 2,1/2+ x dt . (3.2.7)
The next step is to define a distorted Fourier transform from the generalized plane waves constructed in the above paragraph. Since H may have some pure point spectrum in our assumptions, the operator we get is only a partial isometry with range L 2 c (R d ).

Proposition 3.2.6 (Distorted Fourier transform, Theorem 5.1 in [START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF]). There exists a partial isometry

F V from L 2 (R d ) to L 2 c (R d ) which diagonalizes H on H 2 (R d ). For all f ∈ H 2 (R d ), (-∆ + V )f = F * V M | | 2 F V f .
Moreover, for any f in L 2 (R d ) the following representation formula holds in L 2 (R d ):

F V f (ξ) = (2π) -d/2 lim n→∞ |x|<n e(x, ξ)f (x)dx , (3.2.8)
F * V f (x) = (2π) -d/2 lim n→∞ |x|<n e(x, ξ) P c (H)f (ξ)dξ . (3.2.9)
Furthermore, we have

W + = F * V F . (3.2.10)
Thanks to the distorted Fourier transform we are now able to define some analogues of the Fourier multipliers, that commute with H.

Definition 3.2.7. Let m : R d → C be a function in L ∞ . The distorted Fourier multiplier M m (H) := F * V m(ξ)F V (3.2.11)
is a bounded operator on L 2 (R d ), where m(ξ) denotes the operator u ∈ L 2 ξ → mu. We have

∥ M m (H)∥ B(L 2 (R d )) ≤ ∥m∥ L ∞ (R d ) .
It follows from equation (3.2.10) that

W * + M m (H) W + = M m (H 0 ) ,
where M m (H 0 ) is the usual Fourier multiplier by m. Hence, we deduce from the L p boundedness of the wave operator W ± mentioned in Proposition 3.2.

1 that M m (H) is bounded on L p (R d ) if and only if M m (H 0 ) in bounded on L p (R d ).
As a consequence of this principle, we generalize the Fourier multiplier theorem and its variations to the perturbed linear Schrödinger evolution.

Lemma 3.2.8. Let m : R d → C be a function in L ∞ (R d ) supported on a compact set E. Given q in [2 , +∞], there exists a constant C = C(q, V, ∥m∥ L ∞ ) such that for all f in L 2 (R d ), ∥ M m (H)f ∥ L q ≤ C|E| 1 2 -1 q ∥f ∥ L 2 . (3.2.12)
Proof. Let f be a function in L 2 (R d ). We have

| F * mF f (x)| = | E e iξ•x m(ξ)Ff (ξ)dξ| ≤ ∥m∥ L ∞ (R d ) ∥ e(•, ξ)∥ L ∞ |E| 1 2 ∥f ∥ L 2 .
(3.2.13) Then (3.2.12) for M m (H 0 ) (i.e. when V = 0) follows from complex interpolation, and we deduce the estimate for M m (H) from the intertwining property (3.2.1) and from the L p -bound (3.2.2) on W ± .

Similarly, one can generalize the Mikhlin multiplier theorem (see Theorem 3.2 in the book [START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF]) to the perturbed setting. Lemma 3.2.9 (Mikhlin multiplier theorem). Let m : R d \ {0} → C be a multiplier and

1 < p < +∞. Assume that for all α in N d such that 0 ≤ |α| ≤ d-1 2 + 1, | D α m(ξ)| ≤ C α |ξ| -α . (3.2.14) Then, ∥m(H)f ∥ L p (R d ) ≲ C α ∥f ∥ L p (R d ) .
As a consequence of the Mikhlin multiplier theorem, we show that the Sobolev spaces defined with H or H 0 are equivalent. Lemma 3.2.10. Let 1 < p < +∞ and 0 ≤ s. There exists C = C(s, p, d) such that for all f in

S(R d ) C -1 ∥⟨ √ H 0 ⟩ s f ∥ L p ≤ ∥⟨ √ H⟩ s f ∥ L p ≤ C∥⟨ √ H 0 ⟩ s f ∥ L p
Proof. Let q be the conjugated exponent of p. Let us show that ⟨ √ H 0 ⟩ s ⟨ √ H⟩ -s can be extended to a bounded operator in B(L p ).1 We proceed by using complex interpolation and duality in L p . Take f, g in S(R d ) such that the support of f is compact, and note that such functions are dense in L p . Next, we consider the map

F : z → f | e z 2 ⟨ √ H 0 ⟩ z ⟨ √ H⟩ -z g with domain D = z ∈ C | 0 < Re(z) < 2n for a given n ∈ N.
We see that F is a function that is analytic on D and continuous on the closure of D. Moreover, it follows from the Mikhlin multiplier theorem and its distorted version (3.2.14) that for f in L p with 1 < p < ∞ and δ in R we have

|F (e iδ )| ≲ ∥f ∥ L p ∥g∥ L q . Indeed, ∥⟨ √ H 0 ⟩ iδ f ∥ L p ≲ (1 + |δ|) N ∥f ∥ L p , ∥⟨ √ H⟩ iδ f ∥ L p ≲ (1 + |δ|) N ∥f ∥ L p .
Note that the constant can be made independent of δ thanks to the term e -δ 2 that arises from the function e z 2 . In addition, the distorted version of the Mikhlin multiplier theorem and the fact that

V is in S yield ∥⟨ √ H 0 ⟩ 2n ⟨ √ H⟩ -2n ∥ B(L p ) ≲ i+j+k=n ∥V i H k V j ⟨ √ H⟩ -2n ∥ B(L p ) = C n,p < +∞ ,
and we deduce from this that

|F (e 2n+iδ )| ≲ ∥f ∥ L p ∥g∥ L q .
It follows from the three lines lemma, we prove that |F (z)| ≲ ∥f ∥ L p ∥g∥ L q on D n uniformly in n, and the density of functions like f and g in L p and L q yields the desired result.

A straightforward consequence of Lemma 3.2.10 is the generalization to the perturbed setting of the fractional Leibniz rule. Lemma 3.2.11 (Fractional Leibniz rule). Let 1 < p, p 1 , p 2 , q 1 , q 2 < +∞, 1 p = 1 p 1 + 1 p 2 = 1 q 1 + 1 q 2 , and let 0 ≤ s. Assume that f, g are in S(R d ). Then,

∥⟨ √ H⟩ s f g ∥ L p ≲ ∥⟨ √ H⟩ s f ∥ L p 1 ∥g∥ L p 2 + ∥⟨ √ H⟩ s g∥ L q 1 ∥f ∥ L q 2 .
We prove in the next lemma that if m is a radial function, the Fourier multiplier by m is precisely the spectral multiplier defined by the functional calculus on self-adjoint operators.

Lemma 3.2.12. If there exists a Borel function

f : R → C such that m(ξ) = f (|ξ| 2 ) for all ξ in R d , then M m (H) = f (H) . (3.2.15)
Proof. Let us first write the result for the free Laplacian H 0 , that is F * mF = f (H 0 ). By an approximation argument, the proof reduces to the cases where f is a Schwartz function. In this setting, f (H 0 ) is given by the formula

f (H 0 ) = R e itH 0 f (t)dt, where e itH 0 = F * e it| • | 2 F .
By applying the Fourier inversion formula for f , we have that

f (H 0 ) = F * R e it| • | 2 f (t)dt F = F * f (| • | 2 )F = M m (H 0 ) . Hence, Ff (H 0 )F * = M m (H 0 ) .
Also, the intertwining property (3.2.1) and the equation (3.2.10) give

f (H) = W + f (H 0 ) W * + = F * V Ff (H 0 )F * F V = F * V M m F V = M m (H) .

Application to the distorted Littlewood-Paley theory

There exists a nonnegative radial function

ψ in C ∞ (R d ) such that φ = ψ -ψ(2•) is supported on {x ∈ R d | 1 2 ≤ |x| ≤ 2} and satisfies for all x in R d \ 0 N ∈2 Z φ(N -1 x) = 1.
This partition of unity provides frequency decomposition on dyadic annuli for functions f in L 2 (R d ) called the Littlewood-Paley decomposition. We consider such a decomposition for both the flat Fourier transform and the distorted one:

f = N ∈2 Z ∆ N f = ∆ ≤1 f + N ∈2 N ∆ N f, (3.2.16)
where ∆ N := φ(N -1 H 0 ) is the spectral multiplier around frequencies of size N for H 0 as well as

f = N ∈2 Z ∆ N f = ∆ ≤1 f + N ∈2 N ∆ N f, (3.2.17)
where ∆ N := φ(N -1 H). Note that the Bernstein's estimates are still true in the perturbed setting, as a consequence of the intertwining property, and of the L p bound on the wave operators. For 1 ≤ p ≤ q ≤ ∞, s ≥ 0 there exist C p,q and C p,s and such that for all N ≥ 1,

∥ ∆ N f ∥ L q (R d ) ≤ C p,q N d( 1 p -1 q ) ∥ ∆ N f ∥ L p (R d ) , ∥ ∆ N f ∥ W s,p (R d ) ≤ C p,s N s ∥ ∆ N f ∥ L p (R d ) .
(3.2.18) Given f in L 2 (R d ), the associated distorted Littlewood-Paley square function is defined by

Λf (x) = N ∈2 Z | ∆ N f (x)| 2 1/2 .
One can see from Plancherel's theorem that the L 2 (R d )-norm of the Littlewood-Paley square function is equivalent to the L 2 (R d )-norm of the function itself. The so-called Littlewood-Paley square function theorem extends this result to the L p (R d )-norms for 1 < p < ∞. We state a generalized version of this theorem in the perturbed framework.

Proposition 3.2.13. For 1 < p < +∞ and f in L p (R d ), the Littlewood-Paley square function Λf is in L p (R d ), and

∥Λf ∥ L p (R d ) ∼ ∥f ∥ L p (R d ) . (3.2.19)
The constant involved in the above equivalence only depends on p and on the cutoff function ψ used in the Littlewood-Paley decomposition.

We outline the classical proof (see Theorem 5 in the book [START_REF] Stein | Singular Integrals and Differentiability Properties of Functions[END_REF]) that relies on the Mikhlin multiplier theorem from Lemma 3.2.9. The strategy is to randomize the sum ∆f with independent Rademacher variables, and to use Khinchin's estimates to reduce the proof to the L 2 -case. As mentioned above, this case is a straightforward consequence of Plancherel theorem.

Proof. Let us fix 1 < p < +∞ and f in L 2 (R d ) ∩ L p (R d ). Take (ε N ) N ∈2 Z a sequence of independent
Rademacher variables on a probability space (Ω, A, P), i.e. ε N ∼ U( -1, 1 ). Next, we define the random Fourier multiplier for ω in Ω and ξ in R d by

m ω (ξ) = N ∈2 Z ε N (ω)φ(N -1 ξ) .
Since φ(N -1 •) is supported on frequencies of size ∼ N , the random multiplier m ω satisfies the Mikhlin's assumption: for all ξ in R d and α a multi-index,

| D α m ω (ξ)| ≤ N ∈2 Z N -|α| |(D α φ)(N -1 ξ)| ≤ C α |ξ| -|α| .
Besides, m ω is a radial multiplier, and it follows from Lemma 3.2.12 and from Lemma 3.2.9 that

∥m ω (H)f ∥ L p (R d ) ≤ ∥f ∥ L p (R d ) .
(3.2.20)

That being said, the Khinchin's estimate claims that

1 - 1 p 1/2 n |c n | 2 1/2 ≲ E n ε N (ω)c n p 1/p , (3.2.21)
and yields Λf =

N ∈2 Z | ∆ N f | 2 1/2 ≲ E| N ∈2 Z ε N (ω) ∆ N f | p 1/p ≲ E|m ω (H)f | p 1/p .
By taking the L p (R d )-norm on both sides and by using (3.2.20), we have

∥Λf ∥ L p (R d ) ≲ E∥m ω (H)f ∥ L p (R d ) ≲ ∥f ∥ L p (R d ) .
We prove the reversed inequality by duality.

Probabilistic and bilinear

Strichartz estimates in the perturbed setting 3.3.1 Wiener randomization with respect to the distorted Fourier transform

Let us now detail how to adapt the Wiener randomization for the operator -∆ + V and to generalize probabilistic Strichartz estimates. This will be made possible thanks to the distorted Fourier transform presented in the above section. We shall follow the framework detailed by [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF] which can be adapted with no difficulty to the perturbed case.

Wiener decomposition on unit cubes We denote the unit cube [-1 , 1] d by Q 0 and the translated cube centered around n ∈ Z d by Q n = Q 0 + n. Then, we take a well-chosen bump function ψ supported on Q 0 that provides a partition of unity on the frequency space, that is n∈Z d ψ(•-n) ≡ 1, and we define the Wiener's decomposition of a function in L 2 (R d ) as

f = n∈Z d M ψn (H)f , (3.3.1)
where we defined the distorted Fourier multiplier M ψn (H) in (3.2.11). After that, we consider a probability space Ω, A, P and a sequence of mean zero complex valued random variables g n n∈Z d of laws µ n n∈Z d , with uniform bound:

| R e γx dµ n (x)| ≤ e cγ 2 for all n in Z d and γ in R . (3.3.2)
Such a bound is satisfied by Gaussian random variables, Bernoulli variables or any random variables with compactly supported distributions. We also assume that (Re(g n ), Im(g n )) n∈Z d are independent variables.

Definition 3.3.1. The Wiener randomization of a function u 0 in L 2 (R d ) is u ω 0 := n∈Z d g n (ω) M ψn (H)u 0 . (3.3.3)
The Khinchin's inequality asserts that

E| n g n (ω)c n | p 1/p ≲ √ p n |c n | 2 1/2 for (c n ) ∈ ℓ 2 , 1 ≤ p < +∞ , (3.3.4)
and we deduce from it that the randomization procedure improves integrability and preserves regularity. Namely, for all 2 ≤ p < +∞ and s in R there exist some constants 0 < c, 0 < C such that for all 0 < α, u in L 2 (R d ) and v in H s (R d ) we have

P(∥u ω ∥ L p (R d ) > λ) ≤ C e -cλ 2 ∥u∥ -2 L 2 (R d ) , P(∥v ω ∥ H s (R d ) > λ) ≤ C e -cλ 2 ∥u∥ -2 H s (R d ) .
We emphasize that the procedure does not improve regularity provided that the distribution µ n does not concentrate around zero when n goes to ∞, i.e. [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF]). As for the possible gain of decay, we use the independence of the g n n∈Z d 's, and we observe that for ε > 0,

u ω 0 ∈ H s (R d ) \ H s+ (R d ) almost surely when u 0 ∈ H s (R d ) \ H s+ (R d ) (see Lemma B.1 in
E ∥|x| ε u ω 0 ∥ 2 L 2 x = (n,m)∈Z 2d E g n g m |x| 2ε M ψn (H)u 0 M ψm (H)f dx = n∈Z d ∥|x| ε M ψn (H)u 0 ∥ 2 L 2 x (R d )
diverges if we assume that

n∈Z d ∥|x| ε M ψn (H)u 0 ∥ 2 L 2 x (R d ) = +∞ . (3.3.5) Hence, ∥|x| ε u ω 0 ∥ L 2 (R d ) diverges in L 2
(Ω) when u 0 is chosen such that (3.3.5). This indicates that the randomization does not gain decay in general. 2 Still, the initial data ψ 0 = u ω 0 + v 0 in the Cauchy problem (3.1.4) does not decay for a general v 0 in H 1/2 . Besides, we don't use any decay of the data in our arguments to prove Theorem 3.1.1.

Improved probabilistic Strichartz estimates

Let us now recall how to improve the Strichartz estimates (3.2.3) for data randomized according to the distorted Wiener procedure (3.3.3) (see Lemma 2.3 in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]). Proposition 3.3.2 (Improved global-in-time Strichartz estimate). Given a Schrödinger admissible pair (q, r) as in Proposition (3.2.2), a real number r with r ≤ r < +∞ and a function f ∈ H s (R d ) for some s ≥ 0, there exist constants 0 < C, 0 < c such that

P ∥⟨ √ H⟩ s e -itH u ω 0 ∥ L q t (R;L r x (R d )) > λ ≤ C exp -cλ 2 ∥f ∥ -2 H s x (R d ) .
(3.3.6)

Consequently, ∥⟨ √ H⟩ s e -itH u ω 0 ∥ L q t (R;L r x (R d )) < +∞ almost-surely when f is in H s .
The strategy of the proof is to exploit the enhanced space integrability of Wiener randomized data. More precisely, the unit scale frequency components ψ n (H)f benefit from better integrability (uniformly in n) thanks to the distorted Fourier multiplier theorem, while the randomization procedure cancels interference between these different pieces ψ n (H)f . Hence, the space integrability of e -iH u ω 0 is improved and one can apply the deterministic Strichartz estimate for the admissible pair (q, r). The main point of our revisited proof is that the distorted Fourier transform commutes with the perturbed linear flow e -itH , and satisfies the unit-scale Bernstein estimate of Lemma 3.2.8.

Proof. We write the proof for s = 0. The large deviation bound (3.3.6) will be deduced from an estimate on the moments of the random variable ∥ e -itH u ω 0 ∥ L q t (R;L r

x (R d )) . To get such an estimate, we take p with max(q, r) < p and we use Minkowski inequality to get

E ∥ e -itH u ω 0 ∥ p L q t (R;L r x (R d )) 1/p ≤ ∥ e -itH u ω 0 ∥ L q t (R;L r x (R d ;L p ω (Ω))) . 2 Since ∥|x| ε n M ψn u 0 ∥ 2 L 2 ̸ = n ∥|x| ε M ψn u 0 ∥ 2 L 2 x
, we are not able to reproduce the proof of Lemma B.1 in [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF], and to preclude any gain of decay by our randomization procedure almost-surely.

At fixed (t, x)

∈ R × R d , it follows from Khinchin inequality (3.3.4) that ∥ e -itH u ω 0 ∥ L p ω (Ω) = ∥ n∈Z d g n (ω) e -itH M ψn (H)f ∥ L p ω (Ω) ≲ √ p∥ M ψn (H) e -itH f ∥ ℓ 2 n (Z d ) .
Hence, we use Minkowski inequality once again with 2 ≤ min(q, r), and get

E ∥ e -itH u ω 0 ∥ p L q t (R;L r x (R d )) 1/p ≲ √ p∥ M ψn (H) e -itH f ∥ ℓ 2 n (Z d ;L q t (R;L r x (R d ))) . (3.3.7)
Next, we use the distorted Fourier multiplier Lemma 3.2.8 at fixed t with r, r and the unit-scale multiplier ξ → e -it|ξ| 2 ψ(ξ -n) (in particular the volume of the support does not depend on n):

∥ e -itH M ψn (H)f ∥ L r x (R d ) ≲ r, r ∥ e -itH M ψn (H)f ∥ L r x (R d )
. Now, we apply the deterministic global-in-time Strichartz estimate for the perturbed Schrödinger evolution (3.2.3) with the admissible pair (q, r) and we get

∥ e -itH M ψn (H)f ∥ L q t (R;L r x (R d )) ≲ ∥ M ψn (H)f ∥ L 2 x (R d ) . Hence, there exists 0 < C such that E ∥ e -itH u ω 0 ∥ p L q t (R;L r x (R d )) 1/p ≲ √ p ∥ M ψn (H)f ∥ ℓ 2 n L 2 x (Z d ×R d ) ≤ √ p C∥f ∥ L 2 x (R d ) .
To conclude, we apply Markov inequality to get

P ∥ e -itH u ω 0 ∥ L q t (R;L r x (R d ) > λ ≤ √ p Cλ -1 ∥f ∥ L 2 x (R d ) p .
After that we chose p such that √ p := e -1 Cλ -1 ∥f ∥ L 2

x -1 and we distinguish the cases when max(q, r) < p or not.

Given 0 < ε, R we define the set

Ω ε,R = (q, r) ω ∈ Ω | ∥εu ω ∥ L q t L r x (R×R d ) ≤ R , (3.3.8)
where the union is taken over a finite number of pairs (q, r) as in Proposition 3.3.2 that occur in the nonlinear analysis presented in the section 3.4. It follows from (3.3.6) that there exist 0 < c, C such that for all 0 < ε, R

P(Ω \ Ω ε,R ) ≤ C exp(-R 2 ε -2 ∥u 0 ∥ -2 H s ) .
When R is an irrelevant universal constant we denote this set by Ω ε .

Bilinear estimate for the perturbed Schrödinger evolution

Bourgain's bilinear estimate states that given N, M two dyadic numbers with N ≤ M and u 0 , v 0 in L 2 (R d ) localized in the Fourier space, say supp u 0 ⊆ |ξ| ≲ N and supp v 0 ⊆ |ξ| ∼ M , we have

∥(e it∆ u 0 )(e it∆ v 0 )∥ L 2 t,x (R×R d ) ≲ N d-1 2 M -1 2 ∥u 0 ∥ L 2 x (R d ) ∥v 0 ∥ L 2 x (R d ) .
(3.3.9)

We refer to Lemma 5 in [START_REF] Bourgain | Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity[END_REF] for the original proof of this result, and to [START_REF] Colliander | Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3[END_REF] for a more detailed proof. Basically, the proof relies on Plancherel's formula, on the explicit form of the free Schrödinger flow and on the multiplicative property of the flat Fourier transform. As mentioned in the first section, the distorted Fourier transform does not have such a property. That being said, we propose to generalize (3.3.9) to the perturbed case, and we consider the interaction of two initial data localized by perturbed Littlewood-Paley projectors ∆ N , ∆ M introduced in (3.2.17) that evolve under the linear flow e -itH .

Proposition 3.3.3 (Bilinear estimate for perturbed linear Schrödinger evolutions). Given N, M two dyadic integers, we have that for all u 0 , v 0 ∈ L 2 (R d ),

∥(e -itH ∆ N u 0 )(e -itH ∆ M v 0 )∥ L 2 t,x (R×R d ) ≲ N d-1 2 M -1 2 ∥ ∆ N u 0 ∥ L 2 x (R d ) ∥ ∆ M v 0 ∥ L 2 x (R d ) . (3.3.10)
The general strategy of our proof is to reduce (3.3.10) to (3.3.9). To do so, we use a perturbative argument that relies on the Duhamel formula that connects the free evolution to the perturbed one. More precisely, given N, K two dyadic numbers and u = e -itH ∆ N u 0 a solution to the perturbed linear Schrödinger equation with initial data localized around a distorted Fourier frequencies of size N , we have

∆ K u(t) = e it∆ ∆ K ∆ N u 0 -i t 0 e i(t-τ )∆ ∆ K (V u(τ ))dτ . (3.3.11)
Hence, we will need to estimate some space-time Lebesgue norms of terms like ∆ K (V ∆ N e -itH u 0 ).

Semiclassical functional calculus

This section is devoted to the statement of some lemmas used in the proof of Proposition 3.3.3. When studying the interactions of a solution localized around a high frequency N ≫ 1 with other solutions, we introduce the small parameter h = N -1 to place ourselves in the semiclassical analysis framework that provides precise asymptotic expansions with respect to h. This strategy, that can essentially be used when V is smooth, provides an easy but efficient way to intertwine Fourier and distorted Fourier localization. For the convenience of the reader, we detail some notations and results from semiclassical analysis (we refer to [START_REF] Zworski | Semiclassical Analysis[END_REF] for a general presentation). The Weyl quantization of a given symbol

a ∈ C ∞ (R 2d ) is Op h (a)u(x) := (2πh) -2d R d ×R d e i (x-y)•ξ h a( x + y 2 , ξ)u(y)dydξ . (3.3.12)
It defines an operator acting on S(R d ) and for instance

h 2 H 0 = Op h (|ξ| 2 ), h 2 H = Op h (|ξ| 2 + h 2 V ). Moreover, if the symbol a is in the class S(1) := a(h) ∈ C ∞ (R 2d ) | ∀α, β ∈ N d sup h∈[0,1) sup ρ∈R 2d |∂ α x ∂ β ξ a(ρ)| < ∞ ,
we can extend Op h (a) to a bounded operator on L 2 (R d ) thanks to the Calderon-Vaillancourt theorem that yields:

∥ Op h (a)∥ B(L 2 (R d )) ≲ |α|≤6d+2 h |α|/2 ∥∂ α x,ξ a∥ L ∞ (R 2d ) .
(3.3.13) Lemma 3.3.5. For any α ∈ N, there exists C = C(α, V ) > 0 depending on some weighted norms of V and its derivatives 3 such that for any dyadic integer

K = 2 k and f ∈ L 2 x (R d ), we have |k-l|≥3 ∥∆ K (V ∆ L f )∥ 2 L 2 x 1/2 ≤ CK -α ∥⟨x⟩ -2 f ∥ L 2 x . (3.3.18)
Note that we decided to exploit not only the localization in frequency, but also the decay of V in order to use the local smoothing effect later on.

Proof. We prove that there exists C α > 0 depending on weighted norms of the potential V and its derivatives such that for all L = 2 -l with 3 ≤ |k -l|, we have

∥∆ K (V ∆ L f )∥ L 2 x ≤ C2 -2α max(k,l) ∥⟨x⟩ -2 f ∥ L 2 x . (3.3.19)
Let us first consider the case when k + 3 ≤ l. Plancherel's formula yields

∥∆ K (V ∆ L ⟨x⟩ 2 f )∥ L 2 (R d ) = sup ∥g∥ L 2 (R d ) ≤1 | R d ×R d φ(K -1 ξ)φ(L -1 η) V (ξ -η)⟨∇ η ⟩ 2 f (η) g(ξ)dξdη| ,
where we used that F x→η ⟨x⟩ 2 f ∼ ⟨∇ η ⟩ 2 F x→η f . Integrating by parts 4 with respect to η gives

∥∆ K (V ∆ L ⟨x⟩ 2 f )∥ L 2 (R d ) = sup ∥g∥ L 2 (R d ) ≤1 | R d ×R d φ(K -1 ξ)⟨∇ η ⟩ 2 φ(L -1 η) V (ξ -η) f (η) g(ξ)dξdη| . Since φ K (resp. φ L ) is supported on 2 k-1 ≤ |ξ| ≤ 2 k+1 (resp. 2 l-1 ≤ |η| ≤ 2 l+1
), we have that 2 l-2 ≤ |ξ-η| on the support of the integrand. Therefore, for all 0 ≤ δ we have that 1 ≤ 2 -δ(l-2) |ξ-η| δ , and the right-hand-side of the above estimate is less than

≲ 2 -δ(l-2) R d ×R d |φ(K -1 ξ)|ξ -η| δ ⟨∇ η ⟩ 2 φ(L -1 η) V (ξ -η) f (η) g(ξ)|dξdη ≲ 2 -δ(l-2) ∥⟨∇⟩ δ ⟨x⟩ 2 V ∥ L 2 x ∥f ∥ L 2 x ∥g∥ L 2 x ∥φ(K -1 •)∥ L 2 ξ ≤ C(V )2 -δ(l-2)+ d 2 k ∥f ∥ L 2 x ≤ C(V )2 -αl ∥f ∥ L 2
x , by choosing δ ≥ d 2 + 3α, and by using Cauchy-Schwarz and Plancherel. In the case when l + 3 ≤ k, the same computations yield

∥∆ K (V ∆ L ⟨x⟩ 2 f )∥ L 2 (R d ) ≤ C(V )2 -δ(k-2)+ d 2 k ∥f ∥ L 2 x ≤ C(V )2 -αk ∥f ∥ L 2 x , provided that δ ≥ 3d 2 + 3α.
We finish the proof of estimate (3.3.18) by summing over L.

Proof of bilinear estimate for perturbed Schrödinger evolutions We now come to the proof of Proposition 3.3.3 itself. For now on, we use the notation

u(t) := e -itH ∆ N u 0 , v(t) := e -itH ∆ M v 0 .
3 More precisely, our brutal computations give

C ≲ ∥⟨∇⟩ 3d 2 +3α ⟨x⟩ 2 V ∥ L 2
x , but they are far from being optimal. 4 We stress out that the operator

⟨∇ η ⟩ 2 ∼ 1 + d i=1 ∂ 2
ηi is a local operator and has no effect on the frequency localization η ∼ L.

Step 1: Reduction to the case where initial data are flat Fourier localized. First, we reduce Proposition 3.3.3 to the following proposition where the initial data are also localized by flat Fourier multipliers. In what follows, ∆ K is the projector around a frequency K, ∆ ≲N is the projector on frequencies below 2 5 N and ∆ ∼M is a fattened projector around frequency M . Given N, M two dyadic numbers, we write

u(t) := e -itH ∆ N u 0 , v(t) := e -itH ∆ M v 0 ,
where ∆ N , ∆ M are the distorded Fourier multipliers around frequencies N, M as defined in (3.2.17).

Proposition 3.3.6. There exists C > 0 such that for any time interval I ⊆ R and any dyadic integers N, M with N ≤ M , we have

∥uv∥ L 2 t,x (I) ≤ ∥∆ ≲N u∆ ∼M v∥ L 2 t,x (I) + K>2 5 N ∥∆ K u∆ ∼M v∥ 2 L 2 t,x (I) 1/2 + N d-1 2 M -1 2 γ(I) . (3.3.20)
For any interval I and for any partition R = ℓ I ℓ , we have

γ(I) ≤ C∥ ∆ N u 0 ∥ L 2 x ∥ ∆ M v 0 ∥ L 2 x , ℓ γ(I ℓ ) 2 1/2 ≤ C∥ ∆ N u 0 ∥ L 2 x ∥ ∆ M v 0 ∥ L 2 x . (3.3.21)
Proof. We introduce the fattened projector ∆ ∼M , a flat Fourier multiplier φ(M -1 H 0 ) by a smooth cutoff function φ chosen such that (1-φ)φ ≡ 0. Under this assumption, we deduce from Lemma 3.3.4 that for all σ ∈ R and for all α ∈ N,

∥(1 -∆ ∼M ) ∆ M ∥ B(L 2 (R d ,H σ (R d ))) ≲ α,σ M -α . (3.3.22) Then, v is decomposed into v = ∆ ∼M v + (1 -∆ ∼M v), and
uv = u∆ ∼M v + u(1 -∆ ∼M )v . (3.3.23)
We deduce form Cauchy-Schwarz, Sobolev embedding, Strichartz estimate and (3.3.22) that the second term on the right-hand side of (3.3.23) is negligible:

∥u(1 -∆ ∼M )v∥ L 2 t,x (I) ≤ ∥u∥ L 4 t,x (I) ∥(1 -∆ ∼M )v∥ L 4 t,x (I) ≲ N d 12 ∥u∥ L 4 t L 3 x (I) ∥⟨∇⟩ d 12 (1 -∆ ∼M )v∥ L 4 t L 3 x (I) ≤ C α M -α+ d 12 N d 12 ∥ ∆ N u 0 ∥ L 2 x ∥ ∆ M v 0 ∥ L 2 x .
Choosing α = d 12 + 1 2 , writing

γ(I) = (N d-1 2 M -1 2 ) -1 ∥u(1 -∆ ∼M )v∥ L 2 t,x (I) ,
and, by using sub-additivity, we easily obtain (3.3.21). Finally, (3.3.20) follows from a Littlewood-Paley decomposition of u in the first term on the right-hand side of (3.3.23).

Step 2: Replacing the perturbed evolution e -itH by the free one e it∆ . To prove Proposition 3.3.6, it remains to estimate the first two terms on the right-hand side of (3.3.20). The idea is to use Bourgain's estimate (3.3.9) in the free case. To do so, we shall understand how to describe the perturbed evolution e -itH in terms of e it∆ . For this purpose, we see V as a forcing term, and we follow the proof of the inhomogeneous bilinear estimate from [START_REF] Colliander | Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3[END_REF]. Before that, we prove the following lemma that we will use to estimate the Duhamel term that comes from the potential, seen as the forcing term in what follows.

Lemma 3.3.7. For all α ∈ N, there exists C = C(α, V ) such that for all dyadic integer K, N and all u 0 ∈ L 2 x we have

R ∥∆ K V ∆ N e -itH u 0 ∥ 2 L 2 x dτ 1/2 ≤ C K -α ∥ ∆ N u 0 ∥ L 2 x if K ≥ 2 5 N , ∥ ∆ N u 0 ∥ L 2 x if K ≲ N .
(3.3.24)

Proof. In the case when K ≲ N , we just apply the local smoothing (3.2.7). Otherwise, we need to gain a negative power of K. To do so, we fix K > 2 5 N and we do a Littlewood-Paley decomposition of ∆ N e -itH u 0 :

∥∆ K V ∆ N e -itH u 0 ∥ 2 L 2 x ∼ |l-k|≥3 ∥∆ K V ∆ L ∆ N e -itH u 0 ∥ 2 L 2 x + |l-k|≤2 ∥∆ K V ∆ L ∆ N e -itH u 0 ∥ 2 L 2 x = A + B .
Estimate of term A: We use Lemma 3.3.5 applied with f = ∆ N e -itH u 0 to get |l-k|≥3

∥∆ K V ∆ L e -itH ∆ N e -itH u 0 ∥ 2 L 2 x ≤ C δ K -α ∥⟨x⟩ -2 e -itH ∆ N u 0 ∥ 2 L 2
x .

Then, we conclude by the local smoothing estimate (3.2.7) for e -itH that R |l-k|≥3

∥∆ K V ∆ L e -itH ∆ N e -itH u 0 ∥ 2 L 2 x dt ≤ C α K -α R ∥⟨x⟩ -2 e -itH ∆ N u 0 ∥ 2 L 2 x dt ≤ C α K -α ∥ ∆ N u 0 ∥ L 2 x .
Estimate of term B: We use (3.3.17) and the endpoint Strichartz estimate for e -itH , that is the Strichartz estimate 3.2.4 with the admissible pair (2, 2d d-2 ). Let us fix l such that |k -l| ≤ 2 and write

∥∆ K V ∆ L ∆ N e -itH u 0 ∥ L 2 x = sup ∥g∥ L 2 x ≤1 |(∆ K V ∆ L ∆ N e -itH u 0 | g) L 2 x | = |(e -itH ∆ N u 0 | ∆ N ∆ L (V ∆ K g)) L 2 x |
. By Hölder and Sobolev embedding, we have

∥∆ K V ∆ L ∆ N e -itH u 0 ∥ L 2 x ≤ ∥ e -itH ∆ N u 0 ∥ L 2d d-2 x ∥( ∆ N ∆ L )V ∆ K g∥ L d x ≤ ∥ e -itH ∆ N u 0 ∥ L 2d d-2 x ∥( ∆ N ∆ L )V ∆ K g∥ H d-2 2 x . (3.3.25)
Since |k -n| ≥ 5 and |l -k| ≤ 2, we have |n -l| ≥ 3. Therefore, we can apply (3.3.17), and get

∥( ∆ N ∆ L )V ∆ K g∥ H d-2 2 ≲ α K -α ∥V ∆ K g∥ L 2 x ≲ α K -α ∥g∥ L 2
x . Hence, we conclude from (3.3.25) that

∥∆ K V ∆ L ∆ N e -itH ∆ N u 0 ∥ L 2 x ≲ α K -α ∥g∥ L 2 x ∥ e -itH ∆ N u 0 ∥ L 2d d-2 x ≲ α K -α ∥ e -itH ∆ N u 0 ∥ L 2d d-2 x
, and the endpoint Strichartz estimate (3.2.4) for e -itH yields R |k-l|≤2

∥∆ K V ∆ L ∆ N e -itH ∆ N u 0 ∥ 2 L 2 x ≲ α K -α R ∥ e -itH ∆ N u 0 ∥ 2 L 2d d-2 x dt ≲ α K -α ∥ ∆ N u 0 ∥ 2 L 2
x .

This proves the estimate for term B, and finishes the proof of Lemma 3.3.7.

Let us fix an interval I ⊆ R. Without loss of generality, we assume that 0 ∈ I and, given a dyadic integer K, we write Duhamel's formula as follows:

∆ K u(t) = e it∆ ∆ K ∆ N u 0 -i t 0 e i(t-τ )∆ ∆ K (V u(τ ))dτ , ∆ ∼M v(t) = e it∆ ∆ ∼M ∆ M v 0 -i t 0 e i(t-τ )∆ ∆ ∼M (V v(τ ))dτ . (3.3.26)
Then, we write

F (t) = t 0 e -iτ ∆ ∆ K (V u(τ ))dτ, G(t) = t 0 e -iτ ∆ ∆ ∼M (V v(τ ))dτ .
Using decomposition (3.3.26) and developing the product, we obtain

∥∆ K u∆ ∼M v∥ L 2 t,x (I) ≤∥ e it∆ ∆ K ∆ N u 0 e it∆ ∆ ∼M ∆ M v 0 ∥ L 2 t,x (I) + ∥ e it∆ F (t) e it∆ ∆ ∼M ∆ M v 0 ∥ L 2 t,x (I) + ∥ e it∆ ∆ K ∆ N u 0 e it∆ G(t)∥ L 2 t,x (I)
+ ∥ e it∆ F (t) e it∆ G(t)∥ L 2 t,x (I)

= I 1 + I 2 + I 3 + I 4 .

(3.3.27)

Proof of Proposition 3.3.3. It remains to estimate each term in the above decomposition.

Estimate of term I 1 : We can directly use the bilinear estimate (3.3.9) for the free evolution to estimate I 1 , where we have the interaction of two free Schrödinger evolutions of frequency-localized data. For any interval I

∥ e it∆ ∆ K ∆ N u 0 e it∆ ∆ ∼M ∆ M v 0 ∥ L 2 t,x (I) ≤ ∥ e it∆ ∆ K ∆ N u 0 e it∆ ∆ ∼M ∆ M v 0 ∥ L 2 t,x (R) ≲ N d-1 2 M -1 2 ∥∆ K ∆ N u 0 ∥ L 2 x ∥ ∆ M v 0 ∥ L 2 x ≲ C(K, N )N d-1 2 M -1 2 ∥ ∆ N u 0 ∥ L 2 x ∥ ∆ M v 0 ∥ L 2 x ,
where we have from (3.3.17) that for any α, there exists C α such that for all N, K

C(K, N ) = 1 if K ≲ N , C α K -α if K ≥ 2 5 N . (3.3.28)
Moreover, we can use sub-additivity to deduce that for any partition ℓ

I ℓ = R, ℓ ∥ e it∆ ∆ K ∆ N u 0 e it∆ ∆ ∼M ∆ M v 0 ∥ 2 L 2 t,x (I l ) 1/2 ≲ C(K, N )∥ ∆ N u 0 ∥ L 2 x ∥ ∆ M v 0 ∥ L 2 x
as well.

∥ e it∆ F (t) e it∆ ∆ ∼M ∆ M v 0 ∥ L 2 t,x (I) = ∥ e it∆ t 0 e -iτ ∆ ∆ K (V u(τ ))dτ e it∆ ∆ ∼M ∆ M v 0 ∥ L 2 t,x (I) ≤ I ∥ e it∆ e -iτ ∆ ∆ K (V u(τ )) e it∆ ∆ ∼M ∆ M v 0 ∥ L 2 t,x (I) dτ ≲ K d-1 2 M -1 2 I ∥ e -iτ ∆ ∆ K (V u(τ ))∥ L 2 x ∥ ∆ M v 0 ∥ L 2 x dτ ≲ K d-1 2 M -1 2 I ∥∆ K (V ∆ N e -iτ H u 0 )∥ 2 L 2 x dτ 1/2 ∥ ∆ M v 0 ∥ L 2 x ,
where we used the assumption that |I| ∼ 1. If K ≥ 2 5 N , we apply Lemma 3.3.7 to have

K d-1 2 M -1 2 I ∥∆ K (V ∆ N e -iτ H u 0 )∥ 2 L 2 x dτ 1/2 ≲ K d-1 2 -α M -1 2 ∥ ∆ N u 0 ∥ L 2 x ∥ ∆ M v 0 ∥ L 2 x ≲ K -1 M -1 2 ∥ ∆ N u 0 ∥ L 2 x ∥ ∆ M v 0 ∥ L 2 x , by choosing α = d-1
2 -1. Otherwise, K ≤ 2 5 and we use Minkowski, we apply the bilinear estimate (3.3.9) and the local smoothing estimate (3.2.7) to get

∥ e it∆ F (t) e it∆ ∆ ∼M ∆ M v 0 ∥ L 2 t,x (I) ≲ I ∥ e it∆ e -iτ ∆ ∆ ≲N (V u(τ )) e it∆ ∆ ∼M ∆ M v 0 ∥ L 2 t,x (I) dτ ≲ N d-1 2 M -1 2 I ∥V e -iτ H ∆ N u 0 ∥ 2 L 2 x dτ 1/2 ∥ ∆ M v 0 ∥ L 2 x ≲ N d-1 2 M -1 2 ∥ ∆ N u 0 ∥ L 2 x ∥ ∆ M v 0 ∥ L 2 x .
Estimate of term I 3 : Similarly, we apply the bilinear estimate (3.3.9) and Minkowski to get

∥ e it∆ ∆ K ∆ N u 0 e it∆ G(t)∥ L 2 t,x (I) = ∥ e it∆ ∆ K ∆ N u 0 e it∆ t 0 e -iτ ∆ ∆ ∼M (V v(τ ))dτ ∥ L 2 t,x (I) ≲ K d-1 2 M -1 2 ∥∆ K ∆ N u 0 ∥ L 2 x I ∥∆ ∼M (V e -iτ H ∆ M v 0 )∥ L 2 x dτ .
It follows from (3.3.17) that

K d-1 2 ∥∆ K ∆ N u 0 ∥ L 2 x ≲ K -1 ∥ ∆ N u 0 ∥ L 2
x . Moreover, we deduce from the fact that |I| ∼ 1 and from the local smoothing estimate (3.2.7) that

I ∥∆ ∼M (V e -iτ H ∆ M v 0 )∥ L 2 x dτ ≲ I ∥V e -iτ H ∆ M v 0 ∥ 2 L 2 x dτ 1/2 ≲ ∥ ∆ M v 0 ∥ L 2 x .
Estimate of term I 4 : We use Minkowski and (3.3.9) to get

∥ e it∆ F (t) e it∆ G(t)∥ L 2 t,x (I) = ∥ e it∆ t 0 e -iτ ∆ ∆ K (V u(τ ))dτ e it∆ t 0 e -iτ ∆ ∆ ∼M (V v(τ ))dτ ∥ L 2 (I) ≤ I×I ∥ e it∆ e -iτ 1 ∆ ∆ K (V u(τ 1 )) e it∆ e -iτ 2 ∆ ∆ ∼M (V v(τ 2 )) ∥ L 2 t,x dτ 1 dτ 2 ≲ K d-1 2 M -1 2 I ∥∆ K (V ∆ N e -iτ H u 0 ∥ L 2 x dτ I ∥∆ ∼M (V ∆ M e -iτ H v 0 )∥ L 2 x dτ .
Then, we conclude as in case I 2 and I 3 , by using Lemma 3.3.5 and the local smoothing estimate (3.2.7).

We complete the proof of Proposition 3.3.6 in the case when |I| ∼ 1 by summing over K. To prove the global estimate when I = R, we consider a partition ℓ I ℓ made of intervals of size |I ℓ | ∼ 1. Then, we exploit the sub-additivity property of the L 2 norm, that is

∥f ∥ L 2 t (R) ∼ ℓ ∥f ∥ 2 L 2 t (I ℓ ) 1/2
for a given function f ∈ L 2 (R). Moreover, when estimating terms I 2 , I 3 and I 4 , we use that the estimate (3.3.24) from Lemma 3.3.7 is global in time. This is a consequence of the local smoothing effect.

Probabilistic scattering on the continuous spectral subspace

The probabilistic nonlinear a priori estimates and other partial results, such as the critical-weighted strategy presented in this section will be reused in the next when proving Theorem 3.1.1. Here, we chose to state and to prove Theorem 3.1.5 from a pedagogical perspective before addressing the proof of the main theorem. We recall that we search for a solution of (NLS c ), which is (NLS) projected on the continuous spectral subspace, under the form ψ = εu ω + v where u ω = e -itH u ω 0 and where v, that lies in a critical space embedded into L ∞ (R; H sc (R d )), is solution to the cubic NLS equation with a stochastic forcing term

i∂ t v -Hv = P c N (εu ω + v) , v |t=0 = ψ 0 ∈ H sc (R d ) . (3.4.1)
For the sake of completeness, we first recall definitions and essential properties of critical spaces introduced by [HHK09; HTT11]. See also [START_REF] Griebel | Singular Phenomena and Scaling in Mathematical Models[END_REF], p. 49-67) for an expository presentation. These spaces of functions from an interval I ⊆ R to a Hilbert space H are constructed upon V q , U p spaces. Roughly speaking, they can be seen as the extensions of Bourgain spaces X s,b for b = 1/2, they embed into L ∞ (R, H s ) and are well-behaved with respect to sharp cutoff functions in time. Furthermore, they are well suited for global-in-time a priori estimates thanks to the duality argument detailed in section 3.4.1, while Bourgain spaces are rather used for local-in-time estimates.

Critical spaces

For now on, the Hilbert space H is L 2 (R d ) unless otherwise specified. We fix a real number 1 ≤ q < +∞ and an interval I = (a , b) with -∞ ≤ a < b ≤ +∞ and we denote the collection of finite partitions of I by Z:

Z := {t k } k=0...K | a = t 0 < t 1 < • • • < t K = b .

Definitions and embeddings

Definition 3.4.1 (Functions of bounded q-variation). V q (I) is the set of functions v : I → H endowed with the norm ∥v∥ V q (I) := sup

{t k } K-1 k=0 ∈Z K k=1 ∥v(t k ) -v(t k-1 )∥ q H 1/q .
Functions in V q have one-sided limits everywhere, and they may have at most countably many discontinuities. In what follows, we consider the closed subspace V q rc made of right-continuous functions v in V q with lim t→-∞ v(t) = 0. We still write them V q = V q rc . Next we introduce the predual space of V q (in a sense detailed in paragraph 3.4.1), namely the atomic space U p with 1 p + 1 q = 1.

Definition 3.4.2 (Atomic space U p ). A function a : I → H is a p-atom if there exists a partition {t k } k=0...K in Z and {ϕ k } k=0,...,K-1 some elements in H such that

a(t) = K k=1 1 [t k-1 ,t k ) (t)ϕ k-1 , K-1 k=0 ∥ϕ k ∥ p H ≤ 1 .
The atomic space U p (I) is the set of functions u : I → H endowed with the norm

∥u∥ U p (I) := inf ∥(λ j )∥ ℓ 1 | u = j≥1
λ j a j for some U p -atoms (a j ) .

(3.4.2)

Functions in U p are right-continuous, they admit left limits everywhere, and they may have at most countably many discontinuities. As we shall see in the next paragraph, U p is the predual space of V q when 1 p + 1 q = 1.

Proposition 3.4.3 (Embeddings, [HHK09] Proposition 2.2 and Corollary 2.6).

For

1 ≤ p < q < ∞, U p → V p → U q → L ∞ t (I, H) . (3.4.3)
Let us take from [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF] the continuity property of these norms that is particularly crucial when one wants to use some bootstrap argument. Lemma 3.4.4 (Time continuity, see Lemma A.6 in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]). Let J = [a , b), u ∈ U p (J) ∩ C(J; H s x (R 3 )) and v ∈ V q (J) ∩ C(J; H s x (R 3 )). The following mappings are continuous

t ∈ J → ∥u∥ U p [a,t) , t ∈ J → ∥v∥ V q [a,t) .
Duality argument There exists a unique bilinear map B :

U p × V q → C such that B(u, v) = K i=1 (u(t i ) -u(t i-1 ) | v(t i ))
when u is a right-continuous step function with associated partition {t k } K k=0 such that u(t 0 ) := u(a) = 0 and when v is a function in V q . We have

| B(u, v)| ≲ ∥u∥ U q ∥v∥ V p . Moreover, v ∈ V q → B(•, v) ∈ (U p ) *
is a surjective isometry, and

∥v∥ V q = sup u∈U p ,∥u∥ U p ≤1 | B(u, v)|, ∥u∥ U p = sup u∈V q ,∥u∥ V q ≤1 | B(u, v)|. (3.4.4)
The second estimate follows from Hahn-Banach theorem. Furthermore, if ∂ t u ∈ L 1 (I) we have the explicit formula

B(u, v) = I (∂ t u | v)dt .
(3.4.5)

We refer to [START_REF] Hadac | Well-posedness and scattering for the KP-II equation in a critical space[END_REF] and [START_REF] Herr | Global well-posedness of the energycritical nonlinear Schrödinger equation with small initial data in H 1 (T 3 )[END_REF] for proofs and details.

Definition 3.4.5 (Function spaces adapted to linear propagators). (i) We define by U p H (I) := e -itH U p (I) ∩ L ∞ (I; Ran P c ) , V q H (I) := e -itH V q (I) ∩ L ∞ (I; Ran P c ) the critical spaces adapted to the perturbed linear propagator. They are Banach spaces when endowed with norms

∥u∥ U p H := ∥ e itH u∥ U p , ∥v∥ V q H := ∥ e itH v∥ V q .
We have similar definitions for the space U p ∆ and V q ∆ adapted to the free evolution e it∆ . (ii) The spaces DU p and DU p H are defined by

DU p = f = ∂ t u | u ∈ U p , f (a + ) = 0 , DU p H = e -itH f = e -itH ∂ t u | u ∈ U p , f (a + ) = 0 . These spaces are endowed with norms ∥f ∥ DU p = ∥u∥ U p , ∥f ∥ DU p H = ∥ e itH u∥ U p .
Spaces DV q and DV q H are defined in the same way.

Remark 3.4.6. Here, ∂ t has to be understood in the sense of distributions on I. Let us recall that for distributions in one dimension, ∂ t u = 0 implies that u is constant, and thus we note that there is no ambiguity in the above definition, since we imposed the condition lim t→a f = 0.

The definition of DU p H and DV q H are motivated by the need to control the Duhamel integral representation of a solution u to a forced Schrödinger equation. Namely, given f in L 1 t (I, H) and u such that i∂ t u -Hu = f , u| t=0 = u 0 , the Duhamel's formulation reads

u(t) = e -itH u 0 -i t 0 e -i(t-τ )H f (τ )dτ .
Consequently, we get

∥u∥ U p H ≤ ∥u 0 ∥ H + ∥f ∥ DU p H = ∥u 0 ∥ H + ∥ t 0 e iτ H f (τ )dτ ∥ U p .
Furthermore, it follows from the duality argument (3.4.4) that

∥f ∥ DU p H = ∥ e -itH t 0 e iτ H f (τ )dτ ∥ U 2 H = sup ∥v∥ V q ≤1 B t 0 e iτ H f (τ )dτ, v .
Since f is in L 1 t (I, H) we apply the explicit formula for B (3.4.5) to get that

∥f ∥ DU p H = sup ∥v∥ V q H ≤1 I (f | v)dτ . (3.4.6)
The corresponding formula for DV q H can also be deduced from (3.4.4):

∥f ∥ DV q H = sup ∥u∥ U p H ≤1 I (f | u)dτ . ( 3
∥v∥ L q t (R;L r x (R d )) ≲ ∥v∥ V 2 H , (3.4.8) ∥v∥ L p t,x (R×R d ) ≲ ∥⟨ √ H⟩ d 2 -d+2 p v∥ V 2 H .
(3.4.9)

(ii) Bilinear Strichartz estimate: Let N, M be two dyadic numbers,

∥ ∆ N u ∆ M v∥ L 2 t,x (R×R d ) ≲ N d-2 2 N M 1 2 -∥ ∆ N u∥ V 2 H ∥ ∆ M v∥ V 2 H . (3.4.10)
Proof. To prove the embedding (3.4.8), we first deduce from the global-in-time Strichartz estimate (3.2.4) that for any Schrödinger admissible pair (p, q) the mixed Lebesgue space L q t L r x embeds into U q , which embeds into V 2 provided that 2 < q (see Proposition 3.4.3). Estimate (3.4.9) follows from (3.4.8) and from the Bernstein estimate (3.2.18). We refer to Lemma 3.3 [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF] and the references therein to see how to transfer the bilinear estimate (3.3.10) into the space V 2 H .

In order to derive probabilistic nonlinear estimates, we consider the Duhamel integral representation of the solution to (NLS c ). Provided that the nonlinear forcing term

P c N (v + εu ω ) lies in L 1 (R, H d-2
2 ), it follows from the duality argument (3.4.6) that the U 2 H -norm of the solution u to (3.4.1) can be estimated by ∥v∥

U 2 H (H d-2 2 ) ≤ ∥ψ 0 ∥ H 1/2 sup ∥w∥ V 2 H ≤1 R×R d ⟨ √ H⟩ d-2 2 P c N (v + εu ω ) wdtdx .
If we proceed to a Littlewood-Paley decomposition of each term u j that occurs in the forcing term, we are reduced to estimate sums of multilinear integrals of the form

I N 1 ,N 2 ,N 3 ,N 4 = R×R d ⟨ √ H⟩ d-2 2 ∆ N 1 u 1 ∆ N 2 u 2 ∆ N 3 u 3 ∆ N 4 wdxdt , (3.4.11)
where each term u j can be either u ω , v or their complex conjugate for j ∈ 1, 2, 3 , and w is in V 2

H

with norm less than one. Although P c N (v + εu ω ) does not a priori lies in

L 1 (R, H d- 2
2 ), it follows from Bernstein and Strichartz estimates that we have

Π N P c N (v + εu ω ) ∈ L 1 (I, H d-2
2 ) , (3.4.12)

for any dyadic number N as observed in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]. It turns out that the estimates on multilinear integrals obtained by the authors are uniform in N . Hence, we have to estimate Π N v rather than v, and to use a space slightly smaller than U 2 H which keeps track of the frequency cut-off.

Definition 3.4.8. The critical space X d-2

2 (I) is the space made of tempered distributions u :

I → H d-2 2 (R d ) ∩ Ran(P c ) such that ∥u∥ X d-2 2 (I) is finite, where ∥u∥ X d-2 2 (I) := N ∈2 N N d-2 ∥ ∆ N u∥ 2 U 2 H (I) 1/2 .
(3.4.13)

We define as well DX

2 v∥ V 2 H (I) ∼ N ∈2 N N d-2 ∥ ∆ N v∥ 2 V 2 H (I) 1/2 .
(3.4.14)

Probabilistic nonlinear estimates

Thanks to the probabilistic and bilinear improved Strichartz estimates for the perturbed operator obtained in section 3.3 and their transferred versions collected in Proposition 3.4.7 we are now able to reproduce in the perturbed framework the same analysis conducted by [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]. We get the following probabilistic nonlinear estimates.

Proposition 3.4.10. Let 0 < ε, 0 < R and

I = (a, b) with -∞ ≤ a < b ≤ +∞. For all v ∈ X d-2 2 (I) and ω ∈ Ω ε,R (see (3.3.8)) we have sup 2 N N d-2 2 ∥ Π N N (v + εu ω ) ∥ DU 2 H (I) ≲ ∥ψ 0 ∥ H 1/2 + ∥v∥ 3 V 2 H (I) + R 3 , (3.4.15)
as well as the Lipschitz estimate: for all v, v ′ ∈ V 2 H (I), sup

2 N N d-2 2 ∥ Π N N (v ′ + εu ω ) -N (v + εu ω ) ∥ DU 2 H (I) ≲ ∥v∥ 2 V 2 H (I) + ∥v ′ ∥ 2 V 2 H (I) + R 2 ∥v ′ -v∥ V 2 H (I) .
Contribution of terms like v 2 u ω with high-low interactions: We illustrate the case by case analysis performed in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF] for the flat Laplacian, and we transpose it to the perturbed setting. In order to be as brief as possible, we chose to detail one enlightening case. We consider the high-high-low regime, where

v 1 = ∆ N 1 v, v 2 = ∆ N 2 v , u ω 3 = ∆ N 3 u ω , w 4 = ∆ N 4 w, and N 1 ≤ N 1 d-1 3 ≤ N 2 ≤ N 3 , N 4 ∼ N . 5
We refer to ([BOP15a], Proposition 4.1) for the other subcases. As we explained in a previous paragraph, the proof of Proposition 3.4.10 consists in getting some a priori estimates on multilinear integrals of the form (3.4.11). Under the assumption made on the frequencies, the main term is the one where the derivatives fall onto ∆ N 3 v 3 . Hence, it follows from Hölder inequality that 6

| R×R d ⟨ √ H⟩ d-2 2 v 1 v 2 εu ω 3 ∆ N 4 wdxdt| ≲ N d-2 2 3 ∥v 2 ∥ L 2(d+2) d t,x ∥εu ω 3 ∥ L d+2 t,x ∥v 1 w 4 ∥ L 2 t,x ≲ N d-2 2 3 N d-1 2 - 1 N -1 2 + 4 ∥v 2 ∥ V 2 H ∥εu ω 3 ∥ L d+2 t,x ∥v 1 ∥ V 2 H ∥w 4 ∥ V 2 H ≲ C(N 1 , N 2 , N 3 , N 4 ) N d-2 2 2 ∥v 2 ∥ V 2 H N s 3 ∥εu ω 3 ∥ L d+2 t,x N d-2 2 1 ∥v 1 ∥ V 2 H ∥w 4 ∥ V 2 H , with C(N 1 , N 2 , N 3 , N 4 ) = N d-2 2 -s 3 N d-1 2 -0-d-2 2 1 N -d-2 2 2 N -1 2 +0 4 ≲ N d-2 2 -s+ 1 2(d-1) -d-2 2(d-1) -1 2 +0 3 =: N δ-s 3 .
We used the transferred Strichartz estimate (3.4.8) to control the term v 1 and the transferred bilinear estimate (3.4.10) to control the term with v 2 w 4 . Note that under the assumption that

d-1 d+1 • d-2 2 < s, we have δ = d -3 d -1 • d -2 2 + 0 < d -1 d + 1 • d -2 2 < s .
5 For convenience, we define

Λ N = (N 1 , N 2 , N 3 , N 4 ) | N 1 ≤ N 1 d-1 3 ≤ N 2 ≤ N 3 , N 4 ∼ N .
6 For a dyadic number N we shall denote small positive (resp. negative) power of N by N +0 (resp. N -0 ). Therefore, since the highest frequency comes with a negative power we are able to apply Schur's test and to sum over N 1 , N 2 , N 4 . To sum over the dyadic numbers N 3 , we use first Hölder

N 3 ∈2 N N δ-s 3 N s 3 ∥ ∆ N 3 εu ω ∥ L d+2 t,x ≤ N δ-s+0 N 3 ∈2 N N -0 3 d+1 d+2 ∥N s 3 ∆ N 3 εu ω ∥ ℓ d+2 N 3 L d+2 t,x ≲ N δ-s+0 ∥N s 3 ∆ N 3 εu ω ∥ L d+2 t ℓ d+2 N 3 L d+2 x ≲ N δ-s+0 ∥N s 3 ∆ N 3 εu ω ∥ L d+2 t ℓ 2 N 3 L d+2 x ,
N 3 ∈2 N N δ-s 3 N s 3 ∥ ∆ N 3 εu ω ∥ L d+2 t,x ≲ N δ-s+0 ∥⟨ √ H⟩ δ+ εu ω ∥ L d+2 t,x ≲ N δ-s+0 R . (3.4.16)
Hence,

N 3 R×R d |⟨ √ H⟩ d-2 2 v 1 v 2 εu ω 3 w 4 |dxdt ≲ N δ-s+0 R N d-2 2 1 ∥v 1 ∥ V 2 H N d-2 2 2 ∥v 2 ∥ V 2 H ∥w 4 ∥ V 4 H .
It follows from the Schur's test (see [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF], Lemma 3.6) that

Λ N R×R d |⟨ √ H⟩ d-2 2 v 1 v 2 εu ω 3 w 4 |dxdt ≲ R∥N d-2 2 1 v 1 ∥ ℓ 2 N 1 V 2 H ∥N d-2 2 2 v 2 ∥ ℓ 2 N 2 V 2 H ∥w 4 ∥ ℓ 2 N 4 V 2 H
, and the contribution of the integral (3.4.11) in this case is less than

sup ∥w∥ V 2 H ≤1 R∥⟨ √ H⟩ d-2 2 v∥ 2 V 2 H ∥w∥ V 2 H ≲ R∥⟨ √ H⟩ d-2 2 v∥ 2 V 2 H .
Remark 3.4.11. As observed in [START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF], there is at least a portion of ∥⟨ √ H⟩ s εu ω ∥ L p t,x that bounds from above the multilinear integrals where u ω occurs, with p ∈ 4, d + 2, 6 d+2 d+4 (see for instance (3.4.16) where p = d + 2). Hence, it follows from the probabilistic Strichartz estimate (3.3.6) and from the monotone convergence theorem that for all ω ∈ Ω, lim

|I|→0 ∥εu ω ∥ L p t,x (I×R d ) = 0 .
In the vvv-case, we also have

sup ∥w∥ V 2 H ≤1 | I R d 1 I (t)(⟨ √ H⟩ d-2 2 v)v 2 wdxdt| ≤ ∥⟨ √ H⟩ d-2 2 v∥ L 2(d+2) d t,x (I×R d ) ∥v∥ 2 L d+2 t,x (I×R d ) ,
(3.4.17) where the right-hand side of (3.4.17) is uniformly bounded by the global-in-time critical X d-2 2 (R)norm. Hence, the left-hand side of (3.4.17) also goes to zero when |I| → 0 (in order to prove continuity or local well-posedness) or when I ⊂ [t , ∞) with t → +∞ (in order to prove scattering). Consequently, we have in case that We are now ready to extend the probabilistic global Cauchy theory for (NLS) to the case where we have a short-range potential, and when the equation is projected onto the continuous spectral subspace.

Global existence

We denote the interval [0 , ∞) by I and we introduce the Banach space

B R = v ∈ X d-2 2 (I) ∩ C(I, H d-2 2 ) | ∥v∥ X d-2 2 (0,∞) ≤ R
for a fixed small R > 0, as well as the operator

T : v ∈ B R → t → e -itH ψ 0 -i t 0 e -i(t-τ )H P c N (v + εu ω ) dτ .
It follows from the a priori estimates collected in Proposition 3.4.10 that for all v in B R ,

∥T v∥ X d-2 2 (0,∞) ≤ ∥ e -itH ψ 0 ∥ X d-2
2 (0,∞)

+ ∥ P c N (v + εu ω ) ∥ DX d-2 2 (0,∞) ≤ ∥ψ 0 ∥ H d-2 2 + 2C 1 R 3 ,
and that for all v, v ′ in B R ,

∥T (v ′ ) -T (v)∥ X d-2 2 (0,∞) ≤ ∥ P c N (v ′ + εu ω ) -N (v + εu ω ) ∥ DX d-2 2 (0,∞) ≤ 3C 2 R 2 ∥v ′ -v∥ X d-2 2 (0,∞)
.

Hence, T is a contraction mapping on B R provided that

2C 1 R ≤ 1/2, 3C 2 R ≤ 1/2 and ∥ψ 0 ∥ H d-2 2 ≤ R/2 .
With this fixed R = R 0 , we write Ω ε = Ω ε,R 0 the set defined in (3.3.8). It satisfies

P(Ω \ Ω ε ) ≤ C exp -cε -2 ∥u 0 ∥ -2 H s
where c = cR -2 0 is a universal constant.

Continuity and scattering : Let v be a solution to (NLS c ) with random initial data associated with an ω ∈ Ω ε . We need to show that the limit of

I(t) exists in H d-2 2
when t goes to infinity, where

I(t) := t 0 e iτ H P c N (v + εu ω ) dτ .
We use the Cauchy criterion, and chose t 1 , t 2 ∈ (0 , ∞) with, say, t 1 < t 2 . We denote the interval (t 1 , t 2 ) by I. It follows from (3.4.18) in remark 3.4.11 that

∥I(t 2 ) -I(t 1 )∥ H d-2 2 ≤ ∥ P c N (v + εu ω ) ∥ X d-2 2 (I) -→ t 1 ,t 2 →+∞ 0 .
This yields scattering with

v + := ∞ 0 e iτ H P c N (v + εu ω ) dτ ∈ H d-2 2 (R d ) ,
and the same argument applied when |I| → 0 proves that T v is in C(I, H d-2

2 ).

Uniqueness : We use a connectedness argument. Let v 1 , v 2 be two solutions to (NLS c ) in the space

X d-2 2 [0 , ∞). Define A = t ∈ [0 , ∞) | v 1 (t) = v 2 (t)
. By continuity in time, A is closed. Let t ∈ A and, for small τ , we define I = (t -τ , t + τ ) if t > 0, or I = [0 , τ ) otherwise. It follows from remark 3.4.11 that

∥v 2 -v 1 ∥ X d- 2
2 (I) = o(τ ) ∥v 2 -v 1 ∥ X d-2 2 (I)
.

Hence, v 2 = v 1 on I if τ is small enough, which proves that A is open. We conclude that A = [0 , ∞) whenever A is nonempty, and this finishes the proof of Theorem 3.1.5.

Local smoothing and the critical weighted strategy

First, we transfer in U 2 H the local smoothing effect stated in Proposition 3.2.5

Proposition 3.4.12 (Local smoothing in U 2 ). The local smoothing estimate reads in U 2 H (H 1/2 ):

R ∥ P c (u)∥ 2 H 1,-1/2-dt ≲ V ∥u∥ 2 U 2 H (H 1/2 ) . (3.4.19)
The proof is straightforward and follows from the transference principle and from Proposition 3.2.5 with F = 0. Note in particular that the result is true when V = 0 and when the space

U 2 H is replaced by U 2 ∆ . Remark 3.4.13. Estimate (3.4.19) means that U 2 H (H 1/2 ) embeds into L 2 t (R; H 1,-1/2-(R d
)), and we can deduce from this and from the duality argument (3.4.6) that

L 2 t (R; L 2,1/2+ x ) → DV 2 H (H 1/2 ) . (3.4.20) Indeed, provided that f ∈ L 1 (R, L 2
x ), we have

∥⟨ √ H⟩ 1/2 f ∥ DV 2 H = sup ∥u∥ U 2 H ≤1 | R (⟨ √ H⟩ 1/2 f | u)dt| = sup ∥u∥ U 2 H ≤1 | R (f | ⟨ √ H⟩ 1/2 u)dt| ≤ sup ∥u∥ U 2 H ≤1 ∥⟨x⟩ 1/2+ f ∥ L 2 t,x ∥⟨x⟩ -1/2-⟨ √ H⟩ ⟨ √ H⟩ -1/2 u ∥ L 2 t,x ≤ sup ∥u∥ U 2 H ≤1 ∥⟨x⟩ 1/2+ f ∥ L 2 t,x ∥u∥ U 2 H ≲ ∥f ∥ L 2 t (R;L 2,1/2+ x
) .

The general case follows from the density of step functions in V p (see the proof in [START_REF] Griebel | Singular Phenomena and Scaling in Mathematical Models[END_REF], Lemma 4 p.56). Furthermore, embedding (3.4.3) directly yields the other dual embedding

DU 2 H (H 1/2 ) → DV 2 H (H 1/2 ) . (3.4.21)
Remark 3.4.14. In light of the nonlinear analysis performed in section 3.4.2, we shall use the slightly better estimate

R ∥ P c u∥ H 1,-1/2-dt ≲ sup 2 N N 1/2 ∥ Π N u∥ U 2 H .
Let us denote the Schwartz class by S(R d ) and the space of tempered distributions S ′ (R d ), and deduce the above refined estimate from (3.4.19) and duality. Given u ∈ H 1,-1/2-, we have

∥ P c u∥ H 1,-1/2-= sup φ∈S,∥φ∥ L 2 ≤1 |(P c u | ⟨x⟩ -1/2-⟨ √ H⟩φ)| .
Since ⟨x⟩ -1/2-⟨ √ H⟩φ ∈ S(R d ) and u = lim

N →∞ Π N u in S ′ (R d ), we have ∥ P c u∥ H 1,-1/2-= sup φ∈S,∥φ∥ L 2 ≤1 | lim N →∞ Π N u | ⟨x⟩ -1/2-⟨ √ H⟩φ | ≤ sup 2 N ∥ Π N u∥ H 1,-1/2-.
Critical-weighted strategy The next Proposition details the critical-weighted strategy. As explained in the introduction, this strategy makes it possible to handle both linear and nonlinear terms that arise in the stability equation around a ground state. We state two versions of this strategy. The first one is suited to the operator -∆ and has an interest in itself. One can use it when V u is not absorbed by the linear operator, but when it is seen instead as a source term. The second version is suited to the dynamic around the nonlinear ground states, when V is absorbed by the linear operator so that e -itH preserves Ran P c and the decomposition of the phase space from Lemma 3.5.4. This second version plays a key role in the proof of Theorem 3.1.1. See also [START_REF] Koch | Small data scattering and soliton stability in ( Ḣ-1/6 for the quartic KdV equation[END_REF] for a similar approach in the context of Korteweg-de Vries equation.

Proposition 3.4.15. Let u be a solution to the forced7 Schrödinger equation

i∂ t u + (∆ -V )u = f + g , u| t=0 = u 0 . ( 3

.4.22)

There exists C = C(V ) depending on weighted Sobolev norms of V such that for all u ∈ Ran P c (H),

∥u∥ V 2 ∆ (H 1/2 ) + ∥u∥ L 2 t (R;H 1,-1/2-) ≤ C ∥u 0 ∥ H 1/2 + sup 2 N N 1/2 ∥Π N f ∥ DU 2 ∆ + ∥g∥ L 2 t (R;L 2,1/2+ x ) , (3.4.23) ∥u∥ V 2 H (H 1/2 ) + ∥u∥ L 2 t (R;H 1,-1/2-) ≤ C ∥u 0 ∥ H 1/2 + sup 2 N N 1/2 ∥ Π N f ∥ DU 2 H + ∥g∥ L 2 t (R;L 2,1/2+ x
) . (3.4.24)

Remark 3.4.16. We prove the estimate with sup

2 N ∥ Π N f ∥ DU 2 instead of ∥f ∥ DU 2 in order to use (3.4.15).
Proof. Estimate (3.4.24) is slightly easier to prove since we don't have to control the term V u which is absorbed by the left-hand side in (3.4.22). Also, the local smoothing estimates we use are the same for -∆ and -∆ + V provided that u in Ran(P c ) so we only write the proof of (3.4.23). To that end, we pass the term V u on the right-hand side of equation (3.4.22)

i∂ t u + ∆u = f + g + V u, u| t=0 = u 0 .
By definition of the space DV 2 ∆ , Duhamel's formulation gives

∥u∥ V 2 ∆ (H 1/2 ) ≲ sup 2 N ∥Π N f ∥ DV 2 ∆ (H 1/2 ) + ∥g∥ DV 2 ∆ (H 1/2 ) + ∥V u∥ DV 2 ∆ (H 1/2 ) , ≲ sup 2 N N 1/2 ∥Π N f ∥ DU 2 ∆ + ∥g∥ L 2 t (R;L 2,1/2+ x ) + ∥V u∥ L 2 t (R;L 2,1/2+ x
) .

(3.4.25)

Note that we used the dual embedding (3.4.21) to control Π N f in DU 2 ∆ and the local smoothing dual embedding (3.4.20) to control g and V u. As for the weighted norm of u, we decompose it into u = v + w where v, w ∈ Ran P c (H) and are solution to

i∂ t v + ∆v = f , i∂ t w + (∆ -V )w = g -V v , v| t=0 = 0 , w| t=0 = u 0 .
To control u in L 2 t (R; H 1,-1/2-(R 3 )) we first control the weighted norm of v and then the weighted norm of w. To do so, we apply the transferred local smoothing estimate (3.4.19) with V = 0.

∥v∥ L 2 t (R;H 1,-1/2-) ≤ sup 2 N ∥Π N v∥ L 2 t (R;H 1,-1/2-) ≲ sup 2 N N 1/2 ∥Π N v∥ U 2 ∆ = sup 2 N N 1/2 ∥Π N f ∥ DU 2 ∆ . (3.4.26)
Afterwards, we use the fact that w ∈ P c (H) and we apply the local smoothing estimate for H:

∥w∥ L 2 t (R;H 1,-1/2-) ≲ ∥u 0 ∥ H 1/2 + ∥g∥ L 2 t (R;L 2,1/2+ x ) + ∥V v∥ L 2 t (R;L 2,1/2+ x
) .

(3.4.27)

To estimate the second term in the right-hand side, we use the fact that V is localized.

∥V v∥ L 2 t (R;L 2,1/2+ x ) ≲ ∥⟨x⟩ 1+ V ∥ L ∞ x (R d ) ∥v∥ L 2 t (R;L 2,-1/2- x ) ≤ C(V )∥v∥ L 2 t (R;H 1,-1/2-) ≤ C(V )sup 2 N N 1/2 ∥Π N f ∥ DU 2 ∆ , (
(R;H 1,-1/2-) ≲ V ∥u 0 ∥ L 2 x + sup 2 N 2 N ∥Π N f ∥ DU 2 ∆ + ∥g∥ L 2 t (R;L 2,1/2+ x
) .

(3.4.29)

Finally, we deduce from (3.4.26) and (3.4.29) the estimate for u in the weighted space.

∥u∥ L 2 t (R;H 1,-1/2-) ≲ ∥u 0 ∥ H 1/2 + sup 2 N 2 N ∥Π N f ∥ DU 2 ∆ + ∥g∥ L 2 t (R;L 2,1/2+ x
) .

(3.4.30)

It remains to control V u in V 2 ∆ (H 1/2 ) to close the estimate (3.4.25). This is again a consequence of the fact that V is localized, and we proceed as in (3.4.28) to get

∥V u∥ L 2 t (R;L 2,1/2+ x ) ≲ V ∥u∥ L 2 t (R;H 1,-1/2-) . (3.4.31)
We conclude by injecting (3.4.31) and (3.4.30) into (3.4.25) to control u in V 2 ∆ (H 1/2 ).

Probabilistic asymptotic stability for small ground states

For now on, we fix the dimension d = 3 and we make the extra assumption that σ(H) = e 0 ∪ σ c (H) where e 0 < 0 is a simple negative eigenvalue with positive and normalized eigenfunction ϕ 0 . We recall that σ c (H) = [0 , +∞), with no resonance nor eigenvalue at zero. Since complex conjugate and numerical constants play no role in what follows, we might sometimes drop them from the notation for the sake of clarity. We will also drop the dependence on time in the notation and write ∥ D α z Q∥ := sup

Local existence

First, let us briefly transpose the probabilistic Cauchy theory, at least locally in time, to our setting where the Schrödinger operator -∆ + V has some discrete spectrum. More precisely, we prove that under smallness assumptions on the H 1/2 -norm of the initial data ψ 0 , (NLS) still admits a unique local solution under the form

ψ(t) = εu ω (t) + v(t), v ∈ H 1/2 .
Indeed, the discrete part of the solution does not contribute in short time, and we shall be able to reproduce the scheme developed in section 3.4, with the same gain of regularity for the nonlinear part of the solution. Note that by time reversibility, we only consider forward-in-time solutions in what follows.

Proposition 3.5.1 (Local existence). There exist δ 0 > 0 and c > 0 such that for all T ≲ 1, all α 0 ∈ C, ν 0 ∈ Ran P c ∩H 1/2 with ∥v 0 ∥ H 1/2 + |α 0 | < δ 0 , and all ω ∈ Ω ε,R with R = cδ 0 , the Cauchy problem (NLS) with data ψ(t 0 ) = εu ω 0 + ν 0 + α 0 ϕ 0 admits a unique solution ψ on [t 0 , t 0 + T ) under the form

ψ(t) = εu ω (t) + v(t), v ∈ C([t 0 , t 0 + T ); H 1/2 ) .
Uniqueness holds for P c v ∈ X 1/2 (0 , T ) and

(v | ϕ 0 ) ∈ L ∞ t (0 , T ).
We recall that the set of random initial data with improved Strichartz estimates defined in (3.3.8). In light of the above Proposition, we fix R = cδ 0 for now on, and take an initial value u ω 0 that corresponds to some ω in Ω ε,cδ 0 . We also recall that ε is much smaller than R.

Proof. For simplicity, we assume that t 0 = 0. At each time t, we decompose v(t) into v(t) = ν(t) + α(t)ϕ 0 , with ν(t) = P c v(t), and α(t) = (v(t) | ϕ 0 ). Then, ψ is solution to (NLS) if and only if (ν, α) are solution to the coupled system i∂ t ν + (∆ -V )ν = P c N εu ω + ν + αϕ 0 , ν(0) = ν 0 , and i α + e 0 α = (N (εu ω + ν + αϕ 0 ) | ϕ 0 ) , α(0) = α 0 .

It follows from the Duhamel's formulation that solving the above system reduces to find a fixed point for the map Γ = (Γ 1 , Γ 2 ), where

Γ 1 (ν)(t) = e -itH ν 0 -i t 0 e -i(t-τ )H P c N εu ω (τ ) + ν(τ ) + α(τ )ϕ 0 dτ , Γ 2 (α)(t) = e ie 0 t α 0 -i t 0 e i(t-τ )e 0 (N εu ω (τ ) + ν(τ ) + α(τ )ϕ 0 | ϕ 0 )dτ .
Given T > 0 and A to be chosen later on, we search a fixed point for Γ in the Banach space

E T,A = (ν, α) ∈ X 1/2 (0 , T ) × L ∞ ((0 , T ); C) | ∥ν∥ X 1/2 (0,T ) + sup 0≤t≤T |α(t)| ≤ A .
In this setting, we have the following a priori estimates: given (ν, α) ∈ E T,A ,

∥Γ 1 (ν)∥ X 1/2 (0,T ) ≤ C∥ν 0 ∥ H 1/2 + C 1 + T 1/5 A 3 + R 3 , (3.5.1) and sup 0≤t≤T |Γ 2 (α)(t)| ≤ α 0 + C(1 + T ) A 3 + R 3 . (3.5.2)
Under our assumptions, we now fix A = R = 2C 0 δ 0 =: cδ 0 and the above estimates yield

∥Γ 1 (ν)∥ X 1/2 (0,T ) + sup 0≤t≤T |Γ 2 (α)(t)| ≤ C 0 δ 0 + A 3 + T 1/2 ≤ 2C 0 δ 0
provided that δ 0 C 0 < 1/2, and where

C 0 = C 0 (∥ϕ 0 ∥ H 1 , ∥|x| -1/2-ϕ∥ L ∞ x )
. This proves that Γ preserves the space E T,A . Let us briefly explain how we obtained estimates (3.5.1) and (3.5.2). To prove (3.5.1), we first note that the terms without the discrete part αϕ 0 are handled by the analysis conducted in section 3.4.1, globally in time. Therefore, by dropping the complex conjugate from the notations, it remains to control some terms of the form (αϕ 0 ) 3 , (αϕ 0 ) 2 (εu ω + ν), αϕ 0 (εu ω + ν) 2 .

To handle the terms without any power of the linear part u ω , we use the Leibniz rule and we obtain

∥(αϕ 0 ) 3 ∥ DX 1/2 H (0,T ) + ∥(αϕ 0 ) 2 ν∥ DX 1/2 H (0,T ) + ∥(αϕ 0 )ν 2 ∥ DX 1/2 H (0,T ) ≲ T sup 0≤t≤T |α(t)| 3 ∥ϕ 0 ∥ 3 H 1/2 + ∥ν∥ X 1/2 H (0,T ) ≲ T A 3 .
As for the terms with some power of u ω , we can use the Leibniz rule and the local smoothing estimate (3.2.7) as follows. We do a Littlewood-Paley decomposition of each term in the definition of Writing the definition of the X 1/2 -norm: given a fixed N ∈ 2 N , we denote by N 1 , N 2 , N 3 the frequencies at which ϕ 0 , u ω , u ω are localized.

Case αϕ 0 (u ω ) 2 : We consider the worst case where ϕ 0 , which is the smoothest term, comes with the lowest frequency N 1 ≤ N 2 ≤ N 3 . In the high-low-low case, when we have N 2 ≪ N 3 , ∆ N 3 u ω cannot absorb all the derivatives, and we need to use the local smoothing effect. To that hand, we apply Hölder with 1 = 1 2 + 1 5 + 3 10 and get

N 1/2 ∥P N αϕ 0 (εu ω ) 2 ∥ U 2 H (0,T ) ≤ ε 2 N 1/2 3 sup ∥w∥ V 2 H (0,T ) ≤1 | T 0 R 3 α(t) ∆ N 1 ϕ 0 ∆ N 2 u ω ∆ N 3 u ω wdxdt| ≲ sup 0≤t≤T |α(t)|N 1/2 3 ∥ ∆ N 1 ϕ 0 ∆ N 3 εu ω ∥ L 2 t,x [0,T ) ∥εu ω ∥ L 5 t,x [0,T ) ∥w∥ V 2 H ≲ A∥⟨x⟩ 1/2+ ϕ 0 ∥ L ∞ x N 1/2 3 ∥⟨x⟩ -1/2-∆ N 3 εu ω ∥ L 2 t,x (0,T ) ∥εu ω ∥ L 5 t,x [0,T ) .
Then, we use the probabilistic Strichartz estimate (3.5.14) for the L 5 t,x norm and the local smoothing effect (3.2.7) to get

N 1/2 ∥P N αϕ 0 (εu ω ) 2 ∥ U 2 H (0,T ) ≲ A∥ ∆ N 3 εu ω 0 ∥ L 2 x R ≲ AεR ≲ A 3 + R 3 ,
since ε has to be chosen much smaller than R in (3.3.8) for the probability of the bad set of initial data to be small.8

Case (αϕ 0 ) 2 u ω : Similarly, we assume that N 1 ≤ N 2 ≤ N 3 ∼ N and do the same computations with α(t)ϕ 0 instead of εu ω . The only difference is that

∥αϕ 0 ∥ L 5 t,x [0,T ) ≤ AT 1/5 ∥ϕ 0 ∥ L 5 x .
This yields N 1/2 ∥P N (αϕ 0 ) 2 εu ω ∥ U 2 H (0,T ) ≲ T 1/5 A 2 ε ≲ T 1/5 (A 3 + R 3 ) . Then, we obtain (3.5.1) by summing over N 1 , N 2 , N 3 and N as in section 3.4.1. To get estimate (3.5.2), we just use Hölder, the endpoint Strichartz estimate and the embedding L 3 t ((0 , T ); L

4 x ) → X 0 H (0 , T ) sup 0<t<T | t 0 e i(t-τ )e 0 (N εu ω (τ ) + ν(τ ) + α(τ )ϕ 0 | ϕ 0 )dτ | ≲ T 0 R 3 |ν(τ, x) + εu ω (τ, x)| 3 + |α(τ )ϕ 0 (x)| 3 |ϕ 0 (x)|dxdτ ≲ ∥ϕ 0 ∥ L 2 x T 0 ∥ν(τ ) + εu ω ∥ 3 L 6 x dτ + T sup 0<t<T |α(t)| 3 ∥ϕ 0 ∥ 4 L 4 x ≲ C(ϕ 0 ) ∥⟨∇⟩ 1/4 ν∥ 3 L 3 t L 4 x + ∥ε⟨∇⟩ 1/4 u ω ∥ 3 L 3 t L 4 x + T A 3 .
By the embedding L 3 t ((0 , T ); L 4 x ) → X 0 H (0 , T ), the right-hand side is less than

≲ C(ϕ 0 ) ∥ν∥ 3 X 1/4 H (0,T ) + ε 3 ∥u 0 ∥ 3 H 1/4 + T A 3 ≲ C(ϕ 0 )(1 + T ) A 3 + R 3 .
This finishes the proof of estimates (3.5.1) and (3.5.2). The proof of the Lipschitz estimates for Γ follows similarly, and establishes the contraction mapping property for Γ.

Nonlinear ground states

Before we dive into the proof of asymptotic stability we establish the main global estimates on the local flow constructed in the above paragraph, we collect from [GNT04] some properties of the curve of nonlinear ground states that bifurcates from the eigenspace spanned by ϕ 0 . In particular, we will be able to decompose the solution into a ground state plus a radiation term at each time, where the radiation term will be shown to scatter in the last paragraph. First, we recall how the ground states are constructed.

Lemma 3.5.2 (Nonlinear ground states (Lemma 2.1 in [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF])). There exists δ > 0 small enough such that for all z ∈ C with |z| ≤ δ, there exists a nonlinear ground state Q and E solution to

∆ -V + |Q| 2 Q = EQ (3.5.3)
under the form Q(z) = zϕ 0 + q(z), E(z) = e 0 + e(z).

We have uniqueness for (q, e) in the class

(q, e) ∈ (H 2 ∩ Ran(P c )) × R | ∥q∥ H 2 ≤ δ, |e| ≤ δ .
Thanks to the gauge invariance of the nonlinear part,

Q(z e iα ) = Q(z) e iα , E(z) = E(|z|).
In addition, q and its derivatives are small in H 2 (R 3 ):

q = O z 3 , D z Q = O z 2 , D 2 z q = O(z) . (3.5.4)
The first two derivatives of e are also small and satisfy

| D z e| = O(z) , | D 2 z e| = O(1) .
(3.5.5)

The following elliptic estimates on some Sobolev weighted norms of the ground states and its derivatives with respect to the parameter z are crucial, and our arguments heavily rely on the smallness of these quantities.

Lemma 3.5.3 (Weighted elliptic estimates for the derivatives of ground states, see [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF]). There exists δ > 0 such that for all |z| < δ and all k ∈ 1, 2 we have

∥⟨x⟩ k q∥ H 2 (R 3 ) = O z 3 , ∥⟨x⟩ k D z Q∥ H 2 (R 3 ) = O z 2 .
(3.5.6)

We recall that the generalized continuous spectral spaces H c (z) parametrized by z were introduced to encode orthogonality conditions (3.1.7):

H c (z) := η ∈ L 2 (R 3 ) | R 3 ⟨iη, ∂ z 1 Q⟩ dx = R 3 ⟨iη, ∂ z 2 Q⟩ dx = 0 .
In particular, note that Ran(P c ) corresponds to H c (0). The following result (Lemma 2.2 in [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF]) yields a bijection from H c (0) to H c (z). This useful correspondence reduces the dynamic of η in ansatz (3.1.9) to the dynamic of its continuous spectral part ν. Lemma 3.5.4 (Continuous spectral subspace comparison). There exists δ > 0 such that any function ψ ∈ L 2 (R 3 ) can be uniquely decomposed into

ψ = Q(z) + η, η ∈ H c (z) .
If ∥ψ∥ H s ≤ δ for some s ≥ 0 then η ∈ H s and ∥η∥ H s + |z| ≲ ∥ψ∥ H s . Moreover, for all |z| ≤ δ there exists a bijective operator

R(z) : H c (0) → H c (z) u → u + α(z)(u)ϕ 0 .
Here, α(z)u is solution to

Λ(z)α(z)u = ⟨iu, D z Q⟩ , with Λ(z) = 0 -1 1 0 + ⟨ϕ 0 , Im(∂ z 2 q)⟩ -⟨ϕ 0 , Re(∂ z 2 q)⟩ ⟨ϕ 0 , Im(∂ z 1 q)⟩ ⟨ϕ 0 , Re(∂z 1 q)⟩ =: 0 -1 1 0 + γ(z) .
Finally, R(z) -I is compact and continuous in the operator norm on any space Y that satisfies

H -2 ⊂ Y ⊂ H 2 .

Global a priori estimates

Now that we have a local in time solution ψ for small initial data, we need to obtain some global estimates on ϕ to extend it as a global solution and understand to its asymptotic behavior. First, we recall that in a small neighborhood of 0 the solution can be uniquely decomposed into

ψ = Q(z) + η ,
where η satisfies the time-dependent orthogonality conditions η(t) ∈ H c (z(t)). In what follows, the continuous spectral part of the nonlinear part9 of the solution is denoted by ν:

ν = P c (η) -e -itH εu ω 0 ∈ H c (0) .
Then, we deduce from Lemma 3.5.4 that η = R(z) P c η so that the solution decouples into

ψ = Q(z) + η = Q(z) + R(z) εu ω + ν .
In particular, we note that

P c (η) = εu ω + ν, P p (η) = (R(z) -I) P c (η) = α(z) P c (η)ϕ 0 = α(z) εu ω + ν ϕ 0 .
In what follows, we fix δ > 0 as in Lemmas 3.5.3, 3.5.4 and δ 0 > 0 as in the local existence result from Proposition 3.5.1. Then, we take ε and δ ′ < δ such that

∥ν∥ L 2 + |z| ≤ δ ′ =⇒ ∥ψ∥ H s ≤ δ .
The goal is to obtain some global bounds on η in a critical space contained in L ∞ t H

1/2 x and on z to prove that the solutions stay small. Then we bootstrap the local existence result to obtain global existence, and further exploit the global estimates to prove asymptotic stability. We denote by m the following gauge transformation of the parameter z m(z) = z exp i t 0 E(z)(τ )dτ .

(3.5.7)

It turns out that m is the interesting evolution parameter. Indeed, if we assume orthogonality conditions (3.1.7) to hold then m solves the ODE

d dt m(z) = ż + iE(z)z = -A(z, η) -1 ⟨F, D z Q⟩ , m(z)| t=0 = z 0 , (3.5.8)
where for j, k ∈ 1, 2 we have

A(z, η) j,k = ⟨i D j Q, D k Q⟩ + ⟨iη, D j D k Q⟩ = j -k + O δ 2 .
(3.5.9)

The forcing term is

F = F (z, η) = N (Q + η) -|Q| 2 η -Q 2 η .
Remark 3.5.5. As a consequence of (3.5.4), the matrix A is invertible provided that δ is small enough.

Remark 3.5.6. If we drop numerical constants and complex conjugate in the notations, we can write

F = η 3 + Qη 2 .
In particular, we observe that orthogonality conditions (3.1.7) cancelled the terms which are linear in η. Since we do not control the L 1 -in-time norm of the radiation term, we couldn't have handled these linear terms. That's why the time dependent orthogonality conditions are crucial.

Remark 3.5.7. Since E(z) = E(|z|) = E(|m(z)|) we have that z = m m(z) . Hence, m is a bijective operator, so we can recover z from m(z).

To get the evolution equation for ν, we inject the ansatz (3.1.9) into (NLS) and we use the fact that Q is solution to (3.5.3). This yields that the radiation term η solves the equation

i∂ t η + (∆ -V )η = N (Q + η) -N (Q) -i D z Q( ż + iEz) .
By projecting the above equation on the continuous spectral subspace, and noting that P c (D z Q) = D z q, we get

i∂ t ν + (∆ -V )ν = P c N (Q + η) -N (Q) -i D z q ṁ(z) =: f + g , ν| t=0 = ν 0 , (3.5.10)
where we decomposed the forcing term into the sum of a nonlinear term f and a localized term g.

More precisely, we chose to collect in f the higher order nonlinear terms N (ν + εu ω ) as well as the modulation term:

f = P c N (ν + εu ω ) -i D z q ṁ(z) .
On the other hand, g contains the localized lower order terms which involve at least a power of Q or of P p (η):

g = P c Q 2 η + Qη 2 + | P p (η)| 2 P p (η) .
To get some global in time bounds on ν, we then follow the critical-weighted strategy detailed in Proposition 3.4.15, and we are reduced to control f in DU 2 H (H 1/2 ) and g in L 2 t (R; L 2,1/2+

x ). Hence, given an interval I ⊆ R where the solution is defined, we will control ν in the critical-weighted space

X (I) = V 2 H (I; H 1/2 x ) ∩ L 2 t (I; H 1,-1/2- x (R d )) , (3.5.11)
endowed with its natural norm, and ṁ in L 1 t (I). The aim is to get bounds that are independent of I.

Global-in-time a priori estimates on ν and ṁ In the following Lemma, we prove some preliminary estimates on the radiation term η that will be needed in the analysis. We recall that Ω ε,cδ 0 is the set defined in (3.3.8) made of randomized initial data which display some improved Strichartz estimates, with R = cδ 0 (see Proposition 3.5.1).

Lemma 3.5.8 (Preliminary estimates). Take ω ∈ Ω ε,R and let η, z, ν be as in ansatz (3.1.9). For any interval

I ⊆ [0 , ∞), ∥η∥ L 2 (I;L 2,-1/2- x ) ≲ ∥ν∥ X (I) + ε∥u 0 ∥ L 2 (R 3 ) , (3.5
.12)

∥η∥ L 4 t L 4
x (I×R 3 ) ≲ ∥ν∥ X (I) + R .

(3.5.13)

Proof. We can estimate the discrete part of η by its continuous part. Indeed, we recall that

| P p (η)| = |(R -I)(z)(ν + u ω )| ≲ |ϕ 0 ||(ν + u ω | D z Q)| .
(3.5.14) Hence,

∥ P p (η)∥ L 2 t (I;L 2,-1/2- x ) ≤ ∥⟨x⟩ -1/2-ϕ 0 ∥ L 2 x ∥⟨x⟩ 1/2+ D z Q∥ L ∞ x ∥ν + εu ω ∥ L 2 t (I;L 2,-1/2- x ) ≲ ∥⟨x⟩ D z Q∥ L ∞ t (I;H 2 ) ∥ν + εu ω ∥ L 2 t (I;L 2,-1/2- x ) ≲ ∥ν∥ X + ε∥u 0 ∥ L 2 x ,
where we used the local smoothing estimate (3.2.7) to control the weighted norm of the perturbed linear propagation u ω in the last inequality, as well as the fact that

∥u 0 ∥ L 2 x ∼ ∥u ω 0 ∥ L 2 x .
To prove (3.5.13) we apply Hölder in (3.5.14) and Sobolev embedding for ∥ϕ 0 ∥ L 4 (R 3 ) . It comes

∥(R -I)(ν + εu ω )∥ L 4 t L 4 x ≲ ∥ϕ 0 ∥ H 3/4 ∥ν + εu ω ∥ L 4 t L 4 x ∥ D z Q∥ L 4/3 x .
Applying Hölder once again, we get

∥ D z Q∥ L 4/3 x ≤ R 3 ⟨x⟩ -3+ dx 1/3 ∥⟨x⟩ 3/4+ D z Q∥ L 2 x ≲ ∥⟨x⟩ 2 D z Q∥ L 2 x .
Then, it follows from (3.5.6) that

∥(R -I)(ν + εu ω )∥ L 4 t L 4 x ≲ ∥⟨x⟩ 2 D z Q∥ L ∞ t L 2 x ∥ν + εu ω ∥ L 4 t L 4 x ≲ ∥ν∥ L 4 t L 4 x + ε∥u ω ∥ L 4 t L 4
x .

Note that we used the Sobolev weighted estimate (3.5.6) on D z Q. To control the deterministic term ν in L 4 t,x , we use Sobolev embedding, Strichartz estimate for the admissible pair (4, 3) and the transference principle in V 2

H . Since we chose ω ∈ Ω ε,R we can use the improved global-in-time Strichartz estimate (3.3.6) to control the random term u ω and obtain

∥ν + εu ω ∥ L 4 t L 4 x ≲ ∥⟨ √ H⟩ 1/4 ν∥ L 4 t L 3 x + R ≲ ∥⟨ √ H⟩ 1/4 ν∥ V 2 H + R ≲ ∥ν∥ X + R .
This gives both the desired estimate for the discrete part of η and its continuous part ν + εu ω , and finishes the proof of Lemma 3.5.8.

We can now state and prove the main global-in-time a priori estimate.

Proposition 3.5.9 (Global a priori estimates). There exists C = C(ϕ 0 ) such that for all interval I ⊆ R, for all ω ∈ Ω ε,R and

ψ 0 ∈ H 1/2 , if ψ = Q(z) + η = Q(z) + R(z) εu ω + ν , ψ |t=t 0 = εu ω 0 + ψ 0 , is solution to (NLS) on I with ∥ν 0 ∥ H 1/2 + |z 0 | ≤ δ 0 , ∥ν∥ X (I) ≤ δ ′ , sup t∈I |z| ≤ δ ′ , then 10 ∥ν∥ X (I) ≤ C ∥ν∥ 3 X (I) + R 3 + |z 0 | 3 + ∥ ṁ(z)∥ 3 L 1 (I) + ∥ν 0 ∥ H 1/2 , (3.5.15) ∥ ṁ(z)∥ L 1 (I) ≤ C ∥ν∥ 3 X (I) + R 3 + |z 0 | 3 + ∥ ṁ(z)∥ 3 L 1 (I) .
(3.5.16)

Proof of (3.5.15). It follows from weighted-critical estimate (3.4.24) that

∥ν∥ X ≲ ∥ν 0 ∥ H 1/2 + sup 2 N N 1/2 ∥ Π N f ∥ DU 2 H + ∥g∥ L 2 t (R;L 2,1/2+ x
) .

We recall that f = N (ν + u ω ) + i D z q ṁ(z) and we observe that Π N f ∈ L 1 (I, H 1/2 ) as explained in (3.4.12). Hence, we apply the probabilistic nonlinear estimate (3.4.15) on the continuous spectral subspace stated in section 3.4 to get that for all ω ∈ Ω ε,R

sup 2 N N 1/2 ∥ Π N N (ν + u ω )∥ DU 2 H ≲ ∥ν∥ 3 X + R 3 .
To handle the modulation term, we observe that P c (D z Q) = D z Q, and we deduce from the duality argument between U 2 and V 2 detailed in remark 3.4.13 that

∥ D z Q(z) ṁ(z)∥ DU 2 H ≤ sup ∥v∥ V 2 H ≤1 R (D z Q ṁ(z) | v)dt ≤ sup ∥v∥ V 2 H ≤1 ∥v∥ L ∞ t L 2 x ∥ D z Q(z)∥ L ∞ t L 2 x ∥ ṁ(z)∥ L 1 t = ∥ D z Q(z)∥ L 2 x ∥ ṁ(z)∥ L 1 t , where we used the embedding V 2 → L ∞ t (R; L 2 x ). Consequently, sup 2 N N 1/2 ∥ Π N D z Q(z) ṁ(z)∥ DU 2 H ≲ ∥ D z Q∥ H 1/2 ∥ ṁ(z)∥ L 1 t = O z 2 ∥ ṁ(z)∥ L 1 t .
Note that we used (3.5.4) to control ∥ D z Q∥ H 1/2 . Now, we need to estimate the localized lower order terms collected in g, defined in (3.5.10).

∥g∥ L 2 t (R;L 2,1/2+ x ) ≤ ∥ P c (Q 2 η)∥ L 2 t (R;L 2,1/2+ x ) + ∥ P c (Qη 2 )∥ L 2 t (R;L 2,1/2+ x ) + ∥| P p (η)| 2 P p (η)∥ L 2 t (R;L 2,1/2+ x ) .
It follows from (3.5.6) and (3.5.12) that

∥ P c (Q 2 η)∥ L 2 t (R;L 2,1/2+ x ) ≤ ∥⟨x⟩ 1/2+ Q∥ 2 L ∞ x ∥η∥ L 2 t (R;L 2,-1/2- x ) ≲ ∥z∥ 2 L ∞ t ∥ν∥ X + ε∥u 0 ∥ L 2 x .
Similarly, (3.5.6) and (3.5.13) yield

∥ P c (Qη 2 )∥ L 2 t (R;L 2,1/2+ x ) ≤ ∥⟨x⟩ 1/2+ Q∥ L ∞ x ∥η∥ 2 L 4 t L 4 x ≲ ∥z∥ L ∞ t ∥ν∥ X + R 2 ≲ ∥z∥ 3 L ∞ t + ∥ν∥ 3 X + R 3 .
To conclude we recall that

| P p (η)| ≲ |ϕ 0 || ⟨ν + u ω , D z Q⟩ | 3 ,
and therefore

∥| P p (η)| 2 P p (η)∥ L 2 t (R;L 2,1/2+ x ) ≲ ∥ϕ 0 ∥ L 2,1/2+ x R ⟨ν + u ω , D z Q⟩ 6 1/2 ≲ R ⟨ν + u ω , D z Q⟩ 2 1/2 sup R | ⟨ν + u ω , D z Q⟩ | 2 ≲ ∥⟨x⟩ D z Q∥ L 2 x ∥ν + u ω ∥ L 2 t (R;L 2,-1/2- x ) ∥ν + u ω ∥ L ∞ t L 2 x ∥ D z Q∥ L 2 x ≲ ∥ν∥ X + ε∥u 0 ∥ L 2 x 3 ≲ ∥ν∥ 3 X + ε 3 ∥u 0 ∥ 3 L 2
x . Proof of (3.5.16). We recall that

| ṁ(z)| ≲ | η 3 , D z Q | + | Qη 2 , D z Q | .
We apply Cauchy-Schwarz inequality firstly in space and then in time to get that

R | η 3 , D z Q |dt ≤ ∥⟨x⟩ 1/2+ D z Q∥ L ∞ x R ∥⟨x⟩ -1/2-η∥ L 2 x ∥η∥ 2 L 4 t,x dt ≲ ∥⟨x⟩ D z Q∥ H 2 ∥η∥ L 2 t (R;L 2,-1/2- x ) ∥η∥ 2 L 4 t,x
.

By using estimate (3.5.6) on the weighted norm of D z Q = Jϕ 0 + D z Q and estimates of Lemma 3.5.8 on η, we conclude that

R | η 3 , D z Q |dt ≲ ∥ν∥ X + ε∥u 0 ∥ L 2 ∥ν∥ X + R 2 ≲ ∥ν∥ 3 X + ε 3 ∥u 0 ∥ 3 L 2 + R 3 . Similarly, we have R | Qη 2 , D z Q |dt ≤ ∥⟨x⟩ 1/2+ Q∥ L ∞ ∥⟨x⟩ 1/2+ D z Q∥ L ∞ ∥⟨x⟩ -1 η∥ 2 L 2 t L 2 x ≲ ∥⟨x⟩Q∥ H 2 ∥⟨x⟩ D z Q∥ H 2 ∥η∥ 2 L 2 t (R;L 2,-1/2- x ) ≲ ∥z∥ L ∞ t ∥ν∥ 2 X + ε 2 ∥u 0 ∥ 2 L 2 ≲ ∥z∥ 3 L ∞ t + ∥ν∥ 3 X + ε 3 ∥u 0 ∥ 3 L 2 .
This concludes the proof of Proposition 3.5.9.

Proof of Theorem 3.1.1

Now that we have a local solution with some global a priori estimates both on the radiation term η and on the modulation parameter z, we shall be able to prove global well-posedness, and then deduce from these global estimates asymptotic stability.

Global existence and uniqueness It remains to finely tune the parameters that appear in Proposition 3.4.10. First, we take δ 0 such that the local existence result of Proposition 3.5.1 holds true, R = cδ 0 . Then, if the solution exists up to a certain time T we prove that

∥ν∥ X [0,T ) + ∥ ṁ(z)∥ L 1 [0,T ) ≤ δ ′ =⇒ ∥ν∥ X [0,T ) + ∥ ṁ(z)∥ L 1 [0,T ) ≤ δ ′ 2 .
(3.5.17) Indeed, we have from (3.5.15) and (3.5.16) that

∥ν∥ X [0,T ) ≤ C 2(δ ′ ) 3 + (cδ 0 ) 3 + δ 3 0 + δ 0 , ∥ ṁ(z)∥ L 1 [0,T ) ≤ C 2(δ ′ ) 3 + (cδ 0 ) 3 + δ 3 0 .
The a priori estimate (3.5.17) follows if we chose δ 0 even smaller, say such that δ 0 ≤ (4C) -1 δ ′ , and δ ′ ≲ c 3 C -1/2 . Then, we use a continuity argument (see Lemma 3.4.4) and prove that the solution is global, with the bounds ∥ν∥

X (R) + ∥ ṁ(z)∥ L 1 (R) ≤ δ ′ .
Since ṁ(z) lies in L 1 [0 , ∞) the convergence of the modulation parameter follows immediately. There exists z + ∈ C such that lim t→+∞ m(z)(t) = z + .

Note that |z| converges, and hence E(z) is also convergent.

Scattering We turn to the proof of the so-called completeness property of the flow, that is the fact that the radiation part η scatters. To this end, we first prove scattering for ν = P c (η) -εu ω by following the proof of the scattering from Theorem 3.1.5 we made in section 3.4. We recall that ν is solution to (3.5.10), and we prove that the Cauchy criterion is satisfied:

lim t 1 ,t 2 →+∞ ∥ e -it 1 (∆-V ) ν(t 1 ) -e -it 2 (∆-V ) ν(t 2 )∥ H 1/2 (R 3 ) = 0 .
Indeed, by the Duhamel integral formulation,

∥ e -it 1 (∆-V ) ν(t 1 ) -e -it 2 (∆-V ) ν(t 2 )∥ H 1/2 (R 3 ) ≤ ∥f + g∥ DV 2 (t 1 ,t 2 ) .
As observed in remark 3.4.11, it follows from the proof of Proposition 3.5.9 that there exists a constant C depending on some norms of the solution such that

∥f + g∥ DV 2 H (t 1 ,t 2 ) ≤ ∥f + g∥ DU 2 H (t 1 ,t 2 ) ≤ C ∥ν∥ X (t 1 ,+∞) , ∥ ṁ(z)∥ L 1 t (t 1 ,+∞) , ∥u ω ∥ L q t ((t 1 ,+∞);L r
x ) , (3.5.18) and C goes to zero when t 1 goes to +∞. Hence, the Cauchy criterion is satisfied and there exists

ν + ∈ H 1/2 ∩ Ran(P c ) such that lim t→+∞ ∥ν(t) -e -itH εu ω 0 + ν + ∥ H 1/2 (R 3 ) = 0 .
It remains to prove that the discrete part of η goes to zero when t goes to infinity. First, the weak convergence of e -itH εu ω

0 + ν ⇀ t→+∞ 0 in H 1/2 (R 3 )
follows from the Riemann-Lebesgue theorem and the Plancherel formula for the distorted Fourier transform. Then, we use the compactness of the operator R -I from H 1/2 (R 3 ) to C stated in Lemma 3.5.4 to get that

lim t→+∞ P p (η) = lim t→+∞ (R -I) e -itH εu ω 0 + ν = 0 in H 1/2 (R 3 ) .
This concludes the proof of Theorem 3.1.1 with η + = ν + .

Appendix. Weighted elliptic estimates for the ground states

We show how to prove estimates of Lemma 3.5.3 for the ground states Q(z) and its derivative. For the convenience of the reader we recall the equations satisfied by e, q, D e, D z q (see A.4 in [START_REF] Gustafson | Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves[END_REF]), and we omit numerical constants as well as complex conjugate from the notations. The nonlinear ground state Q(z) = zϕ 0 + q and E(z) = e 0 + e are solution to (3.1.1), and they satisfy

(H -e 0 )q = -P c N (Q) + eq , ez = (ϕ 0 | N (Q)) . (3.6.1)
We recall that N (u) = |u| 2 u, D z = (∂ z 1 , ∂ z 2 ) and we use the notation J = D z (z) = (1, i). Then, if we differentiate the above expressions with respect to z we obtain

(H -e 0 ) D z Q = -P c D z N (Q) + q D z e + e D z Q , z D z e + J e = (ϕ 0 | D z N (Q)) , (3.6.2) with D z N (Q) = Q 2 (J ϕ 0 + D z Q) = O z 2 in H 2 .
Proof of Lemma 3.5.3. We first prove the estimate for q in H 1 with weight ⟨x⟩. Given a Schwartz function φ, we have

(H -e 0 )⟨x⟩φ = [-∆, ⟨x⟩]φ + ⟨x⟩(H -e 0 )φ = - 2x • ∇φ ⟨x⟩ - 3 + 2|x| 2 ⟨x⟩ 3 φ + ⟨x⟩(H -e 0 )φ . (3.6.3)
Then, we multiply (3.6.3) by ⟨x⟩φ and we integrate over R 3 to obtain

∥∇(⟨x⟩φ)∥ 2 L 2 -e 0 ∥⟨x⟩φ∥ 2 L 2 = R 3 ⟨x⟩ 2 V (x)+3+ 3 + 2|x| 2 ⟨x⟩ 2 |φ(x)| 2 dx+(⟨x⟩ 2 (H -e 0 )φ | φ) . (3.6.4)
Since the first term on the right-hand side is equivalent to ∥φ∥ 2 L 2 , it suffices to handle the second term. Formally taking φ = q and using (3.6.1), we have

(⟨x⟩(H -e 0 )q | ⟨x⟩q) = (-P c N (Q) + eq | ⟨x⟩ 2 q) = -(N (Q) | ⟨x⟩ 2 q) + (ϕ 0 | N (Q))(ϕ 0 | ⟨x⟩ 2 q) + e∥⟨x⟩q∥ L 2 . Since q = O z 3 in H 2 (see (3.5.4)), we have |(N (Q) | ⟨x⟩ 2 q)| + |(ϕ 0 | N (Q))(ϕ 0 | ⟨x⟩ 2 q)| ≲ |z| 6 ∥⟨x⟩q∥ 2 L 2 + C(∥⟨x⟩ 2 ϕ 0 ∥ L ∞ ) .
Therefore, we deduce from (3.6.4) that

∥∇(⟨x⟩q)∥ 2 L 2 -(e 0 + e + C|z| 6 )∥⟨x⟩q∥ 2 L 2 ≲ ∥q∥ 2 L 2 + |z| 6 C(∥⟨x⟩ 2 ϕ 0 ∥ L ∞ ) .
To conclude we observe that there exists γ > 0 and c γ > 0 such that E(z) = e 0 + e(z) < -c δ for all |z| ≤ δ. Next, we just mention that the estimate for the weight ⟨x⟩ 2 follows similarly, using that

[-∆, ⟨x⟩ 2 ]φ = 4x • ∇φ + 6φ .
Let us now derive the estimate with the weight ⟨x⟩ in H 2 for D z q. To this end, we first differentiate (3.6.3) with respect to x k and we get

(H -e 0 )∂ k (⟨x⟩φ) + (∂ k V )⟨x⟩φ = ∂ k [-∆, ⟨x⟩] φ + ∂ k (⟨x⟩(H -e 0 )φ) .
In addition, we multiply by ∂ k (⟨x⟩φ) on both sides, and we integrate over R 3 to see that

∥∇(∂ k ⟨x⟩φ)∥ 2 L 2 -e 0 ∥∂ k ⟨x⟩φ∥ 2 L 2 = - R 3 V ∂ k (⟨x⟩φ) + ∂ k [-∆, ⟨x⟩] φ ∂ k (⟨x⟩φ)dx + (∂ k (⟨x⟩(H -e 0 ))φ | ∂ k (⟨x⟩φ)) . (3.6.5) Since ∂ k [-∆, ⟨x⟩] φ = -2 ∂ k φ ⟨x⟩ + x • ∂ k ∇φ ⟨x⟩ -x k x • ∇φ ⟨x⟩ 3 + (3 + 2|x| 2 ) 3x k φ ⟨x⟩ 5 - 4x k φ ⟨x⟩ 3 -(3 + 2|x| 2 ) ∂ k φ ⟨x⟩ 3 , we have that | R 3 V ∂ k (⟨x⟩φ) + ∂ k [-∆, ⟨x⟩] φ ∂ k (⟨x⟩φ)dx| ≲ ∥⟨x⟩φ∥ 2 H 1 + ∥φ∥ 2 H 2 .
Moreover, by using (3.5.4), (3.5.5), (3.6.2) and the estimates for q and D z q with weight ⟨x⟩ in H 1 already proved in the first part of the proof, we show that

|(∂ k (⟨x⟩(H -e 0 ))φ | ∂ k (⟨x⟩φ))| ≤ C(ϕ 0 )|z| 4 . Hence, ∥∇(∂ k ⟨x⟩φ)∥ 2 L 2 -(e 0 + e(z))∥∂ k ⟨x⟩φ∥ 2 L 2 ≲ C(ϕ 0 )|z| 4
. This concludes the proof since e 0 + e(z) ≤ -c γ < 0 for γ small enough and |z| ≤ γ. Résumé. Nous montrons le caractère localement bien posé en dessous de l'espace d'énergie pour l'équation de Schrödinger-demi-onde avec une non-linéarité cubique pour presque toutes les données initiales. Pour cette équation, qui est énergie sous-critique, la théorie de Cauchy dans l'espace d'énergie n'a pas encore été établie. En outre, nous montrons que le flot ne peut pas être régulier en dessous de l'espace d'énergie. Afin d'établir une théorie de Cauchy locale probabiliste, nous devons utiliser le calcul paracontrôlé en l'absence d'effet régularisants non-linéaires la deuxième itération de Picard. La preuve est une adaptation de la méthode de Bringmann [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] développée pour une équation d'ondes. Nous discutons par ailleurs du caractère mal-posé de cette équation à basse régularité.

Abstract. We obtain almost-sure local well-posedness below the energy space for the Schrödinger half-wave equation with a cubic nonlinearity. We need to use a refined probabilistic ansatz because of the lack of probabilistic smoothing in the second Picard's iteration, due to the high-low-low nonlinear interactions. The proof is an adaptation of the method of Bringmann [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF]. In addition, we discuss ill-posedness results for this equation. ] to evidence weak turbulence mechanisms. In the defocusing case (µ > 0), global existence and modified scattering is obtained for a class of sufficiently smooth and decaying small initial data on the wave guide R x × T y . In addition, the author shows that the limiting effective dynamics is governed by the Szegő equation on the torus. As a consequence of [GG12a; GG17], this analysis provides arbitrarily small initial data such that for every s > 1 2 and N ≥ 1, the solution u exhibits weak turbulence in the sense that lim sup

t→∞ ∥u(t)∥ L 2 x H s y log(t) N = ∞ , lim inf t→∞ ∥u(t)∥ L 2 x H s y < ∞ .
Subsequently, Bahri, Ibrahim and Kikuchi consider in [BIK20; BIK21] the focusing case µ < 0 and study traveling wave solutions on the wave guide R x × T y . They obtain orbital stability and transverse instability results, subject to the condition of a good Cauchy theory in the energy space. Unfortunately, such a Cauchy theory is yet to be addressed, and not much is known about the global existence of smooth solutions in Sobolev spaces.

The main concern in this paper is to address the local Cauchy problem at low regularity, for initial data in a statistical ensemble. Before presenting our results, let us recall that (NLS-HW) is a Hamiltonian system with a conserved energy

H(u) = 1 2 R 2 |∂ x u| 2 + |D y | 1 2 u 2 dx dy + µ 4 R 2 |u| 4 dx dy .
The L 2 mass is also formally conserved by the flow, and the solutions are left invariant by the scaling symmetry u → u λ (t, x, y) = λu(λ 2 t, λx, λ 2 y) .

(4.1.1)

In light of the conservation laws and of the scaling invariance, the relevant regularity spaces for this equation are the following anisotropic Sobolev spaces H s defined by

H s := L 2 x H s y ∩ H 2s x L 2 y , Ḣs := L 2 x Ḣs y ∩ Ḣ2s x L 2 y .
The scaling (4.1.1) leaves the Ḣ 1 4 -norm invariant. As a consequence, when 0 < s < 1 4 , a low-to-high frequency cascade occurs and there is a short-time inflation of the Ḣs -norm for the regularized solutions [START_REF] Kato | Ill-posedness for the Half wave Schrödinger equation[END_REF]. We make precise the type of ill-posedness by describing a pathological set in Appendix 4.8. When 1 4 < s, the equation is scaling-subcritical. Yet, semilinear local well-posedness is only known in H s when 1 2 < s. Specifically, semilinear well-posedness is obtained in [START_REF] Bahri | Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations[END_REF] from Strichartz estimates with a derivative loss. Furthermore, we prove in Appendix 4.8 that the flow map cannot be of class C 3 when 1 4 < s < 1 2 . To do so, we consider a one-parameter family of traveling wave profiles for the one-dimensional Szegő equation to invalidate some relevant Strichartz estimates. It is therefore not possible to run a contraction mapping argument when 1 4 < s < 1 2 , since otherwise the flow-map would be analytical. The state-of-the-art Cauchy theory results for (NLS-HW) are summarized in the following diagram: Main result As mentioned above, it turns out to be quite challenging to prove semilinear local well-posedness in the energy space, or to run a quasi linear scheme in H s when 1 4 < s < 1 2 . Instead, our goal is to study the probabilistic local well-posedness of equation (NLS-HW) in this range of regularities. Given f 0 ∈ H s (R 2 ) and a sequence of independent normalized Gaussian variables (g k (ω)) k∈Z on a probability space (Ω, A, P), we define a random variable

Norm-inflation [Kat21] Flow map is not C 3 [BGT05a] Local well-posedness [BIK20] H s I L 2 I H 1 4 I H 1 2
ω ∈ Ω → f ω 0 := k∈Z g k (ω)P 1,k f 0 ,
where P 1,k is the partial Fourier projector (in the y-variable) on an interval of unit length centered around k ∈ Z. We refer to (4.2.1) for the precise definition of the relevant anisotropic Wiener randomization procedure we employ, and to Figure 4.2. The considered probability measure on H s is the induced measure by the above random variable. However, due to the lack of dispersion and to the absence of regularizing features (such as bilinear estimates or local energy decay), Picard's iterations have the same regularity as the initial data and the standard probabilistic method, which is discussed below, fails. Instead, we need to consider a refined probabilistic ansatz adapted from the probabilistic quasilinear scheme developed by Bringmann [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] to prove probabilistic well-posedness for a derivative wave equation. For T > 0 and some s and σ in R, we denote

X s,σ T 0 := C t [-T 0 , T 0 ] ; H s (R 2 ) ∩ L 8 t [-T 0 , T 0 ] ; L 4 x W σ,∞ y (R 2 ) . (4.1.2)
We also denote the truncated initial data as

P ≤n f ω 0 := |k|≤n g k (ω)P 1,k f 0 .
Our main result reads as follows.

Theorem 4.1.1 (Probabilistic local well-posedness). Let s ∈ (13/28, 1/2], some suitable 0 < σ < s and f 0 ∈ H s . There exist T 0 > 0 and a sequence

(u n ) n≥1 ∈ C([-T 0 , T 0 ] ; H ∞ ) N converging in expectation to a limiting object denoted u lim n→∞ E ∥u n -u∥ 2 X s,σ T 0 = 0 , (4.1.3)
in such a way that, almost-surely in ω ∈ Ω, there exists T ω > 0 such that for all n ∈ N, u n and u exist in C([-T ω , T ω ]; H s ) and solve (NLS-HW) with initial data P ≤n f ω 0 and f ω 0 , respectively. Remark 4.1.2 (Time of existence). Theorem 1.1 provides a uniform time of existence T ω > 0 for the smooth solutions u n initiated from regularized initial data (P ≤n f ω 0 ) n in the statistical ensemble. This is not a consequence of the known local well-posedness result from [START_REF] Bahri | Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations[END_REF]. Then, the main claim of Theorem 1.1 is the convergence of (u n ) n to a strong solution u in H s , on the time interval [-T ω , T ω ]. Remark 4.1.3 (Regularizing sequence). Note that the regularizing sequence is general and not dyadic. Actually, we first prove the result for dyadic frequencies N and deduce the convergence for general approximations by adapting an argument from [START_REF] Sun | Refined probabilistic global well-posedness for the weakly dispersive NLS[END_REF]. Remark 4.1.4 (Other models). Theorem 1.1 is an adaptation to Schrödinger-type equations of the quasilinear scheme from [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], and allows addressing in the same way other weakly dispersive models. For instance, the case of the one-dimensional half-wave equation and of the Szegő equation are both contained in our analysis by simply forgetting about the variable x everywhere. Remark 4.1.5 (Nonlinearity). We decided to only consider the cubic nonlinearity on R 2 for simplicity. But up to some technicalities, other nonlinearities p ≥ 5 and other geometries (e.g. on the wave guide R × T) would work. Remark 4.1.6 (Sobolev threshold). The threshold s > s 0 with s 0 = 13/28 is a convenient fractional approximation of the exponent s 0 we will actually get in (4.6.1). This is by no means optimal, and it would be interesting to go all the way down to s 0 = 1/4 to cover the whole quasilinear range of exponents.

Background This work is part of the study of nonlinear weakly dispersive equations, in the presence of randomness.

Weakly dispersive equations. There has been a rich activity on the qualitative behavior of solutions to nonlinear evolution equations with weak dispersion. Examples are the half-wave equation [START_REF] Pocovnicu | Traveling waves for the cubic Szegő equation on the real line[END_REF][START_REF] Gérard | A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line[END_REF] and other fractional Schrödinger equations [FL13; GPH15; ST21], the Szegő equation [GG10; GG12b], the Schrödinger equation on compact manifold [START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schröinger equation on surfaces[END_REF], the Schrödinger equation on the Heisenberg group [START_REF] Gérard | Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem[END_REF] for which there is a lack of dispersion in one direction, the Kadomtsev-Petviashvili equations [START_REF] Hadac | Well-posedness and scattering for the KP-II equation in a critical space[END_REF] and many other models. It is in general quite challenging to solve the Cauchy problem for these equations at low regularity. The reason being that in such contexts, dispersive estimates come with a derivative loss. In some special cases where the Hamiltonian equation is integrable in the sense that it has a Lax pair structure providing infinite conservation laws, methods from integrable systems are substituted for the analytical perturbative approach, to study both Cauchy theory and the long-time dynamics.

A first illustrative example one can think of is the Szegő equation, which is known to be globally well-posed in H 1 2 + (T) since [START_REF] Gérard | The cubic Szegő equation[END_REF]. Using the Lax pair structure, the flow map was extended to BMO in [START_REF] Gérard | The cubic Szegő flow at low regularity[END_REF], then the Cauchy problem was recently shown to be globally well-posed even on L 2 + (T) in [GP22] also by using integrable techniques. Note that the cubic Szegő equation is invariant by a

L 4 + (T) → H 1 4
+ (T)-critical scaling. Another model is the derivative Schrödinger equation on the line, which is L 2 -critical with respect to the scaling, but the flow map fails to be uniformly continuous in H s (R) when s < 1 2 [BL01; Tak01]. Nevertheless, [START_REF] Harrop-Griffiths | Global well-posedness for the derivative nonlinear Schröodinger equation in L 2 (R)[END_REF] recently proved from integrable techniques that the equation is globally well-posed in L 2 (R). Previously, integrable techniques were combined with concentration-compactness argument in [START_REF] Bahouri | Global well-posedness for the derivative nonlinear Schrödinger equation[END_REF] to prove global well-posedness in H 1 2 (R). A third model is the Schrödinger equation on the Heisenberg group, for which local well-posedness is only known to hold in adapted Sobolev spaces H s (H 1 ) for s > 2, but is H 1 2 scaling-invariant, moreover, the flow map is known not to be of class C 3 for 1 2 < s < 2. The lack of dispersion comes from a special direction, the vertical direction, where the equation is a transport equation rather than a dispersive one.

For the Grushin Laplacian (a simplified version of the Heisenberg Laplacian), a probabilistic approach was attempted in [START_REF] Gassot | Probabilistic local well-posedness for the Schrödinger equation posed for the Grushin Laplacian[END_REF], proving almost-sure local well-posedness according to a specific randomization procedure which penalizes the direction where there is no dispersion, and gain derivatives in L p spaces. Finally, we stress out that it is sometimes possible to close the derivative gap (namely 1/4 < s < 1/2 in our situation) deterministically by considering more general function spaces than the Sobolev spaces, such as Fourier-Lebesgue spaces, see e.g. [START_REF] Grünrock | Bi-and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS[END_REF] for the derivative NLS equation, and [START_REF] Grünrock | On the wave equation with quadratic nonlinearities in three space dimensions[END_REF] for the wave equation.

Probabilistic Cauchy theory and its limitations. The so-called probabilistic Cauchy theory was initiated by Bourgain [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] and Burq,Tzvetkov [BT08a;[START_REF] Burq | Random data Cauchy theory for supercritical wave equations II: a global existence result[END_REF], and can be sketched as follows. When s < s c is below a critical threshold s c so that instabilities are known to occur, a general initial data in H s may have better integrability properties than expected by the Sobolev embedding. In contexts when the dispersion is strong enough to give some local energy decay or bilinear estimates, or for wave-type equations where the Duhamel formula gains one derivative, one can exploit such an enhanced integrability property to prove that the Picard's iterations are smoother than the linear evolution of the initial data. A rigorous analysis of this observation makes it possible to evidence strong solutions initiated from statistical initial data, according to a non-degenerate probability measure which charges any open set in H s .

A randomized initial data is typically defined from a given function ϕ ∈ H s and its unit-scale frequency decomposition (ϕ ω n ) n in the frequency space (or according to a spectral resolution of the Laplace operator in compact settings), where each mode is decoupled by a sequence (g n (ω)) n of normalized independent Gaussian variables, defined on a probability space (Ω, F, P):

ω ∈ Ω → ϕ ω ∼ n g n (ω)ϕ n , where ϕ ∼ n ϕ n ∈ H s .
Then, for many initial data ϕ ω in a statistical ensemble Σ ⊂ H s which has full measure, one expects to observe a nonlinear smoothing effect for the recentered solution around the linear evolution thanks to the combination of space-time oscillations (dispersion) and probabilistic oscillations (randomization). Namely, the goal is to show that for some ν > s c ,

v(t) := u(t) -e it∆ ϕ ω ∈ C([-T , T ]; H ν ) for all ϕ ω ∈ Σ .
The above recentered solution v is obtained from a fixed point argument at subcritical regularities in H ν and solves the original equation perturbed by stochastic terms stemming from the linear evolution e it∆ ϕ ω . We refer to [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF][START_REF] Burq | Random data Cauchy theory for supercritical wave equations II: a global existence result[END_REF] for an introduction to the probabilistic Cauchy theory for dispersive equations.

Nevertheless, to observe the aforementioned probabilistic smoothing effect, we need to exploit dispersive properties of the equations to gain decay without trading regularity. In equation (NLS-HW), however, there is no dispersion in the y-direction. Therefore, in the low x-frequency regimes, the Strichartz estimates come with a derivative loss, so that we have neither usable bilinear estimate, nor local smoothing estimates at our disposal. Besides, the second Picard iteration of the randomized initial data does not have a better regularity than the initial data. To see this, we consider highlow-low type interactions in high y-frequencies |η| ≫ 1, for an initial data ϕ ω projected at low x-frequencies |ξ| ≲ 1. For simplicity, we assume that there is no dependence in the variable x, then the high-low-low interactions take on the form t 0 e -i(t-τ )|Dy| P |η|≫1 e iτ |Dy| ϕ ω P |η|≤1 e iτ |Dy| ϕ ω P |η|≤1 e iτ |Dy| ϕ ω dτ.

Such interactions are basically transported by the half-wave linear flow, so that every derivative of the second Picard's iteration can fall onto the first linear term. Hence, one can only handle s derivatives instead of the desired ν derivatives. Similarly, Oh [START_REF] Oh | Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation[END_REF] considered the Szegő equation on the circle and proved that the first nontrivial Picard's iterate does not gain regularity compared to the initial data.

To undertake this type of issues, Bringmann [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] developed a refined probabilistic ansatz in a quasilinear setting thanks to paracontrolled calculus. In the present work, we adapt the strategy from [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] to prove almost-sure local well-posedness below the energy space, in the quasilinear regime. We conclude this paragraph by mentioning that the paracontrolled approach was further developed in a spectacular way in a series of papers of Deng, Nahmod and Yue [DNY19; DNY21] and Sun, Tzvetkov [START_REF] Sun | Refined probabilistic global well-posedness for the weakly dispersive NLS[END_REF] bringing tools from random matrix theory and introducing powerful methods such as the random averaging operators and the random tensors. More recently, the paracontrolled approach has taken a step forward, and were successfully implied in the resolution by Bringmann, Deng, Nahmod and Yue [START_REF] Bringmann | Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation[END_REF] of the ϕ 3 4 problem for NLW.

Consequences of Theorem 4.1.1 and perspectives -From the perspective of probabilistic Cauchy theory, equation (NLS-HW) is the first dispersionless Schrödinger-type equation for which we prove probabilistic well-posedness in a quasilinear regime, where the second Picard's iteration does not gain any regularity. Indeed, we provide a measure induced by a mild unit-scale and one-directional randomization procedure in part 4.2.1. In contrast with [START_REF] Gassot | Probabilistic local well-posedness for the Schrödinger equation posed for the Grushin Laplacian[END_REF] on the Schrödinger equation on the Heisenberg group, the randomization does not gain any regularity in L p -spaces and does not penalize the dispersionless direction. We believe that the present approach would also be successful in this context.

-From the deterministic perspective, Theorem 4.1.1 yields a statistical ensemble Σ (a dense fullmeasure set) of initial data leading to strong solutions, in regimes where we show that the equation is semi-linearly ill-posed. The theorem also provides strong solutions in the energy space, which is a first modest step in the comprehension of the long-time Cauchy theory of (NLS-HW), motivated by the rich possible asymptotic behaviors of solutions studied in [START_REF] Xu | Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation[END_REF] in the defocusing case and [BIK21; BIK20] in the focusing case. In contrast to the half-wave equation or the Szegő equation on the line, local existence in the energy space H 1 2 does not follow from a Yudovich argument since the L q -norms are not controlled by the energy when q > 6. We cannot use the conservation of the energy together with a Brezis-Gallouët estimate either in oder to show that in the defocusing case, smooth solutions extend globally in time, since H s is not an algebra when s < 3 4 , and there is no conservation law that controls the H 3 4 -norm.

-In addition, equation (NLS-HW) can serve as a model to study the long-time behavior of probabilistic solutions generated by the paracontrolled decomposition, in the absence of an invariant measure. Indeed, we construct a probabilistic solution in the presence of a conserved energy which is not enough to globalize the solutions since probabilistic information is crucial in the iteration scheme. However, to understand how this information is transported by the flow and to prove quasi-invariance of the measure, one needs dispersion ( [OST18]). Hence, understanding the long-time behavior of the solutions provided by Theorem 4.1.1 requires the combination of energy methods with some measure-theory argument. This is a challenging but interesting problem.

Strategy of the proof

The proof amounts to show the convergence of (u n ) n∈N (whose existence is ensured by Theorem 1.6 in [START_REF] Bahri | Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations[END_REF] recalled in Proposition 4.2.2), on a fixed time interval. First, we establish the convergence along the subsequence (u N ) N ∈2 N by adapting the iterative scheme from [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], which is made up of an induction on frequencies. At every step N of the induction, we consider separately the problematic high-low-low frequency interactions, absorbed in the adapted linear evolution F N studied in Section 4.4 and solution to

i∂ t F N + (∂ 2 xx -|D y |)F N = N (F N , P ≤N γ u N 2 , P ≤N γ u N 2 ) . F N (0) = P N f ω 0 ,
We get probabilistic Strichartz estimates for small times, controlling the L ∞ norm of F N with a loss N γ 2 -σ instead of N 1 2 . To achieve such a goal, we exploit the probabilistic structure of the initial data f ω 0 , which the sum of terms frequency localized in unit-scale intervals and decoupled by independent Gaussian variables. A key ingredient is the independence between the approximate solution u N 2 , which is constructed from the low modes P ≤ N 2 f ω 0 of the initial data, and P N f ω 0 . As for the remainder w N , it solves the equation with zero initial data and stochastic forcing terms but without the singular interaction

N s (F N , u N 2 , u N 2 ) := N (F N , P ≤N γ u N 2 , P ≤N γ u N 2 ) .
In order to get convergence on a fixed time interval, not depending on the iteration step N , we follow [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] by making use of the truncation method from De Bouard and Debussche [START_REF] Bouard | A stochastic nonlinear Schrödinger equation with multiplicative noise[END_REF] sketched in Section 4.3.3. We stress out that we only do this in the half-wave variable y, whereas a deterministic analysis is performed in the Schrödinger variable x. Namely, we exploit the dispersion materialized by the Strichartz estimates and use mixed Lebesgue spaces and T T * -type argument instead of Gronwall inequalities and energy estimates obtained in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF].

Outline of the paper The paper is organized as follows. We first introduce the unit-scale one-directional randomization of the initial in Section 4.2, and prove refined Strichartz estimates for frequency localized functions. Then, we define in Section 4.3 the dyadic subsequence of smooth approximate solutions u N and set up the truncation method. The adapted linear evolution is handled in Section 4.4, and the a priori bounds for the nonlinear remainders are established in Section 4.5. These bounds rely on paracontrolled trilinear estimates that are detailed in Section 4.7. Since the nonlinearity is cubic, we have to control more various types of interactions than in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] where the nonlinearity is quadratic. Finally, in Section 4.6, we show convergence of the approximating sequence (u N ) N to a limit u on a fixed interval, and we prove that it solves the equation (NLS-HW) on a smaller time interval, which is almost-surely nonempty. Then we deduce the convergence of the whole sequence (u n ) n . We conclude by detailing some ill-posedness results for equation (NLS-HW) in Appendix 4.8.

Notation We denote the linear operator A = ∂ 2 xx -|D y |, and the interactions stemming from the cubic nonlinear term by

N (u) = |u| 2 u , N (u 1 , u 2 , u 3 ) = u 1 u 2 u 3 + u 1 u 2 u 3 + u 1 u 2 u 3 .
We write F or F y→η the partial Fourier transform with respect to the half-wave direction y, by η = y the corresponding Fourier frequency, and by F -1 or F -1 η→y the inverse Fourier transform. For fixed α ∈ R, a parameter β satisfies the relation β = α -0 if there exists ε > 0 such that β = α -ε, where ε > 0 can be chosen arbitrarily close to zero independently of the parameters.

Given two Banach spaces E, F , we denote B(E → F ) the set of bounded operators from E to F .

Dispersive estimates with a derivative loss

We say that the pair (p, q), with 2 ≤ p, q ≤ ∞, is admissible if it is a Schrödinger-admissible pair in dimension 1, i.e.

T E = e itA F -1 η→y (m(η)F y→η ) ∈ B(L 2 x,y (R 2 ) → L 2 x,y (R 2 )) , t ∈ R .
Given (p, q) and ( p, q) two admissible pairs as in (4.2.2), and r ∈ [2, ∞], we have the following Strichartz estimates with a derivative loss:

• Bernstein-Strichartz estimate ∥T E f 0 ∥ L p t L q x L r y ≲ |E| 1 2 -1 r ∥f 0 ∥ L 2 x,y , (4.2.3)
• T * -estimate ∥ R e -iτ A F -1 η→y (m(η)F y→η )f 0 (τ ) dτ ∥ L 2 x,y ≲ |E| 1 2 -1 r ∥f 0 ∥ L p ′ t L q ′ x L r ′ y , • T T * -estimate ∥ R 1 0≤τ ≤t e i(t-τ )A F -1 η→y (m(η)F y→η )f 0 (τ ) dτ ∥ L p t L q x L r y ≲ |E| 1 2 -1 r ∥f 0 ∥ L p ′ t L q ′ x L 2 y . ( 4

.2.4)

Notice that when r = 2 there is no loss in the Bernstein estimate and we can choose E = R.

Proof. In the mixed Lebesgue norms, we first use the unit-scale Bernstein estimate in the y-direction to pass from L r y to L 2 y , and then the unitarity of e it|Dy| on L 2 y to reduce the matter to the free Schrödinger evolution on the line. There holds

∥T E f 0 ∥ L p t L q x L r y ≲ |E| 1 2 -1 r ∥ e itA f 0 ∥ L p t L q x L 2 y = |E| 1 2 -1 r ∥∥ e it∂ 2 xx f 0 ∥ L 2 y ∥ L p t L q x .
Subsequently, we use the Minkowski inequality and the Strichartz estimate for e it∂ 2 xx in the x-direction to conclude that

|E| 1 2 -1 r ∥∥ e it∂ 2 xx f 0 ∥ L 2 y ∥ L p t L q x ≲ |E| 1 2 -1 r ∥∥ e it∂ 2 xx f 0 ∥ L p t L q x ∥ L 2 y ≲ |E| 1 2 -1 r ∥f 0 ∥ L 2 x,y .
The T * -estimate follows from duality, and the T T * -estimate is a consequence of the first two estimates, by using the T * estimate with r = 2. We also need an application of the standard Christ-Kiselev argument to localize in time 0 ≤ τ ≤ t.

• The nonlinear remainder w N is solution to (NLS-HW) with a stochastic forcing term and zero initial condition:

(i∂ t + A)w N = N u N -N u N 2 -N F N , P ≤N γ u N 2 , P ≤N γ u N 2 , w N (0) = 0 . (4.3.1)
The key point is that the singular interactions between the high frequencies of the initial data and the low frequencies of the approximate solution u N/2 are removed from the equation satisfied by w N . Therefore, we can expect that a smoothing effect now occurs on w N . Note that at each step we can decompose the solution as a series

u N = u N 2 + F N + w N = u N 0 + N L=2N 0 +1 (w L + F ω L ) . (4.3.2)
The goal is to prove that the series with general term (w N ) N ≥N 0 almost-surely converges in a subcritical space C([-T 0 , T 0 ]; H ν (R 2 )), for some ν > 1 2 and some 0 < T 0 ≪ 1, and the series with general term (F N ) N ≥N 0 converges almost-surely in C([-T 0 , T 0 ]; H s (R 2 )). In addition, we prove that there exists a time T ω > 0 such that the limit of (u N ) solves (NLS-HW) in C [-T ω , T ω ]; H s (R 2 ) .

Functional spaces

In the nonlinear analysis, we need to keep track of the frequency localization of the adapted linear evolution F N . Specifically, in order to avoid the high-low-low interactions in the nonlinear analysis that are responsible for the lack of nonlinear smoothing, we want to ensure that for a given N , the y-frequencies of the functions w N and F N are localized at size |η| ∼ N . To capture such a frequency localization, rigorously stated in Lemmas 4.4.1 and 4.4.2, we fix a parameter D > 0 and some weights

C N,D (M ) := max N M , M N D , C ≤N,D (M ) := max 1, M N D .
Then we set the weighted norms

∥u∥ X N,D [-T ,T ] := M ≥1 C N,D (M )∥P M u∥ L ∞ t ([-T ,T ],L 2 x,y (R 2 )) , (4.3.3) ∥u∥ X ≤N,D [-T ,T ] := M ≥1 C ≤N,D (M )∥P M u∥ L ∞ t ([-T ,T ],L 2
x,y (R 2 )) .

For some r ∈ [2 , ∞], we define as well some frequency-localized mixed Strichartz norms

∥u∥ S r N,D [-T ,T ] := M ≥1 C N,D (M )∥P M u∥ L 8 t ([-T ,T ],L 4 x L r y (R 2 )) , ∥u∥ S r ≤N,D [-T ,T ] := M ≥1 C ≤N,D (M )∥P M u∥ L 8 t ([-T ,T ],L 4 x L r y (R 2 )) .
When r = ∞ we simply write

S N,D := S ∞ N,D , S ≤N,D := S ∞ ≤N,D .
The frequency localization of F ω N around frequencies N stems from the structure of F N . Specifically, each F N is a superposition of functions (F N,k ) N 2 ≤|k|≤N , decoupled by independent Gaussian variables. We observe that F N,k is solution to a linear Schrödinger equation with a potential truncated at frequencies of size ≲ N γ , with initial data P 1,k f 0 for some |k| ∼ N . Hence, at least in short-time, we expect F N,k to be localized at a distance ≲ N γ from the frequency k. In particular,

| supp F(F N,k )| ≲ N γ a.s. (4.3.4)
This key frequency localization would reduce the derivative loss in the Strichartz estimates from Lemma 4.2.1 for F N,k , and therefore for F N by using probabilistic decoupling. To encapsulate (4.3.4), rigorously stated in Lemma 4.4.1, we follow [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] and we consider the recentered Besov spaces

∥f ∥ B ρ,γ k,D [-T ,T ] := M ≥1 c ρ,γ k,D (M )∥P M,k f ∥ L ∞ t ([-T ,T ];L 2 x,y (R 2 )) , c ρ,γ k,D (M ) := M ρ max 1, M N γ D .
(4.3.5) The weight M ρ accounts for an extra gain of derivative ρ = σ -0 on F N,k that comes from the frequency localization of P M,k F N,k . In Section 4.6, we implement a fixed point argument on the nonlinear term w in the space associated with the norm

∥w∥ Y ν N = ∥⟨D y ⟩ ν w∥ X N,α + ∥⟨D y ⟩ ν w∥ X ≤N,D + ∥⟨D y ⟩ σ w∥ S N,α + ∥⟨D y ⟩ σ w∥ S ≤N,D ,
where α, D are parameters, ν > 1 2 is the regularity of the remainder w N determined by 0 < σ = ν -1 2 -0 from the Sobolev embedding H ν y → W σ,∞ y .

The recurrence and the truncation method from de Bouard and Debussche

In contrast with the standard probabilistic well-posedness argument implemented in [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF], which consists in a single fixed point argument for a fixed ω ∈ Ω, we cannot solve the equation pathwise.

The reason is that we need to perform a fixed point argument at each step of the iteration scheme, leading to a sequence of times (T N (ω)) N that would depend intricately on ω. In particular, we cannot ensure that this sequence is uniformly bounded from below in n. Instead, we do have to work in a space L ρ (Ω, E) for some ρ ≥ 1, where E is the space-time functional space in which we establish local existence.

To handle a similar problem arising in the study of a Schrödinger equation with a multiplicative white noise [BD99], de Bouard and Debussche truncated the nonlinearity. This allows to perform a fixed-point argument in some space L ρ (Ω, E) despite the absence of gain of integrability in the ω-variable. We refer to [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] for some discussion on motivation for the truncation method of de Bouard and Debussche. The method consists in fixing a smooth function θ ∈ C ∞ c (R, [-1, 1]) compactly supported in (-2, 2) and such that θ ≡ 1 on [-1, 1], and to essentially reduce the matter to a linear problem by assuming most of the terms to be at most 1 in the nonlinearity. Let F N,θ and

The adapted linear evolution

From now on, and until Section 4.6, we often drop the index θ for simplicity in the notation. At each step n, the adapted linear evolution is decomposed as (4.3.9)

F N = N 2 ≤|k|<N g k (ω)F N,k ,
where each unit block F N,k is the solution to the linear equation

(i∂ t + A)F N,k = θ 2 F,w;≤ N 2 N (F N,k , P ≤N γ u N 2 , P ≤N γ u N 2 ) , F N,k (0) = P 1,k f 0 .
We denote the potential by ϕ := θ F,w;≤ N 2 P ≤N γ u N 2 , and we use the following features of ϕ in this section:

• ϕ is localized at low frequencies ≲ N γ ;

• ϕ satisfies the following a priori estimate, encoded in the truncation defined in (4.3.6)

M ∥⟨D y ⟩ σ ϕ∥ S M,D [-τ ,τ ] ≤ 1 , τ ∈ R ; • ϕ is measurable in the σ-algebra B < N 2 generated by the random variables (g k ) |k|< N 2
. Therefore, ϕ is independent of the Gaussian variables (g k ) N 2 ≤|k|<N .

Almost frequency localization

In what follows, we fix n ≥ 1, N = 2 n , k ∈ Z with N 2 ≤ |k| < N , and the potential ϕ = θ F,w;≤ N 2 P ≤N γ u N 2 is as above. First, we prove the key frequency localization of F N,k at distance ≲ N γ from k, with a gain of ρ = σ -0 derivatives captured by the recentered Besov spaces B ρ,γ k,D defined in (4.3.5). This stems from the localization of the initial condition P 1,k f 0 in a unit-scale interval around k, together with the localization of the potential ϕ.

Lemma 4.4.1 (Frequency localization of the individual blocks). Let D ′′ ≫ 1. Then, for 0 ≤ ρ < σ arbitrarily close to σ, there exist 0 < T 0 and 0 < C such that for every N and N

2 ≤ |k| < N the solution F N,k to (4.3.8) satisfies ∥F N,k ∥ B ρ,γ k,D ′′ [-T 0 ,T 0 ] ≤ C∥P 1,k f 0 ∥ L 2 x,y . (4.4.1) As a consequence, ∥1 N 2 ≤|k|<N F N,k ∥ ℓ 2 k (Z;B ρ,γ k,D ′′ [-T 0 ,T 0 ]) ≤ C∥ P N f 0 ∥ L 2 x,y . (4.4.2)
The proof follows the same lines as the proof of Proposition 4.1 in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF]. Nevertheless, we use the Duhamel formula and the Strichartz estimate in place of a Gronwall argument in order to avoid using the quantity L ∞

x,y (R 2 ) that we do not control.

Proof. The estimate (4.4.2) follows from (4.4.1) by using that

∥1 N 2 ≤|k|<N P 1,k f 0 ∥ ℓ 2 k (Z;L 2 x,y (R 2 )) ≲ ∥ P N f 0 ∥ L 2 x,y .
We prove (4.4.1): given 0 < T 0 ≤ 1, we show that for all D ′′ > 0, there exists C > 0 such that for all k ∈ Z and 0 ≤ T ≤ T 0 ,

∥F N,k ∥ B ρ,γ k,D ′′ [-T ,T ] ≤ ∥P 1,k f 0 ∥ L 2 x,y + CT 1 2 ∥F N,k ∥ B ρ,γ k,D ′′ [-T ,T ] ∥⟨D y ⟩ σ ϕ∥ 2 L 8 t L 4 x L ∞ y . (4.4.3)
We then use the truncation θ to see that the norms of the potential is less than one, and we chose

T 0 > 0 (independent of N ) such that CT 1 2 0 ≤ 1 2 . Let us establish (4.4.3) for fixed k ∈ Z. First, observe that ∥P 1,k f 0 ∥ B ρ,γ k,D ′′ [-T,T ] ≲ ∥P 1,k f 0 ∥ L 2
x,y . Next, it follows from the definition of the Besov norm (4.3.5) and from the Duhamel formula that we need to estimate

M c ρ,γ k,D ′′ (M )∥ t 0 e i(t-τ )A P M,k N (F N,k , ϕ, ϕ) (τ ) dτ ∥ L 2 x,y .
We choose the Strichartz admissible pairs (p, q) = (∞, 2), ( p, q) = (4, ∞) and r = 2, then we apply the T T * -estimate from Lemma 4.2.1 (recall that when r = 2 there is no need to apply the Bernstein estimate, and there is no derivative loss). For fixed M , we have

∥ t 0 e i(t-τ )A P M,k N (F N,k , ϕ, ϕ) (τ ) dτ ∥ L ∞ t L 2 x,y ≲ ∥P M,k N (F N,k , ϕ, ϕ) ∥ L 4 3 t L 1 x L 2 y .
We do a Littlewood-Paley decomposition of F N,k (t) with frequencies K centered around k, and a standard Littlewood-Paley decomposition of ϕ with frequencies L 1 , L 2 . Without loss of generality, we assume that L 2 ≤ L 1 . There holds

∥P M,k N (F N,k , ϕ, ϕ) ∥ L 4 3 t L 1 x L 2 y ≲ K L 2 ≤L 1 ≲N γ ∥P M,k N (P K,k F N,k , P L 1 ϕ, P L 2 ϕ) ∥ L 4 3 t L 1 x L 2 y .
Then, the interactions that contribute to the above sum are the following:

• K ∼ M , • K ≪ M and L 1 ∼ M , • K ≫ M and L 1 ∼ K, Hence, ∥P M,k N (F N,k , ϕ, ϕ) ∥ L 4 3 t L 1 x L 2 y ≲ T 1 2 sup L L σ ∥P L ϕ∥ L 8 t L 4 x L ∞ y 2 K∼M ∥P K,k F N,k ∥ L ∞ t L 2 x,y + K: K≪M ∼L 1 L -σ 1 ∥P K,k F N,k ∥ L ∞ t L 2 x,y + K: K∼L 1 ≫M L -σ 1 ∥P K,k F N,k ∥ L ∞ t L 2 x,y .
It follows from the definition (4.3.5) of weights c ρ,γ k,D ′′ (K) and from the truncation (4.3.8) that sup

L L σ ∥P L ϕ∥ L 8 t L 4 x L ∞ y ≲ 1 . Therefore, ∥P M,k N (F N,k , ϕ, ϕ) ∥ L 4 3 t L 1 x L 2 y ≲ T 1 2 K∼M ∥P K,k F N,k (t)∥ L 2 x,y + T 1 2 K≪M ∼L 1 K -ρ L -σ 1 + L 1 ∼K≫M K -ρ max 1, K N γ -D ′′ L -σ 1 ∥F N,k (t)∥ B ρ,γ k,D ′′ [-T ,T ] .
Next, we multiply by c ρ,γ k,D ′′ (M ) = M ρ max 1, M N γ D and we sum over M to obtain

∥F N,k (t)∥ B ρ,γ k,D ′′ [-T ,T ] ≲ ∥P 1,k f 0 ∥ L 2 x,y + T 1 2 M ∼N c ρ,γ k,D ′′ (M )∥P M,k F N,k ∥ L ∞ t L 2 x,y + T 1 2   M ≲N γ M ρ-σ   ∥F N,k ∥ B ρ,γ k,D ′′ [-T ,T ] .
The assumption that σ > ρ allows us to conclude. Now, we prove that F N is essentially localized at frequency N . This turns out to be crucial in the estimates for the remainder w N , in the situations when we want to avoid the high-low-low frequency interactions. We recall that such a frequency localization is captured by the weighted norm (4.3.3).

Lemma 4.4.2 (Frequency localization for the adapted evolution). Let D ′′ > 0. There exists

T 0 > 0 such that F N satisfies ∥⟨D y ⟩ s F ω N ∥ X N,D ′ [-T 0 ,T 0 ] ≲ ∥ P N f ω 0 ∥ H s (R 2 ) .
Proof. Given M ≫ 1 and 0 ≤ T ≤ 1, we need to estimate

M s C N,D ′ (M )∥P M F N ∥ L ∞ t ([-T,T ];L 2
x,y ) .

We recall that F N is solution to the truncated equation (4.3.7), and that ϕ

(t) = θ F,w;≤ N 2 P ≤N γ u N 2 (t).
-Case when M ∼ N . In contrast with [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], we do not use an energy estimate. Instead, we proceed as in the proof of Lemma 4.4.1 in order to avoid using L ∞

x,y . We start from the Duhamel formula, and we estimate

F N (t) = P N e itA f ω 0 + t 0 e i(t-τ )A N (F N , ϕ, ϕ)(τ ) dτ .
From the Bernstein-Strichartz T T * -estimate in Lemma 4.4.1, with (p, q) = (∞, 2), ( p, q) = (4, ∞) and r = 2, we obtain

∥F N ∥ L ∞ t L 2 x,y ≲ ∥P N f ω 0 ∥ L 2 x,y + ∥N (F N , ϕ, ϕ)∥ L 4 3 t L 1 x L 2 y .
Thanks to Hölder's inequality and to the truncation, we know that on [-T, T ],

∥N (F N , ϕ, ϕ)∥ L 4 3 t L 1 x L 2 y ≲ T 1 2 ∥F N ∥ L ∞ t L 2 x,y ∥ϕ∥ 2 L 8 t L 4 x L ∞ y ≲ T 1 2 ∥F N ∥ L ∞ t L 2
x,y .

Choosing T 0 small enough we conclude that if T ≤ T 0 ,

∥F N ∥ L ∞ t L 2 x,y ≲ ∥P N f ω 0 ∥ L 2 x,y .
Therefore, when M ∼ N , we deduce that

M s ∥P M F N ∥ L ∞ t L 2 x,y ≲ N s ∥F N ∥ L ∞ t L 2 x,y ≲ ∥⟨D y ⟩ s P N f ω 0 ∥ L 2 x,y .
-Case when M ≁ N . In this case, we first fix k with N 2 ≤ |k| < N and we perform a dyadic decomposition in frequency around k for F N,k . We see that when M ≪ N ,

∥P M F N,k ∥ L ∞ t L 2 x,y ≲ K∼N ∥P K;k F N,k ∥ L ∞ t L 2 x,y ≲ N N γ -D ′′ ∥F N,k ∥ B ρ,γ k,D ′′ ,
whereas when N ≪ M , we have

∥P M F N,k ∥ L ∞ t L 2 x,y ≲ K∼M ∥P K;k F N,k ∥ L ∞ t L 2 x,y ≲ M N γ -D ′′ ∥F N,k ∥ B ρ,γ k,D ′′ .
Hence,

∥P M F N,k ∥ L ∞ t L 2 x,y ≲ max(M, N ) N γ -D ′′ ∥F N,k ∥ B ρ,γ k,D ′′ .
We deduce that if D ′′ is large enough with respect to D ′ , then

C N,D ′ (M )∥P M F N (t)∥ L ∞ t L 2 x,y ≲ (M N ) -D ′ N 2 ≤k<N |g k (ω)|∥F N,k ∥ B ρ,γ k,D ′′ .
Using Lemma 4.4.1 we deduce that

C N,D ′ (M )∥P M F N (t)∥ L ∞ t L 2 x,y ≲ (M N ) -D ′ N 2 ≤k<N |g k (ω)|∥P 1,k f 0 ∥ L 2 x,y .
From the Cauchy-Schwarz inequality and using that D ′ can be taken large enough to control the extra N 1 2 -loss coming to the sum over k and the factor M s , we conclude

M s C N,D ′ (M )∥P M F N (t)∥ L 2 x,y ≲ (M N ) -D ′ 2 ∥ P N f ω 0 ∥ L 2 x,y .

Strichartz estimates for the adapted linear evolution

In this part, we undercut the derivative loss in the Strichartz estimates (4.2.3) for the adapted linear evolution F N . To do so, we take advantage of the frequency localization of the function F N,k together with probabilistic decoupling.

Lemma 4.4.3 (Strichartz estimate for the individual block). Let 0 < ρ < σ as in Lemma 4.4.1, let δ = γ(σ -ρ) > 0 (arbitrarily small as ρ is close to σ), and let T 0 > 0 small enough. For all D ′ > 0, there exists D ′′ > D ′ such that for all σ ′ > σ and N 2 ≤ |k| < N , 0 < T ≤ T 0 , we have

∥⟨D y ⟩ σ ′ F N,k ∥ S r N,D ′ [-T ,T ] ≲ r ∥⟨D y ⟩ σ ′ P 1,k f 0 ∥ L 2 x,y + T 1 2 N σ ′ +γ( 1 2 -1 r )-γσ+δ ∥F N,k ∥ B ρ,γ k,D ′′ [-T ,T ] . Proof. We need to estimate M C N,D ′ (M )M σ ′ ∥P M F N,k ∥ L 8 t ([-T ,T ];L 4 x L r y (R 2 )) .
Given a dyadic integer M ≫ 1 and 0 ≤ t ≤ T , the Duhamel integral formulation of the truncated equation (4.3.7) reads P M F N,k (t) = e itA P M P 1,k f 0 + i t 0 e i(t-τ )A P M N (F N,k , ϕ, ϕ) (τ ) dτ .

First, we estimate the linear evolution of the unit-scale block P M P 1,k f 0 , which is nonzero only when M ∼ N . To do so, we use the Bernstein-Strichartz estimate (4.2.3) from Lemma 4. In what follows, without loss of generality, we assume that L 1 ≥ L 2 . The frequency support in the variable η of the Duhamel term has length | supp η F y→η N (P K,k F N,k , P L 1 ϕ, P L 2 ϕ) | ≲ max(L 1 , K) .

We use the Bernstein-Strichartz T T * -estimate (4.2.4) from Lemma 4.2.1 with (p, q) = (8, 4), (p, q) = (4, ∞) and |E| ≲ max(L 1 , K), and get Let us first consider the contribution of the terms with K ≪ N . In such a case, only the terms with M ∼ N contribute. Moreover, using the truncation (4.3.8), so that ∥⟨D y ⟩ σ ϕ∥ L 8

t L 4 x L ∞ y ≲ 1, M σ ′ L 2 ≤L 1 ≤N γ K≪N max(L 1 , K) 1 2 -δ ∥P M N (P K,k F N,k , P L 1 ϕ, P L 2 ϕ) ∥ L 4 3 t L 1 x L 2 y ≲ 1 M ∼N N σ ′ L 2 ≤L 1 ≤N γ K≪N max(L 1 , K) 1 2 -δ K -ρ max 1, K N γ -D ′′ L -σ 1 L -σ 2 c ρ,γ k,D ′′ (K)∥P K,k F N,k ∥ L ∞ t L 2
x,y .

When K ≤ N γ we get by comparing K and L 1 that

1 M ∼N N σ ′ K,L 1 ,L 2 ≤N γ max(L 1 , K) 1 2 -1 r K -ρ L -σ 1 L -σ 2 ∥F N,k ∥ B ρ,γ k,D ′′ ≲ 1 M ∼N N σ ′ +γ( 1 2 -1 r -σ)+δ ∥F N,k ∥ B ρ,γ k,D ′′ .
On the other hand, when K ≥ N γ , taking D ′′ ≫ 1 2 yields

N γ ≤K≪N K 1 2 -1 r -ρ K -D ′′ ≲ N γ( 1 2 -1 r -ρ-D ′′ ) ,
so that

1 M ∼N N σ ′   N γ ≤K≪N L 1 ,L 2 ≤N γ K 1 2 -1 r -ρ K N γ -D ′′ L -σ 1 L -σ 2   ∥F N,k ∥ B ρ,γ k,D ′′ ≲ 1 M ∼N N σ ′ +γ( 1 2 -1 r -σ)+δ ∥F N,k ∥ B ρ,γ k,D ′′ .
Summing over M ∼ N , in which case C N,D ′ (M ) ∼ 1, and taking the L 4 3 -norm in time for T ≤ 1, we obtain the desired bound for the contributions with K ≪ N :

M ∼N C N,D ′ (M )M σ ′ L 1 ,L 2 ≤N γ K≪N max(L 1 , K) 1 2 -δ ∥P M N (P K,k F N,k , P L 1 ϕ, P L 2 ϕ) ∥ L 4 3 t L 1 x L 2 y ≲ T 1 2 N σ ′ +γ( 1 2 -1 r -σ)+δ ∥F N,k ∥ L ∞ t B ρ,γ k,D ′′ .
Next, we handle the contribution of the terms with K ≳ N , where only the terms with K ≳ max(N, M ) actually contribute. We shall use the localization around frequency K ∼ N γ of F N,k imposed by the norm B ρ,γ k,D ′′ to prove smallness: for sufficiently large D ′′ depending on D ′ , Since

F N = N 2 ≤|k|<N g k (ω)F N,k ,
The conditional Khintchine's inequality and the mutual independence between the (g k )'s and the (F N,k )'s yield

E |⟨D y ⟩ σ ′ P M F N | p B N 2 1 p ≤ √ p ∥1 N 2 ≤|k|<N ⟨D y ⟩ σ ′ P M F N,k ∥ ℓ 2 k .
Plugging into (4.4.2),

∥⟨D y ⟩ σ ′ F N ∥ L p ω S r N,D ′ ≲ ∥1 N 2 ≤|k|<N C N,D ′ (M )⟨D y ⟩ σ ′ P M F N,k ∥ L p ω ℓ 1 M L 8 t L 4 x L r y ℓ 2 k .
Thanks to the Cauchy-Schwarz inequality, we can go from ℓ 1 M to ℓ 2 M up to replacing the weight C N,D ′ by C N,D ′ +1 . Using the Minkowski inequality again, we obtain

∥⟨D y ⟩ σ ′ F N ∥ L p ω S r N,D ′ ≲ √ p ∥C N,D ′ +1 (M )⟨D y ⟩ σ ′ P M F N,k ∥ L p ω ℓ 1 M L 8 t L 4 x L r y ℓ 2 k ≲ √ p ∥C N,D ′ +1 (M )⟨D y ⟩ σ ′ P M F N,k ∥ ℓ 2 k L p ω ℓ 2 M L 8 t L 4 x L r y ≲ √ p ∥C N,D ′ +1 (M )⟨D y ⟩ σ ′ P M F N,k ∥ ℓ 2 k L p ω S r N,D ′ +1
.

In the last estimate, we used that ℓ 1 M ⊂ ℓ 2 M . Finally, we apply the refined Strichartz estimate for the individual blocks F N,k (Lemma 4.4.3). The frequency localization stated in Lemma 4.4.1 yields

∥⟨D y ⟩ σ ′ F N ∥ L p ω S r N,D ′ ≲ √ p∥⟨D y ⟩ σ ′ P 1,k f 0 ∥ ℓ 2 k + √ pT 1 2 N σ ′ +γ 1 2 -1 r -γσ+δ ∥F N,k ∥ ℓ 2 k B ρ,γ k,D ′′ ≲ √ p ∥ P N f 0 ∥ L 2 x H σ ′ y + √ pT 1 2 N σ ′ +γ 1 2 -1 r -γσ-s+δ ∥ P N f 0 ∥ L 2
x H s y . Meanwhile, the estimate for p ≤ min(4, r) follows from Hölder's inequality. Next, we pass from S N,D ′ to S r N,D ′ with r ≫ 1 by using the Sobolev embedding in y, and by loosing therefore an arbitrarily small power of N 1 r (see Step 3 in the proof of Proposition 4.4 from [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF]). Here we can choose N δ .

Estimates on the nonlinear term w N

For fixed N , we construct the local solution to the truncated equation on a time interval [-T 0 , T 0 ] that does not depend on N . Proposition 4.5.1. Assume that 0 < σ < ν -1 2 , 0 < σ < σ ′ < s and 0 < γ < 1. Fix α > 0 such that max(ν -σ ′ , σ) < α < ν. Let D = D(s, ν, σ ′ , σ, α) > 0 and D ′ ≥ D ′ (s, ν, σ ′ , σ, α, D) > 0 be sufficiently large. There exist T 0 = T 0 (s, ν, σ ′ , σ, α, D, D ′ ) > 0 smaller than the T 0 in Lemmas 4. To prove such a proposition we do a contraction mapping argument, which is a consequence of the a priori estimates stated below.

Contraction mapping argument

As a consequence of the a priori estimates (4.5.1), we see that Γ (defined in (4.5.2)) maps Y ν N into Y ν N . We now prove that it is a contraction mapping in the Banach space Y ν N i.e. that for all z, w ∈ Y ν N [-T, T ] when T ≤ T 0 ,

∥Γ(z) -Γ(w)∥ Y ν N ≤ 1 2 ∥z -w∥ Y ν N .
To achieve this goal, we adapt the proof in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], inspired by [START_REF] Bouard | A stochastic nonlinear Schrödinger equation with multiplicative noise[END_REF], to the trilinear case . Since the linear terms are handled as above, it only remains to consider the quadratic and cubic terms. Let t z := sup 0 ≤ t ≤ T : ∥z∥ Y ν N [-t,t] ≤ 2 . The time t w is defined similarly. We observe from a continuity argument (both on the norm and on z, w) that for all t > t z (resp. t > t w ), we have θ z;N (t) = 0 (resp. θ w;N (t) = 0). Without loss of generality, we assume that t z ≤ t w and we encapsulate the nonlinear terms under consideration in Λ: Λ(z, w)(t) = Λ 1 (z, w)(t) + Λ 2 (z, w)(t) + Λ 3 (z, w)(t) , Λ 1 (z, w)(t) = θ z;N (t) 2 |z| 2 z(t) -θ w;N (t) 2 |w| 2 w(t) , Λ 2 (z, w)(t) = θ F ;N (t) θ z;N (t)N (z, z, F N )(t) -θ w;N (t)N (w, w, F N )(t) , Let us consider the cubic term Λ 1 (z, w). We decompose the nonlinear interactions as in [START_REF] Bouard | A stochastic nonlinear Schrödinger equation with multiplicative noise[END_REF]: Here, we used that θ z;N (τ ) = 0 when τ ∈ (t z , t w ]. Then, we deduce from the multilinear estimates (4.7.5), (4.7.7), (4.7.11) and (4.7.15) that t 0 e i(t-τ )A Λ 1 (τ ) dτ

Conclusion

We now show that the sequence (u N ) N tends to a limit u on some time interval [-T 0 , T 0 ], that u is solution to equation (NLS-HW) on a random time interval [-T, T ], and therefore finish the proof of Theorem 4.1.1. We refer to [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], Appendix A regarding the measurability properties of the functions F N and w N .

Optimizing the constraints

We collect the main constraints resulting from the nonlinear analysis.

• Constraint coming from the adapted linear evolution F N in Strichartz type spaces S N,D ′ :

σ ′ + γ 2 -s -γσ < 0 . • Contribution of term F N (P >N γ F ≤ N 2 )u N 2 in Y ν N : ν -s -γσ ′ < 0 . • Contribution of term F N (P >N γ w ≤ N 2 )u N 2 in Y ν N :
ν -σ ′ -γσ < 0 .

This reduces to the minimization problem for s under the constraints

               σ ′ -s -γσ + γ 2 < 0 ν -γσ ′ -s < 0 ν -σ ′ -γν < 0 σ -ν < -1 2 σ -σ ′ < 0 .
Discretizing γ ∈ (0 , 1) and using a linear programming solver leads to an approximate optimal solution (s 0 , σ ′ 0 , σ 0 , ν 0 ) = (0.464131, 0.154581, 0.077290, 0.577291) , γ = 0.732232 . (4.6.1)

We chose s > s 0 , σ = σ 0 -, σ ′ = σ ′ 0 -and ν = ν 0 -, and we now establish Theorem 4.1.1.

4.6.2 Convergence of (u N ) N ∈2 N First, we prove the convergence (4.1.3) in expectation along the dyadic subsequence (u N ) N ∈2 N up to the time T 0 > 0 which is chosen so that Proposition 4.5.1 holds true, and yields the a priori estimate on the truncated solution. In what follows T is fixed, with 0 < T ≤ T 0 , and the norms are taken on the slab Ω × [-T , T ] × R 2 . We recall from (4.3.2) that for some N 0 sufficiently large, the approximate solution reads

u N = u N 0 + N M =N 0 (w M + F M ) , ,
We first prove the convergence of the series F M and w M in the space L 2 ω (Ω; X s,σ T ), where the Sobolev-Strichartz space X s,σ T is defined in (4.1.2). Thanks to a completeness argument, it is enough to prove that the partial sums form a Cauchy sequence. Let 0 ≤ N -≤ N + < ∞. We first address the adapted linear evolution, controlled in L 2 ω (Ω; C t ([-T 0 , T 0 ]; L 2 x H s y )):

N + M =N - ⟨D y ⟩ s F M L 2 ω L ∞ t L 2
x,y

≲ N + M =N - ⟨D y ⟩ s P N F M L 2 ω L ∞ t ℓ 2 N L 2 x,y ≲ N + M =N - ⟨D y ⟩ s P N F M L 2 ω ℓ 2 N L ∞ t L 2
x,y .

From the definition (4.3.3) of the spaces X M,D ′ , we have

N + m=N - ⟨D y ⟩ s F M L 2 ω L ∞ t L 2
x,y

≲ N + m=N - max N M , M N -D ′ ∥⟨D y ⟩ s F M ∥ X M,D ′ L 2 ω ℓ 2 N .
Applying Cauchy-Schwarz's inequality, and then Lemma 4.4.2, the right-hand-side of the above inequality is bounded by r.h.s ≲

N + M =N - max N M , M N -D ′ ∥ P M f ω 0 ∥ L 2 x H s y L 2 ω ℓ 2 N ≲   N + M =N - ∥ P M f 0 ∥ 2 L 2 x H s y   1 2
.

Similarly, convergence in L 2 ω (Ω; L 8 t ([-T 0 ; T 0 ]; L 4 x W σ,∞ y

)) follows from the probabilistic Strichartz estimate in Proposition 4.4.4

∥⟨D y ⟩ σ F M ∥ L 2 ω L 8 t L 4 x L ∞ y ≲ T 1 2 M σ+ γ 2 -s-γσ+2δ ∥P M f 0 ∥ H s ,
where σ + γ 2 -s -γσ + 2δ < 0 when ρ is chosen close enough to σ since σ < σ ′ . To establish the convergence of the nonlinear remainders (w N ) N , we use the a priori estimate from Proposition 4.5.1. Namely,

∥w M ∥ Y ν M ≲ T 1 2 M ν-σ ′ -γν ∥⟨D y ⟩ σ ′ F M ∥ S M,D ′ + T 1 2 M ν-s-γσ ′ ∥⟨D y ⟩ s F M ∥ X M,D ′ ,
where ν -σ ′ -γν < 0 and ν -s -γσ ′ < 0. Using again Lemma 4.4.2 and Proposition 4.4.4, we get that for a very small parameter ε > 0,

∥⟨D y ⟩ σ w M ∥ L 2 ω L 8 t L 4 x L ∞ y + ∥⟨D y ⟩ ν w M ∥ L 2 ω L ∞ t L 2 x,y ≲ ∥w N ∥ Y ν M ≲ T 1 2 M -ε ∥ P M f 0 ∥ L 2
x H s y , so that the sum over M converges.

(t) = θ F ;N (t) = θ w;N (t) = 1, so that for every N , u N is an actual solution to (NLS-HW) with initial data P ≤N f ω 0 . We conclude by passing to the limit N → ∞ in the Duhamel formula.

The H 2s

x L2 y part

We prove that for each N ∈ 2 N the sequence (u N ), a priori constructed in C([-T 0 , T 0 ]; L 2 x H s y ), actually converges to u in C([-T 0 , T 0 ]; H s ). We simply use the T T * -estimate to have a priori estimates of the H 2s

x L 2 y -norm of u N :

∥u N ∥ L ∞ t H 2s x L 2 y ≲ ∥P N f ω 0 ∥ H 2s x L 2 y + ∥⟨D x ⟩ 2s |u N | 2 u N ∥ L 4 3 t ([-T 0 ,T 0 ];L 1 x L 2 y ) ≲ ∥P N f ω 0 ∥ H 2s x L 2 y + T 1 2 0 ∥u N ∥ L ∞ t H 2s x L 2 y ∥u N ∥ 2 L 8 t L 4 x L ∞ y ≤ C∥P N f ω 0 ∥ H 2s x L 2 y + 1 2 ∥u N ∥ L ∞ t H 2s x L 2 y ,
for T 0 sufficiently small, and where we used the truncation argument of De Bouard-Debussche to control the L 8 t L 4 x L ∞ y -norm of u N = u N (ϕ (i) ) 1≤i≤3 occurring in the equation (4.3.10) satisfied by w n .

Π hi,lo,lo (ϕ (1) , ϕ (2) , ϕ (3) ) = N 2 ,N 3 ≪N 1 N (P N 1 ϕ (1) , P N 2 ϕ (2) , P N 3 ϕ (3) ) , Π lo,hi,lo (ϕ (1) , ϕ (2) , ϕ (3) ) =

N 1 ,N 3 ≪N 2 N (P N 1 ϕ (1) , P N 2 ϕ (2) , P N 3 ϕ (3) ) , Π lo,hi,hi (ϕ (1) , ϕ (2) , ϕ (3) ) =

N 1 ≪N 2 ∼N 3 N (P N 1 ϕ (1) , P N 2 ϕ (2) , P N 3 ϕ (3) ) , Π hi,hi (ϕ (1) , ϕ (2) , ϕ (3) ) =

N 3 ≲N 1 ∼N 2 N (P N 1 ϕ (1) , P N 2 ϕ (2) , P N 3 ϕ (3) ) .

To estimate the norm L 4 3

t L 1 x L 2 y of the Duhamel term, we use the Hölder inequality, putting one term in L ∞ t L 2 x,y , the other two in L 8 t L 4 x L ∞ y , and gaining a factor T 1 2 . Recall that any small loss of derivative in y is harmless, since our goal is to prove a result for some s > s 0 . We multiply by N 0 + to account for such a 0 + -loss. At the end the loss are collected in the factor N δ , where δ > σ -ρ > 0 is arbitrarily small. Note that by homogeneity, the above inequalities are still true without the assumption ∥⟨D y ⟩ σ ϕ∥ S ≤N,D ≲ 1, up to multiplying the upper bound by ∥⟨D y ⟩ σ ϕ∥ S ≤N,D . We only make this assumption for shortness of notation. we deduce that

∥K ρ ∥ 2 Ḣ s 2 y = R R |h| 1-s 2 1 ((y + h) 2 + ρ 2 )(y 2 + ρ 2 )
dh dy .

We make the change of variable y = ρu and h = ρv to get

∥K ρ ∥ 2 Ḣ s 2 y = ρ -1-s R R |v| 1-s 1 ((u + v) 2 + 1)(u 2 + 1) du dv ∼ C ′ ρ -1-s , Moreover, ∥f ∥ Ḣs x L 2 y = ∥ e it∂xx G∥ Ḣs x ∥K ρ ∥ L 2 y = ρ -1 2 1 1 + y 2 dy 1 2 ∼ C ′′ ρ -1 2 , so that ∥f ∥ Ḣs ∼ ρ→0 C ′ ρ -1 2 -s 2 .
Since inequality (4.8.1) has to hold as ρ → 0, we see that necessarily, ρ -3 4 ≲ ρ -1 2 -s , therefore

3 4 ≤ 1 2 + s 2 ,
leading to the result 1 2 ≤ s.

Pathological set

An adaptation of the arguments from [START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schröinger equation on surfaces[END_REF] implemented in [START_REF] Kato | Ill-posedness for the Half wave Schrödinger equation[END_REF] implies the following ill-posedness result.

One can see that S defines a dense subset of H s . We make use of the following precise upper bounds in H m , m ≥ 0, of the initial data regularized by convolution, see Lemma 2.4 in [START_REF] Camps | Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations[END_REF].

∥ρ εn k * v 0,l ∥ H m 1 x H m 2 y ≲ n m 1 +2m 2 -s k-1 , ∞ l=k ∥ρ εn k * v 0,l ∥ H m 1 x H m 2 y ≲ n m 1 +2m 2 k n -s k+1 n k n k+1 3 2 .
Note that this estimate is not valid for every ε < ε n k , but only when ε ≥ ε n k . As a consequence, we do not have a uniform control on the time of existence of the smooth solution with initial data ρ ε * f in L 2

x H m y for m > 1/2. It remains to study the time evolution of ρ εn k * f , by comparing it to the ODE profile ρ εn k * v 0,k only. An adaptation of the proof of Proposition 2.6 in [START_REF] Camps | Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations[END_REF] leads to the following result. Let u εn k be the maximal solution of (NLS-HW) with initial data ρ εn k * f in L 2

x H 1 y , and T * (ε n k ) the maximal time of existence. Note that according to [START_REF] Bahri | Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations[END_REF], Theorem 1.6, the Cauchy problem for (NLS-HW) is locally well-posed in this space. Moreover, T * (ε n k ) only depends on the norm of the initial data in L 2

x H 1 y . Therefore, the norm of the solution must blow up when t → T * (ε n k ). We establish that this does not happen before the norm inflation time t n k by running a perturbative analysis.

Proposition 4.8.7 (Growth of perturbed initial data). Fix 0 < γ < β < 1. Let 0 ≤ s < 1 4 . Let n = n k . Then the solution u εn is well-defined in C([0, t n ], L 2

x H 1 y ). Moreover, there holds ∥u εn (t n ) -ρ εn * e itA u 0 -v εn n (t n )∥ L 2

x H s y ≤ C, as a consequence, ∥u εn (t n )∥ L 2

x H s y ≳ log(n) 2s(β-γ)-γ . As a consequence, the set S is a subset of the pathological set P.

Chapter 5

Lack of Regularisation in the second Picard iteration for NLS on S 2 with random initial data joint work with Nicolas Burq, Mickaël Latocca, Chenmin Sun and Nikolay Tzvetkov [START_REF] Burq | Lack of regularization in the first Picard iteration for NLS on S 2[END_REF].

Résumé. Nous considérons l'équation de Schrödinger cubique avec une renormalisation de Wick (NLS) sur la sphère bidimensionnelle, avec des données initiales aléatoires qui peuvent être distribuées selon la mesure de Gibbs. Nous montrons que la seconde itération de Picard n'améliore pas la régularité des données initiales, à l'échelle des espaces de Sobolev classiques. Ceci contraste fortement avec la même équation de Schrödinger renormalisée sur des tores bidimensionnel, pour laquelle nous savons grâce aux travaux de Bourgain que la seconde itération de Picard gagne une demi-dérivée. Notre preuve repose sur l'identification d'une partie singulière de la non-linéarité. Nous montrons que cette partie singulière est responsable d'un phénomène de concentration sur un grand cercle (à savoir une géodésique fermée stable), qui tout effet régularisant non linéaire dans la seconde itération de Picard.

Abstract. We consider the Wick ordered NLS on the two-dimensional sphere, with random initial data, possibly distributed according to the Gibbs measure. We show that the first nontrivial Picard iteration does not improve the regularity of the initial data in the scale of the classical Sobolev spaces. This is in sharp contrast with the Wick ordered NLS on the two-dimensional tori, a model for which we know from the work of Bourgain that the second Picard iteration gains one half derivative. Our proof relies on identifying a singular part of the nonlinearity. We show that this singular part is responsible for a concentration phenomenon on a large circle (i.e. a stable closed geodesic), which prevents any regularization in the second Picard iteration.

The {g n } n≥0 are identically independent normalized Gaussian variables with complex values. According to the Weyl's asymptotics, we know that λ n = O(n) and, as a consequence, ϕ ω defined by (5.1.1) converges in L 2 (Ω; H 0-(M)).

On the other-hand, local well-posedness a priori holds in H s (M ) for s > 1 2 , and we do not know how to construct a dynamics on the support of the Gibbs measure. In the case of the torus T 2 , Bourgain prove local well-posedness up to scaling-critical space, namely for all s > 0, by using bilinear estimates and Fourier restriction spaces. Then, Bourgain proved in [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] well-posedness for almost-every initial data on the support of the Gibbs measure, by using a linear-nonlinear decomposition, and by observing a probabilistic nonlinear smoothing effect. Specifically, Bourgain showed that the second Picard's iteration has regularity H The positive result follows from a bilinear estimate that requires 1 4 -derivatives, and which is optimal for the certain spherical harmonics. In the other hand, the instabilities are due to the presence of spherical harmonics which concentrated on a great circle. The quasi-linearity of (NLS) was further studied in [Ban04; BGT10], proving that oscillating H s -normalized sequences of initial data are not linearly propagated.

Regularity of the second Picard's iteration

In light of the above discussion, the deterministic threshold for the local Cauchy theory is far from the regularity of the Gibbs measure. Moreover, it is not known how to construct strong solutions in H s (S 2 ) when 0 < s < 1 4 . In this work, we consider randomized initial data, and we prove that the second Picard's iteration does not gain any regularity. This is in sharp contrast with the case of the torus, and precludes the construction of strong solutions by using the Bourgain linear-nonlinear decomposition.

On the other hand, we propose a self-contained proof of the gain of one-half derivative of the second Picard's iteration in the case of irrational tori.

In order to make our statement precise, we introduce some notation. Denote by where ϕ ω is given by (5.1.1). The subscript M marks the dependence of this object on the underlying manifold considered. Let us denote by T 2 β the torus T 2 endowed with the metric dx 2 + β -2 dy 2 , and by S 2 the Euclidean sphere in R 3 . With these notations at hand, we prove the following theorem. We give some comments.

• The bound (5.1.3) should be viewed as a gain of 1/2-derivative by the Picard iteration, compared to the H 0-regularity of the initial data ϕ ω . Even if this result is contained in the work of [START_REF] Fan | 2D-defocusing nonlinear Schrödinger equation with random data on irrational tori[END_REF], it allows us to stress out the existence of radically different behaviors for (NLS) with random initial data below L 2 on S 2 and on T 2 . We propose a self-contained proof of (5.1.3) in Section 5.5.

• In contrast to the case of the tori, the second Picard iterate for (NLS) on the sphere does not improve the regularity of initial data in H α-1 2 -0 (S 2 ). Moreover, the divergence for the quadratic moment of ∥Π N I 2 S (t, ϕ ω )∥ L 2 (S 2 ) N ≥1 is likely to yield the almost sure divergence of this random process, but we do not have a self-contained elementary proof of this result.

Our result indicates that proving the existence of a dynamics below H 1 4 (S 2 ) does not follow from a linear-nonlinear decomposition. In addition, the structure of the solution associated to ϕ ω would not be u(t) = e it∆ ϕ ω + {more regular terms} , (5.1.4) which is in sharp contrast with the work of [START_REF] Deng | Invariant Gibbs measures and global strong solutions for nonlinear Schröodinger equations in dimension two[END_REF]. Nevertheless, we have hope to use the random averaging operators to paracontrol the worst frequency interactions, and to prove almost-sure local well-posedness for some α < 3 4 , breaking therefore the deterministic threshold.

Organization of the article

The proof of Theorem 5.1.2 in the case of the sphere is given in Section 5.4 and mostly relies on a concentration property of Proposition 5.3.1, which is proven in Section 5.3. We also recall preliminary estimates and notations in Section 5.2. Finally, we prove (5.1.3) in Section 5.5.

Notation and preliminaries

Spherical harmonics

Spherical coordinates (ρ, θ, φ) are defined by (x 1 , x 2 , x 3 ) = (ρ sin(θ) cos(φ), ρ sin(θ) sin(φ), ρ cos(θ)) .

The polar coordinate θ ∈ (0 , π) is the colatitude, φ ∈ (0 , 2π) is the azimut and ρ ∈ R + is the radial distance, so that S Let us now gather some results that we will use in the proof of Theorem 5.1. (5.2.3) See Lemma 2.1 in [START_REF] Burq | Probabilistic Sobolev embeddings, applications to eigenfunctions estimates[END_REF] for a proof. This crucially exploits the invariance by rotation of the sphere S 2 . This indeed implies that the kernel

K n (x, y) := |k|≤n Y n,k (x)Y n,k (y)
of the orthogonal projector on E n is invariant under the action of any rotation. Since rotations act transitively on S 2 we deduce that K n (x, x) does not depend on x, and the result follows from integrating in x over S 2 .

Randomization : Gaussian measure on spherical harmonics of degree n

Fix n ∈ N and let E n be the space of spherical harmonics of degree n :

E n = Span C Y n,k | k ∈ -n, . . . , n .
Recall that we denoted by λ n = n(n + 1) the n th eigenvalue of ∆ 2 S with multiplicity [n] = 2n + 1. Given g n,k (ω) 1≤n,|k|≤n a sequence of identically distributed standard complex Gaussian random variables, the randomization Y n,k .

Next we will prove that random spherical harmonics e ω n 1≤n defined above satisfy some uniform bounds in Lebesgue spaces.

Lemma 5.2.3. Let q ≥ 2, then there exists a constant C q , not depending on n, such that for all n ≥ 1 there holds

E ∥e ω n ∥ 2 L q x ≤ C q .
(5.2.6)

Proof. Let n ≥ 0. One observes that for fixed x ∈ S 2 ,

|k|≤n g n,k (ω)Y n,k (x)
is a Gaussian variable of mean 0 and variance k≤n |Y n,k (x)| 2 . It follows from (5.2.3) that e ω n (x) is a normalized Gaussian variable. Therefore, there exists C > 0 independent of q, n, x such that

∥e ω n (x)∥ L q Ω ≤ C √ q .
Then thanks to the Hölder inequality, Fubini theorem and that S 2 has finite volume, we obtain

E ∥e ω n ∥ 2 L q
x ≤ ∥e ω n (x)∥ 2 L q (Ω;L q (S 2 ) ≤ C √ q .

This concludes the proof of (5.2.6) with C q = C √ q.

Concentration of spherical Harmonics

It follows from the expression of the Laplace operator on the sphere (5.2.2) that v n,k is solution on (0 , π) to -sin(θ) d dθ 2 v n,k (θ) + k 2 -n(n + 1) sin 2 (θ) v n,k (θ) = 0.

(5.3.1)

This equation turns out to display some ellipticity away from the equator, in the regime where |k| is close to n. We shall exploit this property in a quantitative way by the use of semiclassical functional calculus, and get a large family of spherical harmonics that are concentrated near the equator.

Proposition 5.3.1. Let 0 < δ ≪ 1. There exists C δ > 0 such that for all k ∈ {-n, . . . , n} with n 1 -δ 2 ≤ |k| we have

∥Y n,k ∥ L 2 (|x 3 |>2δ) ≤ C δ n .
Remark 5.3.2. In fact, one has ∥Y n,k ∥ L 2 (|x 3 |>2δ) ≤ C δ,K n -K for all K ∈ N, but we choose to present a self-contained proof for the crude bound O(n -1 ), which already fits our needs.

Proof. In order to use semiclassical functional calculus and prove this proposition, we shall extend v n,k to a function defined on the whole real line R. One way of doing so would be to use the symmetries of Y n,k to periodize v n,k . Still, we prefer to avoid boundary conditions and we rather make the change of variable 1 y(θ) = arctanh(cos(θ)) .

Note that y : (0 , π) → R is a C ∞ -diffeomorphism and that sin 2 (θ) = 1 -cos 2 (θ) = 1 -tanh 2 (y) = cosh(y) -2 .

This yields dy = -sin(θ) -1 dθ = -cosh 2 (y) sin(θ)dθ .

(5.3.2)

In light of the definition of the scalar product (5.2.1) on S 2 , we define Proof. The key point is that P 0,h,α is elliptic in the region | tanh(y)| > 2δ and the proof follows from the standard parametrix construction of elliptic operators. Since we only need to invert P h,α in the elliptic region up to order 1, the proof is more straightforward and self-contained. Recall that for all y ∈ supp(χ), ξ ∈ R and √ 1 -δ 2 < α we have p 0 (y, ξ, α) = ξ 2 + α 2 -cosh(y) -2 ≥ δ 2 2 .

(5.3.7)

Let us write ϕ α (y) = α 2 -cosh -2 (y), φ α (y, z) = 1 0 ϕ ′ α (tz + (1 -t)y)dt so that p 0 (y, ξ, α) = ξ 2 + ϕ α (y), ϕ α (y) -ϕ α (z) = (y -z)φ α (y, z).

We introduce the symbol q(y, ξ, α) := χ(y) p 0 (y, ξ, α) .

It follows from (5.3.7) that for all k ∈ N there exists 0 < C k such that for all √ 1 -δ 2 < α ≤ 1 and ξ ∈ R, sup y∈supp(χ)

|∂ k ξ q(y, ξ, α)| ≤ C k δ -(2+k) ⟨ξ⟩ -(2+k) .

(5.3.8)

The semiclassical pseudodifferential operator Q h,α associated with q is defined via its Schwartz kernel Q h f (y) = R K q (y, z)f (z)dz, K q (y, z) = 1 2πh R e i(y-z)ξ h q(y, ξ)dξ .

Observe that (z -y)e i(y-z)ξ h = ih∂ ξ (e i(y-z)ξ h

) .

(5.3.9)

Hence, integrating by parts twice yields that (y -z) 2 K(y, z) = -h 2π R ∂ 2 ξ q(y, ξ, α) e i(y-z)ξ h dξ , and we deduce from (5.3.8) that

|K(y, z)| ≤ Cδ -4 h (y -z) 2 R ⟨ξ⟩ -4 dξ ≤ Cδ -4 h (y -z) 2 .
Moreover, we have the trivial bound Since the computations are the same when changing the role of y and z, we prove that there exists C > 0 such that for all 0 < h ≤ 1 and 0 < δ < 1, It follows from Schur's test that Q h is uniformly bounded in L 2 (R) for 0 < h ≤ 1, with ∥Q h ∥ L 2 (R)→L 2 (R) ≤ Cδ -4 .

|K(y, z)| ≤ h -1 δ -2 R ⟨ξ⟩ -2 dξ ≤ Cδ -4 h -1 .
(5.3.10)

Then we check directly from integration by part that the Schwartz kernel of the operator Recall that p 1 (y, hD y ) = tanh(y)h∂ y -cosh(y) -2 . From the a priori estimate on v n,k , obtained by multiplying (5.3.5) by the complex conjugation of v n,k and integrating over R, we have

R h := Q h • P 0,h
∥p 1 (y, hD y ) v n,k ∥ L 2 (R) ≤ Ch .
for all 0 < h ≤ 1. Therefore, ∥P 0,h v n,k ∥ L 2 (R) ≤ Ch .

From Lemma 5.3.3, there exists C δ such that

∥χ v n,k ∥ L 2 (R) = ∥Q h P 0,h v n,k -R h v n,k ∥ L 2 (R) ≤ (C δ + C)h .
It follows that

∥Y n,k ∥ L 2 (|x 3 |>2δ) = ∥ v n,k ∥ L 2 (| tanh(y)|>2δ) ≤ ∥χ v n,k ∥ L 2 (R) ≤ C δ h ,
which completes the proof of Proposition 5.3.1.

Proof. Recall that if g is a standard complex Gaussian variable then

E[g 2 ] = E[g 2 ] = E[|g| 2 g] = E[|g| 2 g] = 0 .
Since the u ω n 's are linear combinations of independent standard complex Gaussian variables, we also have To prove this, we expand

E[(u ω n ) 2 ] = E[(u ω n ) 2 ] = E[|u ω n | 2 u ω n ] = E[
E (I | II) L 2 (S 2 ) = 1≤n 1 ,n 2 ,n 3 ,q,m≤N n 2 ̸ =n 1 ,n 3 2 S E u ω n 1 u ω n 2 u ω n 3 u ω m |u ω q | 2 -∥u ω q ∥ 2 L 2 (S 2 )
dx .

In light of (5.4.3), (5.4.4) and from the independence of the u ω n n≥1 , we see from a case by case analysis that expectation of each term appearing in the above sum vanishes provided that n 2 ̸ = n 1 and n 2 ̸ = n 2 . Similarly, and (5.4.2) follows from the triangle inequality.

E (I | III) L 2 (S 2 ) =
Lemma 5.4.3. There exits C > 0 such that for all N ≥ 1 and t ∈ R, E ∥Π N III(t)∥

n 2s n 6α 1 E |(|e ω n 1 | 2 e ω n 1 | Y n,k ) L 2 (S 2 ) | 2 .
We used that the contribution of the terms when n ≫ n 1 , in the semiclassical regime, is negligeable. Then, it follows from the Plancherel's formula that

∥Π N (III)∥ 2 L 2 (Ω;H s (S 2 )) = 1≤n 1 n 2s-6α 1 E ∥e ω n 1 | 2 e ω n 1 ∥ 2 H α-1 2 (S 2 ) ≤ 1≤n 1 n 2s-6α 1 E ∥e ω n 1 ∥ 6 L 6 (S 2 ) .
As a consequence of Lemma 5.2.3, there exists C not depending on p such that

E ∥e ω n 1 ∥ 6 L 6 (S 2 ) ≤ C .
Therefore the series converges provided that s < α -1 2 + 2α, and the lemma is proved. Note that this term gains one derivative when the initial data is supported on the Gibbs measure.

At this stage, we proved in Lemma 5.4.2 and in Lemma 5.4.3 that there exists C > 0 such that for all N ≥ 1, t ∈ R and s ≤ α -1 2 , ∥I N (t, II)∥ 2 L 2 (Ω;H s (S 2 )) -C ≤ ∥Π N I 2 S (t, ϕ ω )∥ 2 L 2 (Ω;H s (S 2 )) .

To get the divergence claimed in Theorem 5.1.2 it remains to estimate from below the quadratic moment of the H α-1 2 (S 2 )-norm of the Duhamel integral I N (t, II). To do so we shall use the concentration property for high order spherical harmonics of Proposition 5.3.1. Proposition 5.4.4. There exist η > 0 and N 0 ≥ 0 such that for all N ≥ N 0 and t ∈ R we have .

η N 0 ≤n≤N 1 n ≤ ∥I N (t,
We expand e -it ′ ∆ 2 S (II) on the orthonormal basis of L 2 (S 2 ) made of spherical harmonics :

Π N e -it ′ ∆

2

Y n 1 ,r (x)

n 1 2 1 Y n,k (x)dx × 1 n 2 -n ′ 1 2 S 2 |e ω n ′ 2 | 2 -∥e ω n ′ 2 (y)∥ 2 L 2 (S 2 ) n ′ 2 2α g n ′ 1 ,r ′ (ω) n ′ 1 1 2 e n ′ 1 ,r ′ (y) n ′ 1 α
Y n,k (y)dy .

(5.4.9)

Observe that E(g m,r ) = 0, E(g 2 m,r ) = 0, E(|e ω m | 2 -∥e ω m ∥ 2 L 2 (S 2 ) ) = 0. and that the laws of e ω n (x) and e ω n (y) are the same as a consequence of rotational invariance. Hence, it follows from the independence of the g m,r that the only terms that remain in the series (5.4.9) after taking the expectation are the ones with n 1 = n ′ 1 , n 2 = n ′ 2 , r = r ′ . To estimate from below the right-hand side from (5.4.8), we are thus reduced to handle the series Let us now estimate from below the right hand side of (5.4.11). We use the explicit expression of e ω 1 , and we get rid of a well-chosen set in order to exploit the concentration of Y 1,1 near the equator. Besides, recall that

e ω 1 = 1 √ 3 g 1,1 (ω)β(x 1 + ix 2 ) + g 1,-1 (ω)Y 1,-1 + g 1,0 (ω)Y 1,0 ,
where β ≥ 1 is the renormalization constant such that β 2 Note that P(S A,ε ) > 0. Moreover, for ω ∈ S ε we have Indeed, we see from the definition of Λ N 0 that for each n ≥ N 0 ,

|e ω 1 | 2 -∥e ω 1 ∥ 2 L 2 (S 2 ) = 1 3 |g ω 1,1 | 2 β 2 |x 1 + ix 2 | 2 + O(ε) - 1 3 |g ω 1,1 | 2 + |g ω 1,-1 | 2 + |g ω 1,0 | 2 = 1 3 |g ω 1,1 | 2 β 2 |x 1 + ix 2 | 2 -1 + O(ε) .
# k | (n, k) ∈ Λ N 0 ≥ n 1 -1 -δ 2 ≳ n δ 2 .
This proves that the right hand side in the above estimate displays a logarithmic divergence, and concludes the proof of Proposition 5.4.4 by taking η ∼ αP(S ε )δ 2 > 0.

First iteration for the torus

In this appendix, we consider the cubic NLS on the general torus

T 2 β = R 2 /(2πZ) 2 ,
endowed with the metric g = dx 2 1 + β -2 dx 2 2 for some β > 0. The cubic NLS on T 2 β is written as

i∂ t v + ∆ β v = |v| 2 v,
(5.5.1)

where ∆ β = ∂ 2 x 1 +β 2 ∂ 2 x 2 . Hence from now on we will only concentrate on (5.5.1). For n = (k, m), n ′ = (k ′ , m ′ ) ∈ Z 2 , we denote by Q(n, n ′ ) = kk ′ +β 2 mm ′ the associate quadratic form and Q(n) := Q(n, n). Then e it∆ β = F -1 x e -itQ(•) F x . In light of (5.1.1), we consider initial data of the form ϕ ω (x) = We write the Wick-ordered nonlinearity N (u ω )(t) =

(n 1 ,n 2 ,n 3 )∈Z 2 n 2 ̸ =n 1 ,n 3 e it(Q(n 1 )-Q(n 2 )+Q(n 3 )) ⟨n 1 ⟩⟨n 2 ⟩⟨n 3 ⟩ g n 1 (ω)g n 2 (ω)g n 3 (ω) e i(n 1 -n 2 +n 3 )•x = (n 1 ,n 2 ,n 3 ,n)∈Γ e itΦ(n) ⟨n 1 ⟩⟨n 2 ⟩⟨n 3 ⟩ g n 1 (ω)g n 2 (ω)g n 3 (ω) e in•x , where the constraint set is

Γ = {(n 1 , n 2 , n 3 , n) ∈ (Z 2 ) 4 : n = n 1 -n 2 + n 3 , n 2 ̸ = n 1 , n 2 ̸ = n 3 } .
and the resonant function Φ defined on Γ is

Φ(n) := Q(n 1 ) -Q(n 2 ) + Q(n 3 ) -Q(n 1 -n 2 + n 3 ) = 2Q(n 1 -n 2 , n 2 -n 3 ).
The first Picard iteration is written as

I T 2 β (t, ϕ ω ) = t 0 e i(t-t ′ )∆ β N (u ω (t ′ ))dt ′ ,
We shall now prove the gain of regularity (5.1.3) claimed in Theorem 5.1.2.

Proof. For fixed t, direct computation, independence assumption as well as the observation that E[g 2 ] = 0 for standard complex guassian functions yields that

E ∥I T 2 β (t, ϕ ω )∥ 2 H s x ∼ n=(n 1 ,n 2 ,n 3 )∈Γ ⟨n 1 -n 2 + n 3 ⟩ 2s ⟨n 1 ⟩ 2 ⟨n 2 ⟩ 2 ⟨n 3 ⟩ 3 ⟨Φ(n)⟩ 2 ,
(5.5.2)

To compute the right side of (5.5.2), we decompose dyadically |n j | ∈ N j , and assume that N (1) ≥ N (2) ≥ N (3) is the non-increasing order of N 1 , N 2 , N 3 . It would be sufficient to obtain an inequality of the form

(N 1 N 2 N 3 ) -2
|n j |∼N j ,j=1,2,3 n 2 ̸ =n 1 ,n 3 ⟨n 1 -n 2 + n 3 ⟩ 2s ⟨Q(n 1 -n 2 , n 2 -n 3 )⟩ 2 ≲ (N (1) ) -δ .

(5.5.3)

The left side of (5.5.3) can be majorized by Then Q(m 1 , m 0 ) can only take a discrete number of values. In these situations, we have (the same dimension treatment as in [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF])

#{m 1 ∈ Z 2 : |m 1 | ∼ N 1 , |Q(m 1 , m 0 ) -l| ≤ δ} ≲ N 1 .
Therefore, without loss of generality, we may assume that both ξ 0 and η 0 are non zero. Denote by m β = (ξ 0 , β 2 η 0 ) = diag(1, β 2 )m 0 . Then it is reduced to estimate the cardinality of the set

S δ := z ∈ Z 2 : z • m β |m β | - l |m β | ≤ δ |m β | .
We observe that S δ is a rectangle with side length ∼ N 1 and width ≤ 2δ. Since 2δ < 1/2, we see that #S δ ≲ N 1 .

Thus the contribution of (5.5.4) in the sum is less then N 2s-1 1 , and the associated dyadic summation over N 1 ≫ N 2 , N 3 converges provided that s < 1 2 . • Case 2: N 2 ≫ N 1 , N 3 :

In this case, |Q(n 1 -n 2 , n 2 -n 3 )| ∼ N 2 2 . Coming back to (5.5.4), the range of the sum of l is |l| ≥ N 2 2 . Hence by the crude estimate M N 1 ,N 2 ,N 3 ≲ (N 1 N 2 N 3 ) 3 , the contribution of (5.5.4) is bounded by

|l|≥N 2 2 N 2s 2 l 2 ≲ N 2s-2 2 .
Then associated dyadic summation over N 2 ≫ N 1 , N 3 converges, provided that s < 1.

For the rest situations, the argument is the same as for Case 1. For example, if N 1 ∼ N 2 ≫ N 3 (N 2 ∼ N 3 ≫ N 1 ), we can fix n 2 , n 3 (n 1 , n 2 ), and do the same manipulation as in Case 1.

This completes the proof of (5.1.3) in Theorem 5.1.2.

Titre:

  Approches probabilistes pour des équations de Schrödinger non-linéaires............................... Mots clés : équation de Schrödinger non-linéaire, problème de Cauchy, données initiales aléatoires, diffusion, instabilités Résumé : Cette thèse a pour objet l'étude d'équations de Schrödinger non-linéaires avec des données initiales aléatoires, dans des régimes singuliers. Les équations de Schrödinger non-linéaires interviennent dans de nombreux modèles ondulatoires pour décrire l'enveloppe d'un paquet d'ondes qui oscillent rapidement. Quant à elle, l'étude statistique de ces ondes non-linéaires dans des régimes singuliers est motivée par la présence d'instabilités qui compromettent l'analyse des trajectoires individuelles.

Title:

  Probabilistic approaches for nonlinear Schrödinger equations..................................................... Keywords : nonlinear Schrödinger equation, Cauchy problem, random initial data, scattering, instabilities

  dispersive equations with random data . . . . . . . . . . . . . . . . . . . . 1 0.1.1 Generalities and main questions . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.1.2 A brief history of probabilistic Cauchy theory . . . . . . . . . . . . . . . . . . 6 0.1.3 Globalization of probabilistic flows and generic dynamics. . . . . . . . . . . . 14 0.2 Presentation of the results in this manuscript . . . . . . . . . . . . . . . . . . . . . . 24 0.2.1 On the Cauchy theory in scaling-supercritical regimes . . . . . . . . . . . . . 24 0.2.2 Long-time dynamics under stochastic perturbations . . . . . . . . . . . . . . 28 0.2.3 Probabilistic Cauchy theory for weakly dispersive equations . . . . . . . . . . 41 0.3 Synthèse en Français . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 0.1 Nonlinear dispersive equations with random data 0.1.

  b n (ω)c n e inx , x ∈ T , is almost-surely in L p (T) for every p ∈ [2 , ∞).

Figure 1 :

 1 Figure 1: The colored stripes indicate the microlocal support of the elementary blocks P 1;k in the Wiener randomization.

Figure 3 :

 3 Figure 3: Cauchy theory for equation (NLS-HW) in the scale of Sobolev spaces.

3 2 +

 2 (R 3 ), see Proposition 1.2.10. Hence, the reason for the technical constraint on p ∈ 3, 5 in the statement of the second part of Theorem 1.1.4.

t 0 e 2 +

 02 i(t-τ )∆ F (τ )dτ .Lemma 1.3.6 (Trilinear estimates, see Proposition 3.1 in[START_REF] Bényi | Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS[END_REF][START_REF] Bényi | On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on R d , d ≥ 3[END_REF]). Let 1 4 < s < 1 2 and 1 2 < σ < 2s. There exist C > 0, b = 1 and θ = 0 +

1 2 ≤

 2 σ < 1, by directly performing a contraction argument and obtaining global spacetime bounds.

2. 2 . 1

 21 The free Schrödinger evolution and critical spaces Proposition 2.2.1 (Strichartz estimates in dimension d = 3). Let u 0 ∈ L 2

( 2 . 0 e

 20 2.14)Hence, we need to study the following oscillatory integral I(x, ρ, φ) = π i|x|ρ cos θ ψ k (ξ(ρ, θ, φ)) sin θdθ .(2.2.15)

Definition 2 .4. 2 .

 22 Given a time-interval J = [t 0 , b] ⊂ [0 , T * ), we define the Strichartz norm of a frequency truncated function v by

  4.4 Local smoothing and the critical weighted strategy . . . . . . . . . . . . . . 184 3.5 Almost-sure asymptotic stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 3.5.1 Local existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 3.5.2 Nonlinear ground states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 3.5.3 Global a priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 3.5.4 Proof of Theorem 3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

2 2

 2 Theorem 3.1.5. Assume that H has no resonance nor eigenvalue at zero. Let s d = d-1 d+1 • s c and s ∈ (s d , s c ]. For all ω ∈ Ω ε and ψ 0 ∈ H d-(R d ) ∩ Ran(P c ) small enough, the Cauchy problem

d- 2 2

 2 (I) by replacing in (3.5.11) the U 2 H (I)-norm by the DU 2 H (I) one. Remark 3.4.9. In the case of V 2 H it follows from Definition 3.4.1 and from the almost orthogonality property of Littlewood-Paley decomposition that ∥⟨ √ H⟩ d-2

  lim I ∥N (v + εu ω )∥ DU 2 H (I) = 0 whenever |I| → 0 or I ⊂ [t , ∞) with t → +∞ . (3.4.18) 3.4.3 Proof of Theorem 3.1.5

t

  ∥ D α z Q(z(t))∥, as well as O(z) := O(sup t |z(t)|).

  -posedness for the Schrödinger half-wave equation joint work with Louise Gassot and Slim Ibrahim [CGI22].

Contents 4 . 1

 41 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4 . 1 :

 41 Figure 4.1: Cauchy theory for equation (NLS-HW) in the scale of Sobolev spaces.

  2.1 with (p, q) = (8, 4) and |E| ≲ 1. There holds∥P M e itA P 1,k f 0 ∥ L 8 t L 4 x L r y ≲ 1 M ∼N ∥P 1,k f 0 ∥ L 2 x,y . To handle the nonlinear part, we perform a Littlewood-Paley decomposition of ϕ: ϕ ∼ 1≤L ≤ N γ P L ϕ , and a Littlewood-Paley decomposition of F N,k recentered around k: F N,k ∼ 1≤K P K,k F N,k .

L 2 ≲L 2

 22 ≤L 1 ≤N γ K ∥ t 0 e i(t-τ )A P M N (P K,k F N,k , P L 1 ϕ, P L 2 ϕ) (τ ) dτ ∥ L 8 t L 4 x L r y ≤L 1 ≤N γ K max(L 1 , K) 1 2 -1 r ∥P M N (P K,k F N,k , P L 1 ϕ, P L 2 ϕ) (τ ) dτ ∥ L 4 3 t L 1 x L 2 y .

  4.1 and 4.4.2, and a unique solution w N ∈ Y ν N ([-T 0 , T 0 ]) to the truncated equation (4.3.10). In addition, for all 0 < T < T 0 ,∥w N ∥ Y ν N ([-T ,T ]) ≲ δ T 1 2 N ν-σ ′ -γν ∥⟨D y ⟩ σ ′ F N ∥ S N,D ′ + T 1 2 N ν-s-γσ ′ ∥⟨D y ⟩ s F N ∥ X N,D ′ .(4.5.1)

Λ 3 2 ) 2 )

 322 (z, w)(t) = θ F,w;≤ N 2 (t) θ z;N (t)N (z, z, u N (t) -θ w;N (t)N (w, w, u N (t) .

  t-τ )A 1 [0,tz] (τ ) θ z;N (τ ) -θ w;N (τ ) |z| 2 z(τ ) dτ t-τ )A 1 [0,tz] (τ )θ w;N (τ ) |z| 2 z(τ ) -|w| 2 w(τ ) dτ Y ν N + t 0 e i(t-τ )A 1 (tz,tw] (τ ) θ z;N (τ ) -θ w;N (τ ) |w| 2 w(τ ) dτ Y ν N .

22

  ∥θ z;N -θ w;N ∥ L ∞ t ∥1 [0,tz] z∥ 3 ∥θ z;N -θ w;N ∥ L ∞ t ∥1 [tz,tw] w∥ 3 w∥ Y ν N .The quadratic interactions are controlled analogously. Hence, Γ is a contraction mapping on Y ν N for sufficiently small T 0 , and this concludes the proof of Proposition 4.5.1.

Proposition 4.7. 2 (

 2 Trilinear estimates with high-low-low type interactions). Let ϕ be of type v or G, and assume that max(ν-σ ′ , σ) < α < ν, D > α and 2D + s + σ ′ < D ′ . Assuming that ∥⟨D y ⟩ σ ϕ∥ S ≤N,D ≲ 1, we have ∥I Π hi,lo,lo (F, P >N γ v, ϕ) ∥ Y ν N ≲ T 1 2 N ν-σ ′ -γν+δ ∥⟨D y ⟩ σ ′ F ∥ S N,D ′ ∥⟨D y ⟩ ν v∥ X ≤N,D , (4.7.1) ∥I Π hi,lo,lo (F, P >N γ G, ϕ) ∥ Y ν N ≲ T 1 2 N ν-s-γσ ′ ∥⟨D y ⟩ s F ∥ X N,D ′ ∥⟨D y ⟩ σ ′ G∥ S ≤N,D ′ , (4.7.2) ∥I Π hi,lo,lo (F, F, ϕ) ∥ Y ν N ≲ T 1 2 N ν-s-σ ′ ∥⟨D y ⟩ s F ∥ X N,D ′ ∥⟨D y ⟩ σ ′ F ∥ S N,D ′ , (4.7.3) ∥I Π hi,lo,lo (w, G, ϕ) ∥ Y ν N ≲ T 1 2 ∥⟨D y ⟩ ν w∥ X N,α ∩X ≤N,D ∥⟨D y ⟩ σ ′ G∥ S ≤N,D ′ , (4.7.4) ∥I Π hi,lo,lo (w, v, ϕ) ∥ Y ν N ≲ T 1 2 ∥⟨D y ⟩ ν w∥ X N,α ∩X ≤N,D ∥⟨D y ⟩ σ v∥ S ≤N,D . (4.7.5)

T 1 2 2 N 2 x Ḣ s 2 y

 1222 Proposition 4.7.3 (Trilinear estimates with low-high-low type interactions). Let ϕ be of type v or G, and assume that max(ν -σ ′ , σ) < α < ν, D > α and 2D + s + σ ′ < D ′ . Assuming that ∥⟨D y ⟩ σ ϕ∥ S ≤N,D ≲ 1, we have∥I Π lo,hi,lo w, G, ϕ ∥ Y ν N ≲ ∥⟨D y ⟩ ν w∥ X N,α ∥⟨D y ⟩ σ ′ G∥ S ≤N,D ′ , (4.7.6) ∥I Π lo,hi,lo w, v, ϕ ∥ Y ν N ≲ T 1 2 ∥⟨D y ⟩ σ w∥ S N,α ∥⟨D y ⟩ ν v∥ X ≤N,D , (4.7.7) ∥I Π lo,hi,lo F, G, ϕ ∥ Y ν N ≲ T 1 2 N ν-σ ′ -s ∥⟨D y ⟩ s F ∥ X N,D ′ ∥⟨D y ⟩ σ ′ G∥ S ≤N,D ′ , (4.7.8) ∥I Π lo,hi,lo F, v, ϕ ∥ Y ν N ≲ T 1 -σ ′ +0 ∥⟨D y ⟩ σ ′ F ∥ S N,D ′ ∥⟨D y ⟩ ν v∥ X ≤N,D . (4.7.9) with ∥K ρ ∥ 4 L 4 (Ry) = R 1 (y 2 + ρ 2 ) 2 dy = ρ -3 R 1 (v 2 + 1) 2 dv .This implies that as ρ → 0,∥e itA f ∥ L 4 ([0,1]×R 2 x,y ) = ∥e it∂xx G∥ L 4 ([0,1]×Rx) ∥K ρ ∥ L 4 (Ry) ∼ Cρ -3 4 .Next, let us look at the L component of the Ḣs -norm of f ρ . It reads|K ρ (y + h) -K ρ (y)| 2|h| 1+s dh dy .Since K ρ (y + h) -K ρ (y) = -h (y + h + iρ)(y + iρ) ,

.

  Lemma 4.8.6 (Norm inflation of the lollipop by convolution). Let m 1 , m 2 ∈ N, k ≥ k 0 ≥ 1. If m 1 = m 2 = 0, there holds k-1 l=k 0 ∥ρ εn k * v 0,l ∥ H m 1 If m 1 + 2m 2 ≥ 1, we have k-1 l=k 0

1 2 -

 2 (T 2 ).The cubic Schrödinger equation on the sphere S 2In a series of works, Burq, Gérard and Tzvetkov [BGT02; BGT04; BGT05a; BGT05b; BGT10] investigated the Cauchy problem for the cubic Schrödinger equation on S 2 . They proved the following.Theorem 5.1.1 ([START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schröinger equation on surfaces[END_REF]). The Cauchy problem (NLS) is uniformly locally well-posed in H s (S 2 ) when s > 1 4 . When s ≤ 1 4 , however, instabilities occurs and the flow-map fails to be of class C 3 .

Π

  N u = n : λn≤N (u, φ n ) L 2 (M ) φ nthe orthogonal projection on eigenfunctions with eigenvalues ≤ N . We shall write :|u| 2 u := |u| 2 u -2∥u∥ 2 L 2 (M ) uthe Wick-ordering nonlinearity in a renormalized version of (NLS). Note that we pass from the Wick-ordering equation to (NLS) from a gauge transformation e it∥u∥ 2 L 2 , by using the conservation of the mass. The randomized initial dataϕ ω = n g n (ω) ⟨λ n ⟩ α φ n lies in H α-1 2 -(M) almost-surely.The case of the Gaussian free field, and hence of the Gibbs measure, corresponds to α = 1 2 . Then, the first Picard iterate of ϕ ω readsI M (t, ϕ ω ) = t 0e i(t-t ′ )∆g : |e it ′ ∆g ϕ ω | 2 e it ′ ∆g ϕ ω : dt ′ ,

2 H 2 )E I T 2 β(t, Π N ϕ ω ) 2 H

 2222 Theorem 5.1.2. Fix t ∈ R. There exist N 0 ∈ N and η > 0 such that for all N ≥ N 0 and α ≥ 1 2 ,η|t| ln(N ) ≤ E ∥I S 2 (t, Π N ϕ ω )∥On the other hand, for all ε > 0 there exists C > 0 such that sup N ∈N

f-∆ S 2 2 )

 22 2 = x ∈ R 3 | |ρ| = 1 . With this coordinates system, we have(f | g) L 2 (S 2 ) (θ, φ)g(θ, φ) sin(θ)dθdφ .(5.2.1) Note that this system of coordinate is singular on the axis (Ox 3 ). The Laplace operator on the sphere writes It is a selfadjoint positive operator that admits the following spectral resolution. For n ∈ N and k ∈ -n, . . . , n , we denote by Y n,k the spherical harmonics of degree n and order k, namelyY n,k (θ, φ) = e ikφ v n,k (θ), v n,k (θ) = c n,k L n,k (cos(θ)),where L n,k (cos(θ)) is the associated Legendre function of degree n and order k, and c n,k is a normalization constant.Proposition 5.2.1 (Spherical harmonics, see[START_REF] Stein | Introduction to Fourier Analysis on Euclidean Spaces[END_REF]). Spherical harmonics Y n,k 1≤n ,|k|≤n form an orthonormal basis of L 2 (S 2 ) made of eigenfunctions of the Laplace operator on S 2 . They satisfy-∆ S 2 Y n,k = λ 2 n Y n,k , n ∈ N * , k ∈ {-n, . . . , n},with λ n = n(n + 1). The multiplicity of the eigenvalue λ 2 n is [n] = 2n + 1.

2 .

 2 Lemma 5.2.2. For any x ∈ S 2 there holds|k|≤n |Y n,k (x)| 2 = [n] .

  Gaussian measure on E n . This randomization also yields a Gaussian measure on H α-1 2 -(S 2 ) induced by the random variable ω → ϕ ω = , and without any loss of generality, we do as if λ n = [n] = n. By doing so, the random initial data readsϕ ω =

v

  n,k (y) = cosh(y) -1 v n,k (θ(y)) ,and (5.3.2) yields that v n,k belongs to L 2 (R) with ∞ -∞ | v n,k (y)| 2 dy = π 0 |v n,k (θ)| 2 sin(θ)dθ = 1 .(5.3.3)

  Therefore, there exists 0< C such that for all z ∈ R R |K(y, z)|dy ≤ Cδ -4 R min h (y -z) 2 , h -1 dy ≤ Cδ -4 h |s|≥h ds s 2 + h -1 |s|≤h ds ≤ Cδ -4 .

  sup y∈R R |K(y, z)|dz + sup z∈R R |K(y, z)|dy ≤ Cδ -4 .

  ,α -χ is given byK(y, z) = 1 2πh R χ(y)(ϕ α (z) -ϕ α (y)) ξ 2 + ϕ α (y) e i(y-z)ξ h dξ .Integrating by parts according to (5.3.9) providesK(y, z) = -iφ α (y, z)χ(y) 2π R ∂ ξ 1 ξ 2 + ϕ α (y) e i(y-z)ξ h .Besides, iterating the integration by parts leads to|K(y, z)| ≤ Cδ -4 min{1, h 2 |y -z| 2 } .Hence, sup z∈R R |K(y, z)|dy + sup y∈R R |K(y, z)|dz ≤ Cδ -4 h and it follows from Schur's test that∥R h ∥ L 2 →L 2 ≤ C δ h .This concludes the proof of Lemma 5.3.3 with C δ ≤ Cδ -4 . Now we finish the proof of Proposition 5.3.1. Note that v n,k solves the equation P 0,h v n,k = -(hp 1 (y, hD y ) + h 2 ) v n,k .

|u ω n | 2

 2 u ω n ] = 0 .(5.4.3) Moreover, we see from the definition of the random spherical harmonics e ω n n≥1 thatE[|u ω n | 2 -∥u ω n ∥ 2 L 2 (S 2 ) ] = 0 . (5.4.4)Therefore, we claim that for all t ∈ R,E (I | II) L 2 (S 2 ) = E (I | III) L 2 (S 2 ) = 0.(5.4.5)

2 S E u ω n 1 u ω n 2 u ω n 3 u ω m |u ω m | 2 2 L 2 ( 2 L 2 (

 222222 1≤n 1 ,n 2 ,n 3 ,m≤N n 2 ̸ =n 1 ,n 3 dx = 0 .This proves (5.4.5). We deduce from this thatE I N (t, I) | I N (t, II) L 2 (S 2 ) = E I N (t, I) | I N (t, II) L 2 (S 2 ) = 0 . Thus ∥Π N I S 2 (t, ϕ ω )∥ Ω;H α-1 2 (S 2 ))= ∥I N (t, I) + I N (t, II)∥ Ω;H α-1 2 (S 2 ))

S 2 |x 1 +ix 2 |

 22 2 dx = 1. Then, given 0 ≤ ε ≪ 1 we define S ε = ω ∈ Ω | 18 ≤ |g 1,1 (ω)| 2 , |g 1,-1 (ω)| 2 + |g 1,0 (ω)| 2 ≤ ε .

Write a = 2 S 2 S

 22 |e ω 1 (x)| 2 -∥e ω 1 ∥ L 2 (S 2 ) |Y n,k (x)| 2 dx, b = O(ε) |Y n,k (x)| 2 dx = O(ε) .

n∈Z 2 1 2 e

 22 ⟨n⟩ g ω n e in•x,where ⟨n⟩ = (1 + Q(n)) 1/2, andu ω (t, x) = n∈Z itQ(n) ⟨n⟩ g ω n e in•x

(N 1 N 2 N 3 ) - 2 l∈Z(N (1) ) 2s ⟨l⟩ 2 |n j |∈N j n 2 ̸ 2 1

 22222 =n 1 ,n 3 1 |Q(n 1 -n 2 ,n 2 -n 3 )-l|≤δ(5.5.4)≤(N 1 N 2 N 3 ) -2 (N (1) ) 2s sup l∈Z |n j |∈N j n 2 ̸ =n 1 ,n 3 1 |Q(n 1 -n 2 ,n 2 -n 3 )-l|≤δ , (5.5.5)for some number 0 < δ < 1 to be fixed later. Now let us estimate the quantity (for fixed l ∈ Z and δ < 1/4)M N 1 ,N 2 ,N 3 := |n j |∼N j n 2 ̸ =n 1 ,n 3 1 |Q(n 1 -n 2 ,n 2 -n 3 )-l|≤δ :(5.5.6)• Case 1 N 1 ≫ N 2 , N 3 : in this case, (5.5.6) can be majorized by(N 2 N 3 ) 2 sup n 2 ,n 3 :n 2 ̸ =n 3 n 1 :|n 1 |∼N 1 ,n 1 ̸ =n |Q(n 1 -n 2 ,n 2 -n 3 )-l|≤δ .For fixed n 2 , n 3 , we denote bym 1 = n 1 -n 2 , m 0 = n 2 -n 3 , |m 1 | ∼ N 1 ≫ |m 0 |. It is sufficient to estimate the number of m 1 ∈ Z 2 , |m 1 | ∼ N 1 such that |Q(m 1 , m 0 ) -l| ≤ δ.Denote by m 0 = (ξ 0 , η 0 ), if η 0 = 0 or ξ 0 = 0, Q(m 1 , m 0 ) ∈ β 2 Z or Q(m 1 , m 0 ) ∈ Z, with respectively.

  Contents 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 1.2 Pathological set of data where norm inflation occurs . . . . . . . . . . . . . . . . . . 77 1.2.1 Unstable profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 1.2.2 Pathological set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 1.2.3 Perturbative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 1.3 Generic well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 1.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 1.3.2 Probabilistic local well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . 92 1.3.3 Approximate solutions by convolution and convergence . . . . . . . . . . . . . 95

	1.1 Introduction

Our model will be the semi-linear Schrödinger equation on R 3

  .4.7) It remains to estimate the contributions of the terms on the right-hand side in (2.4.7). Each term Iu is decomposed into Iu = If + Iv. To estimate the terms Iv in L 2 t L 6 x , we use the endpoint Strichartz estimate, while we need Sobolev embedding to estimate Iv in L ∞ t L 6 x , respectively. As for the terms If , we only use the definitions from Section 2.2. This concludes the proof of Lemma 2.4.3.

  Contents 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 3.1.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 3.1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.1.3 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 3.2 Spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 3.2.1 Wave operators and the intertwining property . . . . . . . . . . . . . . . . . . 159 3.2.2 Distorted Fourier multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.2.3 Application to the distorted Littlewood-Paley theory . . . . . . . . . . . . . . 164 3.3 Bilinear and Strichartz estimates in the perturbed case . . . . . . . . . . . . . . . . . 166 3.3.1 Wiener randomization with respect to the distorted Fourier transform . . . . 166 3.3.2 Bilinear estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 3.4 Almost-sure scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 3.4.1 Critical spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3.4.2 Probabilistic nonlinear estimates . . . . . . . . . . . . . . . . . . . . . . . . . 181 3.4.3 Proof of Theorem 3.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3.

  To estimate the others terms in (3.3.27), we shall use Lemma 3.3.5 and exploit the local smoothing effect to get some global in time estimates. For now on we fix an interval I of size |I| ∼ 1, and see how to handle terms I 2 , I 3 and I 4 .Estimate of term I 2 : We use Minkowski and the bilinear estimate (3.3.9) for the free Schrödinger

	equation to get

  . . . . . 202 4.2 Set up and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4.2.1 Dispersive estimates with a derivative loss . . . . . . . . . . . . . . . . . . . . 210 4.2.2 Local existence above the energy space . . . . . . . . . . . . . . . . . . . . . . 211 4.3 Iteration scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 4.3.1 The refined Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 4.3.2 Functional spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 4.3.3 The recurrence and the truncation method from de Bouard and Debussche . 213 4.4 The adapted linear evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 201 4.4.1 Almost frequency localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 4.4.2 Strichartz estimates for the adapted linear evolution . . . . . . . . . . . . . . 219 4.5 Estimates on the nonlinear term w N . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4.5.1 A priori trilinear estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 4.5.2 Contraction mapping argument . . . . . . . . . . . . . . . . . . . . . . . . . . 225 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 4.6.1 Optimizing the constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 4.6.2 Convergence of (u N ) N ∈2 N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 4.6.3 The limit is a strong solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 4.6.4 The H 2s x L 2 y part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 4.6.5 Limit of the whole sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 4.7 Paracontrolled trilinear estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 4.8 Remarks on ill-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 4.8.1 Semilinear ill-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 4.8.2 Pathological set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

	4.1 Introduction

We consider the Cauchy problem at low regularity for the cubic Schrödinger half-wave equation on

R 2 i∂ t u + ∂ xx -|D y | u = µ|u| 2 u , (t, x, y) ∈ R 1+2 , |D y |u(ξ, η) = |η| u(ξ, η) , u t=0 = u 0 , (NLS-HW) with µ ∈ R * .

The study of this equation is motivated by mathematical interests, as a model for understanding weakly dispersive equations that are not completely integrable, but for which some global solutions have unbounded Sobolev norms. Background on the Schrödinger half-wave equation This model was first introduced by Xu [Xu17

  Lemma 4.2.1 (Bernstein -Strichartz estimates). Given a Borel function m : E → C with support in a bounded measurable set E ⊂ R of finite Lebesgue measure |E| < ∞, we define T E to be the linear evolution frequency-localized in E:

	2 p	+	1 q	=	1 2	.	(4.2.2)

,

  We use the orthogonality of the spherical harmonics Y n,k to expand∥Π N (III)∥ 2 L 2 (Ω;H s (S 2 )) =and further expand the square, which after an application of Fubini's theorems leads to∥Π N (III)∥ 2 L 2 (Ω;H s (S 2 )) =Y n,k (x)Y n,k (y) dx dy .

		|k|≤n 1≤n 1 ≤N	n 2s E	N n 1 =1 S 2	|e ω n 1 (x)| 2 e ω 1 n 3α n 1 (x)	Y n,k (x) dx	2
	1≤n≤N |k|≤n	1 1≤n 1 ,n ′	n 2s (n 1 n ′ 1 ) 3α	S 2	E |e ω n 1 (x)| 2 e ω n 1 (x)|e ω n ′ 1	(x)|e ω 1 n ′	(y)
	By rotational invariance the law of e ω n 1 (x) and e ω n ′ 1 independence of e ω n 1 (x) and e ω n ′ 1 (y) when n 1 ̸ = n ′ 1 that (y) are the same. Hence, we obtain from the
	∥Π 1≤n≤N	1≤n 1	n 6α 1 n 2s	E	S 2	|e ω n 1 (x)| 2 e ω n 1 (x)Y n,k (x) dx	2
		|k|≤n
		=
		1≤n 1 1≤n≲n 1
				|k|≤n
							2 H α-1 2 (S 2 )	≤ C.
	Proof. Recall that
		III = -	1≤n 1 ≤N	e itn 2 1	|e ω n 1 | 2 e ω n 1 n 1 3α .

N (III)∥ 2 L 2 (Ω;H s (S 2 )) =

  II)∥Proof. Recall that we defined∥I N (t, II)∥ L 2 (Ω;H s (S 2 )) = E ∥Π N t 0 e i(t-t ′ )∆ 2 S (II)(t ′ )dt ′ ∥ 2

	L 2 (Ω;H α-1 2 (S 2 ))	.	(5.4.6)
		1/2
		H s (S 2 )

  Y n,k (x)dx Y n,k . (5.4.7) By applying the Plancherel formula and using the fact that e it∆ 2 S is a unitary operator on L 2 (S 2 ), we have n 1 ,r (ω)Y n 1 ,r and we develop the absolute value in (5.4.8) :

			2 S (II)(t) =				e it ′ (n 2 -n 2 1 )
						1≤n≤N |k|≤n	1≤n 1 ,n 2 ≤N	|e ω n 2 | 2 -∥e ω n 2 (x)∥ 2 L 2 (S 2 )	e ω n 1 (x)
									S 2	n 2α 2	n α 1
		t
	∥Π N	0	e i(t-t = n≤N |k|≤n	n 2s |	1≤n 1 ,n 2 ≤N	e it(n 2 -n 2 1 ) -1 i(n 2 -n 2 1 ) S 2	|e ω n 2 | 2 -∥e ω n 2 (x)∥ 2 L 2 (S 2 ) n 2α 2	e ω n 1 (x) 1 n α	Y n,k (x)dx| 2 . (5.4.8)
	Then, we expand e ω n 1 =	1 1 n 2 1 |r|≤n 1
	| ∼ 1≤n 1 ,n 2 ≤N 1≤n 1 ,n 2 ,n ′ e it(n 2 -n 2 1 ) -1 i(n 2 -n 2 1 ) S 2 1 1 ,n ′ 1 |r|≤n 1 , |r ′ |≤n ′ 2 ≤N ⟨n -n 1 ⟩ 2	|e ω n 2 | 2 -∥e ω n 2 (x)∥ 2 L 2 (S 2 ) n 2α 2 |e ω n 2 | 2 -∥e ω n 2 (x)∥ 2 L 2 (S 2 ) e ω n 1 (x) n α 1 g n 2 ,r (ω) Y n,k (x)dx| 2 S 2 1 n 2α 2 n 2

′ )∆ (II)(t)dt∥ 2 L 2 (S 2 ) = ∥Π N t 0 e -it∆ (II)(t)dt∥ 2 H s (S 2 )

g

  ,r (ω)Y n 1 ,r (x)Y n,k (x)dx| 2 L 2 (S 2 ) |Y n,k (x)| 2 dx| 2 , (5.4.10)where we made a crude estimate by keeping in the series the terms where n 1 = n, r = k and n 2 = 1. Moreover, we only keep the terms where 2 ≤ n to exploit the independence between e ω 1 and g n,k (ω). Since the g n,k are standard Gaussian variables, we haveE |g n,k (ω)| 2 = 1 and taking s = α -1 2 yields ∥I N (t, II)∥ 2 L 2 (Ω;H s (S 2 )) ≳ -∥e ω 1 ∥ 2 L 2 (S 2 ) |Y n,k (x)| 2 dx| 2 .(5.4.11)

	1≤n≤N |k|≤n	1≤n 1 ,n 2 ≤N |r|≤n 1 g n 1 ≳ n 2s n 2(α+ 1 2 ) 1 1 n 2 -n 2 1 2 E | S 2 |e ω n 2 | 2 -∥e ω n 2 ∥ 2 L 2 (S 2 ) n 2α 2 2≤n≤N n 2(s-α-1 2 ) E |g n,k (ω)| 2 | S 2 |e ω 1 | 2 -∥e ω 1 ∥ 2
		|k|≤n
		2≤n≤N	1 n 2 E |	S 2	|e ω 1 | 2
		|k|≤n

  Then, for ω ∈ S ε the inequality |a+ b| 2 ≥ -∥e ω 1 ∥ L 2 (S 2 ) |Y n,k (x)| 2 | 2 |x 1 + ix 2 | 2 -1 |Y n,k (x)| 2 dx| 2By collecting estimates (5.4.11) and (5.4.15), we obtain that∥I N (t, II)∥ 2 L 2 (Ω;H s (S 2 )) ≳E 1 Sε(ω) ∥Π N t 0 e i(t-t ′ )∆ S 2 (II)(t ′ )dt ′ ∥ 2 L 2 (S 2 ) -∥e ω 1 ∥ 2 L 2 (S 2 ) |Y n,k (x)| 2 dx| 2 .It follows from (5.4.15) that ∥I N (t, II)∥ 2 L 2 (Ω;H s (S 2 )) ≳ αP(S ε )

	|	S	2	|a| 2 2 n 2 E 1 Sε(ω) | |g 1,1 (ω)| 2 S 18 | 2 2≤n≤N |k|≤n 1 (n,k)∈Λ N 0 1 (x)| 2 ≥ |e ω ≳ n≤N 1 n 2 E 1 Sε(ω) | β 2 ≳	-|b| 2 yields S 2 |e ω 1 | 2 -∥e ω 1 ∥ 2 L 2 (S 2 ) |Y n,k (x)| 2 dx| 2 S 2 |e ω 1 | 2 (n,k)∈Λ N 0 n≤N 1 n 2 ≳ αP(S ε )δ 2 N 0 ≤n≤N 1 n .

-O ε 2 | 2 S |Y n,k (x)| 2 dx| 2 .

(5.4.12)

For the cubic case, the Wick-ordering reads : |u| 2 u := |u| 2 u -2∥u∥ 2 L 2 u. We use a gauge transformation and the conservation of the L 2 -mass to pass from the cubic nonlinearity, to the Wick-ordering nonlinearity. See, e.g.,[START_REF] Oh | A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations[END_REF] for discussions on Wick-ordering renormalization in the context of dispersive equations.

We denote the paracontrolled term by C referring to the common terminology used in[START_REF] Deng | Invariant Gibbs measures and global strong solutions for nonlinear Schröodinger equations in dimension two[END_REF], which denotes the term C the colored Gaussian variables, alluding to the Gaussian structure of the linear evolution f ω , altered by a low-frequency potential.

We say that a function g is paracontrolled by f when there exists a function h such that g = Π hi f Π lo h + a smoother remainder.

-s φ(nx) , v ε n (0) := ρ ε * v n (0) ,

This reminds the probabilistic nonlinear smoothing. . . It is also a manifestation of the semi-linearity of the equation. For oscillating data, this means that the nonlinear solution is, in some sense, well-approximated by the linear evolution of the same initial data (in contrast with scattering results where the final state is different). We refer, e.g., to[START_REF] Burq | High frequency solutions of the nonlinear Schrödinger equation on surfaces[END_REF] for a precise discussion on this aspect of semi-linear problems.

This in sharp contrast with the problems addressed in Part III, where the structure of the adapted linear evolution deteriorates as time evolves. Hence, it is not clear how to iterate the refined paracontrolled probabilistic Cauchy theory, except when there is an invariant measure, for which the globalization argument of Bourgain adapts. (See section 0.1.3)

Notice that we have a power of N . Understanding how to use the subcritical-scaling in H σ , would enable us to assume that E ≲ 1 and certainly lower the threshold for s .

In this argument, we use V 2 instead of U 2 for subtle technical reasons. From Proposition 3.4.12, the weighted space continuously embeds into U 2 , and therefore, by duality, it is the V 2 -norm of the Duhamel term that captures the local smoothing effect. However, it is not clear that doing things the other way around would work.

Another strategy, used by[START_REF] Gassot | Probabilistic local well-posedness for the Schrödinger equation posed for the Grushin Laplacian[END_REF] for the Grushin Laplacian, would be to cook up a randomization procedure that penalizes some problematic features of the equation, and to remove therefore the singular interactions.

The study of such a model is rather driven by the mathematical analysis of weakly dispersive equations with anisotropy. Let us note however that (NLS-HW) can be seen as a special case of the fractional Ginzburg-Laudau equation in media with fractal properties[START_REF] Tarasov | Psi-series solution of fractional Ginzburg-Landau equation[END_REF].

By randomized initial data we suggest that the associated measure in H s is not degenerate. Indeed, there exist randomization procedures that allow one to gain derivatives in L p -spaces, and this is not what we want here.

Still, we only prove this convergence for dyadic truncation, and it is not easy to generalize this convergence for any truncation. Sun and Tzvetkov did it in[START_REF] Sun | Refined probabilistic global well-posedness for the weakly dispersive NLS[END_REF].

This is a major obstruction in the probabilistic Cauchy theory, and we need to better understand the random structure of the colored terms in order to go all the way down to the support of the Gibbs measure.

This statement is sometimes referred as conditional uniqueness, whereas unconditional uniqueness in H s (M ) holds when it is possible to prove uniqueness in C([t 0 , T ], H s ) and not just in X T .

If p is not an odd integer we further assume that ⌊s⌋ + 1 < p, so that the nonlinearity makes sense in H s .

In[START_REF] Lindblad | A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations[END_REF], the norm inflation is due to another concentration phenomenon related with the Lorentz invariance of the wave equation.

This assumption is technical, and might be removed. It guarantees that the bubbles constructed in paragraph 1.2.1 are C ∞ .

We refer to[START_REF] Colliander | Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R 3[END_REF] for the proof of global well-posedness and scattering in the energy space of the quintic (p = 5) defocusing Schrödinger equation.

Note that when s = 0 the lower bound goes to zero as n goes to infinity. This is in agreement with the mass conservation.

The lower bound on s stems from the contributions of the high-high-low and of the high-low-low interactions in the first Duhamel iteration. Besides, by refining the recentering method in[START_REF] Bényi | Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on R 3[END_REF], the regularity threshold was lowered to 1 5 < s. In[START_REF] Dodson | Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation[END_REF], a different approach based on local smoothing in lateral spaces improves the regularity threshold in dimension 4, and can certainly be adapted in dimension 3. Note also that s needs to be strictly grater than the threshold because of the summation over dyadic blocks.

In order to have a reasonable measure, we ask for some non-degeneracy conditions on f 0 . Namely, we want to ensure that supp µ is dense in H s , and that µ(H σ \ H s ) = 0 for all σ > s. To do so, it is sufficient to assume that ∥P 1,k f 0 ∥ L 2 > 0 for all k ∈ R 3 . We refer to Lemma B.1 in[START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF] and[START_REF] Bényi | Higher order expansions for the probabilistic local Cauchy theory of the cubic nonlinear Schrödinger equation on R 3[END_REF] for further details.

Note that the only features of this specific approximate identity by convolution we used is that it commutes with e it∆ , and that it is uniformly bounded in L p -spaces.

Some stronger restrictions will appear in the analysis, and we shall only use (2.1.7) when σ > 6 7 . The high-low method from Bourgain[START_REF] Bourgain | Scattering in the energy space and below for 3D NLS[END_REF] already settle global existence at these range of regularities, and the original version of the I-method from[START_REF] Colliander | Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R 3[END_REF] yields the global spacetime estimates (2.1.7) and scattering.

One can also perform an extra microlocal randomization to decouple the orientation of each block that appears in (2.1.8). For instance, one can multiply in the physical space each block by some independent Gaussian vectors on the sphere, and make therefore the initial data as generic as possible.

Note that u, hence w, is in X σ (J). This follows from the uniqueness part of Theorem 2.1.2 combined with a contraction mapping argument in X σ (J). Moreover, ∥u∥ X σ (J) ≤ CE.

Note that (4, 3) is an admissible pair. Hence, ∥f ∥ L 4t L 3 x (R) ≤ C∥f 0 ∥ L 2x for every ω.

We also have similar terms, but the analysis does not depend on the complex conjugates at this stage.

we sum over the indices, and∂ j f = 3 j=1 ∂ xj f .

The proof provides the same result for the operator ⟨ √ H⟩ s ⟨ √ H 0 ⟩ -s .

Here, f (resp. g) has to be thought of as a nonlinear term controlled in the critical space (resp. a linear and localized term controlled in a weighted L 2 space).

In this case, note that the estimates do not depend on T .

We emphasize that the so-called linear part of the solution u ω (t) = e -itH u ω 0 lies in H c (0) at each time t, since u 0 is in H c (0) and the randomization preserves this space, as well as the flow e -itH .

We recall that δ ′ < δ is chosen such that ∥ψ∥ H s ≤ δ. This forces the solution to remain small and allows us to apply Lemma 3.5.3 and Lemma 3.5.4.

+ F N + w N . The rest of the argument follows as above.

This change of variable can be deduced from the requirement of sin θ∂ θ = -∂ y .

Remerciements

Acknowledgements

The authors would like to thank Chenmin Sun and Nikolay Tzvetkov for presenting them the problem and the "tanghuru" construction. They also thank Nicolas Burq and Frédéric Rousset for helpful discussions, and Rémi Carles and Tristan Robert for relevant references.

Acknowledgements

The author would like to thank his thesis supervisors Nicolas Burq and Frederic Rousset for suggesting him this problem and for giving him many helpful comments related to this work.

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1929284 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the program "Hamiltonian Methods in Dispersive and Wave Evolution Equations". The authors would like to thank the organizers of the program, and all the staff at ICERM for their great hospitality. S. Ibrahim is supported by the NSERC grant No. 371637-2019.

Pathological set of data where norm inflation occurs

The purpose of this section is to prove Theorem 1.1.4. First, we follow the strategy from [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] and we study the unstable profile, or the bubble, regularized by an approximate identity. Our key observation can be stated as follows. When convoluted with an approximated identity (ρ ε ), a Moreover, given a and b two symbols in the class S(1) there exists a symbol c also in S(1) such that Op h (a) • Op h (b) = Op h (c). In addition, c has an explicit semiclassical asymptotic expansion of the form (3.3.15)

We can now state an approximation result of spectral multipliers by pseudo-differential operators. This result comes from [START_REF] Robert | Autour de l'Approximation Semi-Classique[END_REF], and was extended to more general multipliers in [START_REF] Sogge | Fourier Integrals in Classical Analysis[END_REF]. We chose to write a version presented in the context of compact manifolds in [START_REF] Burq | Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds[END_REF], Proposition 2.1. The proof relies on the so-called Helffer-Sjöstrand formula and can be rewritten in our setting with no difficulty.

Lemma 3.3.4. Let η in C ∞ c (R) be a smooth cutoff function. There exists a sequence of symbols (c j ) j≥0 in S(1) such that for every h ∈ (0 , 1], n ∈ N and 0 ≤ σ ≤ n,

(3.3.16)

Besides, for (x, ξ) ∈ R 2d c 0 (x, ξ) = η(|ξ| 2 ), c j (x, ξ) = k≥2 1 (k -1)! η (k-1) (|ξ| 2 )q j,k (x, ξ) ,

where the finite sum representing c j is made of functions q j,k which are polynomials of degree less than 2(k -1) in the frequency variable ξ, and they are Schwartz functions with respect to the space variable x.

We emphasize that supp c j ⊆ (x, ξ) ∈ R 2d | |ξ| 2 ∈ supp η . Hence, we can approximate ∆ N := φ(N -1 H) by pseudo-differential operators supported on frequencies of size N . Since the Fourier multiplier ∆ K := φ(K -1 H 0 ) is already a pseudo-differential operator localized on frequencies of size K, we can use the semiclassical asymptotic expansion (3.3.14) to see that ∆ K ∆ N is negligible when 1 ≪ |K -N |. Namely, for all s, σ ∈ R and 0 < α there exists C s,σ,α such that for all N = 2 n and K = 2 k with 3 ≤ |k -n|, (n,k) .

(3.3.17)

Therefore one can intertwine localization with respect to spectral Littlewood-Paley multiplier for H and H 0 up to a negligible term. The next lemma, written in light of equation (3.3.11), encapsulate the above discussion.

Part III

Quasilinear probabilistic Cauchy theory 4.2 Set up and preliminaries

Since the standard Strichartz estimates for the one-dimensional free Schrödinger evolution provide control on the norm L ∞ x (R) without losing derivatives, we only need to consider randomized initial data in the y-direction as in Figure 4.2. We fix φ ∈ C ∞ c (R, [0, 1]) supported in [-1, 1] and equal to 1 on [-1 2 , 1 2 ] such that {φ(• -k) : k ∈ Z} form a partition of unity, and set ψ(x) = φ(x) -φ x 2 . For dyadic M , the spectral projector P M,k around k ∈ Z of size M for y-frequencies is defined as follows (see Figure 4.2). For f 0 ∈ L 2 (R 2 ) we set

Given a dyadic integer N ≥ 1, P N := P N,0 denotes the Littlewood-Paley projector at frequencies N 2 ≤ η ≤ 2N in the y-direction only, and P N denotes a similar fattened projector using a function ψ with a slightly larger support. Let (g k ) k be a sequence of complex-valued independent normalized Gaussian variables. Given f 0 ∈ H s (R 2 ), we associate a random function obtained by the Wiener randomization

This randomization induces a non-degenerate probability measure on H s (R 2 ), densely supported (see [START_REF] Burq | Random data Cauchy theory for supercritical wave equations I: local theory[END_REF] Lemma B.1). In addition, for and N ∈ 2 N , we set

Finally, we emphasize that the family

2M

Figure 4.2: Support of the projection P M,k in the frequency space (R ξ , R η ).

Local existence above the energy space

We recall the best know local well-posedness result for equation (NLS-HW).

Proposition 4.2.2 (Local well-posedness above the energy space, [BIK20] Theorem 1.6). Let s > 1 2 . For every f 0 ∈ H s (R 2 ), there exists T = T (∥f 0 ∥ H s ) > 0 such that equation (NLS-HW) admits a unique local solution in C((-T, T ), H s ).

It is shown in [START_REF] Bahri | Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations[END_REF] that local well-posedness actually holds in L 2

x H s y . The proof follows from a fixed point argument in

), for some T > 0 depending on the H s (R 2 )-norm of the initial data. The use of the Strichartz space

)) requires that s > 1 2 because of the lack of dispersion in the y-direction. Consequently, we have no control on the growth of the H s -norm of the solution since we do not have access to the conserved energy, which is at the level of H

Iteration scheme

The refined Ansatz

We construct the solution to (NLS-HW) by iteration on the frequencies, that we regroup as dyadic packets. Let N ∈ 2 N , and let u N be the maximal solution 1 to (NLS-HW) with initial data localized at frequencies at most N

Then, we consider at each step n the adapted linear-nonlinear decomposition v N = F N + w N , defined as follows.

• The adapted linear evolution F N is solution to a linear Schrödinger half-wave equation that encapsulates the bad nonlinear interactions, with initial data P N f ω 0 localized at frequency ∼ N :

The parameter 0 < γ < 1, to be determined later in the analysis, separates two ranges of frequency scales N ≫ N γ .

1 Observe that the local well-posedness result from [START_REF] Bahri | Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations[END_REF], recalled in Proposition 4.2.2, implies existence and uniqueness for u N in H 1 2 + (R 2 ) up to a time T (n) > 0. In Theorem 4.1.1 we claim that (u N ) converges almost-surely in H s (R 2 ) to a strong solution to (NLS-HW) up to a time T ω > 0 almost-surely in the randomization.

w M,θ to be determined later as the solutions to some equation. The cutoff functions are denoted

We define by induction F N,θ and w N,θ as solutions to the following truncated equations. First,

and similarly, the blockF N,k,θ is solution to the decomposed truncated equation

Then, the truncated nonlinear remainder w N,θ solves

where the nonlinearity N θ (w N,θ ) is given by

Finally, we denote the truncated approximate solution

We prove the convergence of (u N,θ ) N on a time-interval [-T 0 , T 0 ]. Subsequently, we use a priori estimates and a continuity argument to remove the truncation, and to get a solution to the original equation (4.3.1) on a sub-interval [-T ω , T ω ] with 0 < T ω ≤ T 0 almost-surely. This is the matter of Section 4.6, once we solved the fixed point for the truncated equations and proved the convergence of the approximate solutions (u N,θ ) N in L 2 (Ω; E). there holds

and the term M σ ′ are absorbed by the factor N M

-10(D ′ +1) : we can sum over M to get

This concludes the proof of the refined Strichartz estimates for the unit blocks F N,k .

Next, we infer probabilistic Strichartz estimates on the whole adapted linear evolution F N from the above estimate on the individual blocks F N,k , combined with probabilistic decoupling (we recall that the high frequencies of the initial data are independent of the low frequencies). The proof follows the same lines as in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], Proposition 4.4. Proposition 4.4.4 (Probabilistic Strichartz estimates). Let F N be a solution to the truncated equation (4.3.7), s > 0, D ′ > 0 and T 0 be as in Lemmas 4.4.1 and 4.4.2. There holds that for all 0 < T ≤ T 0 , p ≥ 1 and σ ′ > 0,

where δ = γ(σ -ρ) > 0.

The contributions to the power of N can be explained as follows:

Strichartz with freq. localization

• N -γσ gain frequency localization .

Proof. The idea is to use the independence between

-measurable thanks to the assumptions on ϕ. In order to use Minkowski's inequality in the large deviation estimates, we first prove the result with the norm S r N,D ′ with r ≫ 1 instead of S N,D ′ (which denotes S ∞ N,D ′ ). We condition on B N

2

, take p ≥ r and deduce from Minkowski's inequality that

A priori trilinear estimates

We denote the Duhamel integral by where we recall that N θ was defined in (4.3.11). Applying the trilinear estimates from Section 4.7, together with the truncation of the nonlinearity, we obtain the following a priori estimates on the nonlinear component w N .

Proposition 4.5.2 (A priori estimate on w N ). Under the same constraints on the parameters as in the statement of Proposition 4.5.1, for 0 ≤ T ≤ T 0 , there holds

Let us briefly comment the proof of the above estimate. Notice that the new terms of the approximate solution u N constructed at step N , which are either F := F N or w := w n , appear at least once in the nonlinear forcing term (because N (u N

2

) is removed from the nonlinearity). Then, the strategy to obtain optimal a priori estimates on w N is to use the T T * estimate from Lemma 4.2.1 and the refined probabilistic estimate from Proposition 4.4.4. Specifically, when we have trilinear mixed terms (i.e some functions of type F N interact with others of type w N ), we place F N in L ∞ y and w n in L 2 y , respectively. Surprisingly, we do so even if F N has the highest frequency, in order to reduce the loss of derivatives from the deterministic Strichartz estimates of Lemma 4.2.1. This is implemented precisely in the paracontrolled trilinear estimates collected in Section 4.7.

Proof. Up to permutation, we always assume that a term of type w N or F N is at the first position. We collect the different contributions depending on the nature of the terms. We place the terms u N/2 (denoted ϕ in Section 4.7) in the last position. Thanks to the truncation, they satisfy

We have from (4.7.3), (4.7.8), (4.7.12) and (4.7.16) that

Contribution of w N F N F N . We have from (4.7.4) and (4.7.6), (4.7.10), (4.7.14) that

Contribution of w N w N F N . We have from (4.7.5), (4.7.7), (4.7.11) and (4.7.15) that

Contribution of w N w N w N . We have from (4.7.5), (4.7.7), (4.7.11) and (4.7.15) that

We have from (4.7.4), (4.7.6), (4.7.10), (4.7.14) or from (4.7.5), (4.7.7), (4.7.11), (4.7.15) depending on the nature of the first factor u N

We have from (4.7.5), (4.7.6), (4.7.10) and (4.7.14) that

We have from (4.7.4), (4.7.6), (4.7.10) and (4.7.14) that

We have from (4.7.3), (4.7.8), (4.7.12) and (4.7.16) that

. It follows from the definition of the adapted linear evolution F N that these terms only contribute to the equation satisfied by w N when one of the two terms u N/2 is truncated at frequencies > N γ . We split this term into

There holds

Indeed, we use (4.7.1) for the high-low-low -type interactions. For the other interactions, we use (4.7.8), (4.7.12), (4.7.16) when u N 2 is of type G, and (4.7.9), (4.7.13), (4.7.17) when u N 2 is of type v, respectively. Similarly, we have

which follows from (4.7.2) for the high-low-low type interactions, and (4.7.8), (4.7.12), (4.7.16), or (4.7.9), (4.7.13), (4.7.17) depending on the nature of u N

Limit of the whole sequence

Once the sequence (u N ) N ∈2 N is constructed and its convergence established, we define the general approximating sequence (u n ) n∈N by iteration: given n ∈ N and N such that N 2 < n ≤ N , we set

where the adapted linear evolution F n solves

with initial data

Note that (u N ) N is a subsequence of (u n ) n . The truncated versions F n,θ and w n,θ are defined as in (4.3.7) and (4.3.10), respectively. For fixed n ∈ N and N ∈ 2 N such that N 2 < n ≤ N , we can prove the same estimates for F n,θ and w n,θ that we get for F N,θ (in Section 4.4) and w N,θ (in Section 4.5). We deduce from these estimates and from the convergence of (u N,θ ) N ∈2 N that (u n,θ ) n∈N is a Cauchy sequence. Finally we conclude that the whole sequence (u n ) n is convergent to the limit u of (u N ) N in X s,σ T 0 , moreover it is a solution to equation (NLS-HW) with initial data P ≤n f ω 0 on [-T ω , T ω ] by construction.

Paracontrolled trilinear estimates

In order to establish the trilinear estimates, we will perform a Littlewood-Payley decomposition. After we cut in frequencies, we use in various situations the following decomposition.

Lemma 4.7.1. For every ϕ (1) , ϕ (2) , ϕ (3) such that the upper bound is finite, we have

Proof of Lemma 4.7.1. This follows from the definition of the norms in Section 4.3. To control the Strichartz-norms we use the T T * -Bernstein-Strichartz estimate from Lemma 4.2.1 with (p, q) = (4, ∞), and either (p, q) = (∞, 2) and r = 2 or (p, q) = (8, 4) and r = ∞ respectively. We have the yfrequency localization on an interval E = M 2 ≤ |η| ≤ 2M of length |E| ≲ M . Moreover, we use that ν = σ + 1 2 + 0 in the second case.

We later use this Lemma for some given functions ϕ (1) , ϕ (2) , ϕ (3) of type F, G, w, v, where

Next, we perform a Littlewood-Paley decomposition of each function, and we conduct a case-bycase analysis of each paracontrolled term defined below, depending on the nature of the functions Proposition 4.7.4 (Trilinear estimates with low-high-high type interactions). Let ϕ be of type v or G, and assume that max(ν -σ ′ , σ) < α < ν and 2D+s+σ ′ < D ′ . Assuming that ∥⟨D y ⟩ σ ϕ∥ S ≤N,D ≲ 1, we have

(4.7.13) Proposition 4.7.5 (Trilinear estimates with high-high interactions type). Let ϕ be of type v or G, and assume that max(ν -σ ′ , σ) < α < ν and 2D+ν +s+σ ′ < D ′ . Assuming that ∥⟨D y ⟩ σ ϕ∥ S ≤N,D ≲ 1, we have

The general strategy is as follows. The first term ϕ (1) is of type w or F . Hence, it is always localized at frequencies ∼ N , in the sense that they belong to the spaces X N,α and S N,α . When it is of type F , and the other terms are of type w, we place F it in L ∞ to exploit the improved stochastic bounds from Proposition 4.4.4.

Proof of Proposition 4.7.2. It is sufficient to put the third term in S ≤N,D with σ derivatives, and to perform the same analysis as in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF] Section 7 to handle the first two terms. We only detail the first case.

•Proof of (4.7.1). For fixed M ≥ 1, we have

y . Therefore, we obtain from the definitions of the norms and the assumption

It remains to multiply by C N,α (M ) + C ≤N,D (M ), and to sum over M . Since D > α, we get

Similarly, in all the other high-low-low -type frequency interactions estimated in Proposition 4.7.2, we put the third term in L 8 t L 4

x L ∞ y , and we gain nothing from it. As for the quadratic interaction of the first two terms, the analysis is exactly as in [START_REF] Bringmann | Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation[END_REF], and we do not write it here.

Proof of Proposition 4.7.3. For fixed M ∈ 2 N , note that we have by definition

•Proof of (4.7.6). In this case, ϕ (2) = G and ϕ (1) = w. On the other hand, the second term ϕ (3) := ϕ, localized at frequencies ≲ N , has no importance, and we always place it in L ∞ with σ derivatives in y. We do not gain anything from it. There holds

Note that we used the assumption α < ν to obtain

Then, we multiply by C ≤N,D (M )+C N,α (M ) M ν , and using that α > ν -σ ′ , D > α and D ′ > D +ν, we sum over M to conclude

•Proof of (4.7.8). We proceed similarly in this case, where ϕ (1) = F and ϕ (2) = G. Once again, we use the localization of F at frequencies ∼ N , which is stronger than the localization of w. There holds

Then, we observe that

, provided D ′ > 2D + ν + σ ′ + s. Hence, we can multiply by C ≤N,D (M ) + C N,α (M ) M ν , sum over M and obtain (4.7.8).

•Proof of (4.7.7). This case is easier since the derivatives fall onto the term v which is in H ν . We have ϕ (2) = v and ϕ (1) = w. We place the high frequency term in L 2 in order to absorb M σ , and we end up with

Since α > σ, we observe that the sum over N 1 is bounded by N -α M α-σ when M ≲ N and N -σ when M ≫ N , so that

.

We deduce (4.7.7) after multiplying by C ≤N,D (M ) + C N,α (M ) and summing over M , making the norm ∥⟨D y ⟩ ν v∥ X ≤N,D appear.

•Proof of (4.7.9). This proof is identical than the one above. Observe that we only used the localization of w at frequency N in the Strichartz space S N,α , and the localization of F is even better. Indeed, we have

Since D ′ > σ ′ , we have

Since D ′ > σ ′ + α, we deduce (4.7.9) after multiplying by C ≤N,D (M ) + C N,α (M ) and summing over M , making the norm ∥⟨D y ⟩ ν v∥ X ≤N,D appear.

Proof of Proposition 4.7.4. In this situation, inequality (4.7.18) becomes restricted to the indices

•Proof of (4.7.10). Now, we observe that

Then, according to (4.7.19), the sum over N 1 is bounded by N -α . Moreover, one can check that

therefore we have

Estimate (4.7.14) comes from multyplying by C ≤N,D (M ) + C N,α (M ) and summing over M , using that α > ν -σ ′ .

•Proof of (4.7.11). Similarly, we write

Since α > σ, we observe that the sum over N 1 is bounded by N M -α M -σ when M ≲ N and N -σ when M ≫ N . Moreover, the sum over N 2 is bounded as in (4.7.20) by replacing D ′ by D and σ ′ -0 by ν, which leads to

It remains to multiply by C ≤N,D (M ) + C N,α (M ) and sum over M to get the norm ∥⟨D y ⟩ ν v∥ X ≤N,D .

•Proof of (4.7.12). We adopt the same strategy but we put the first term in the adapted space X N,D ′ after taking s derivatives:

M -s when M ≲ N and N -s when M ≫ N . Moreover, the sum over N 2 is bounded as in (4.7.20) and we get

•Proof of (4.7.13). Finally, we put the term F in S N,D ′ :

The sum over N 1 is bounded as above by N M -D ′ M -σ ′ when M ≲ N and N -σ ′ when M ≫ N , and the sum over N 2 is bounded as in (4.7.20) by

Proof of Proposition 4.7.5. In this situation, inequality (4.7.18) becomes restricted to the indices

•Proof of (4.7.14). Now, we observe that

Then, since N 2 ∼ N 1 and α < ν, we observe that

Estimate (4.7.14) comes from multyplying by C ≤N,D (M ) + C N,α (M ) and summing over M .

•Proof of (4.7.15). The proof is identical as the proof of (4.7.14), indeed we only need to replace G by v, D ′ by D dans σ ′ by σ in the argument.

•Proof of (4.7.16). We adopt the same strategy but we put the first term in the adapted space X N,D ′ after taking s derivatives:

Then we have

After multiplying by C ≤N,D (M ) + C N,α (M ) the sum over M is bounded by N ν-s-σ ′ since 2D ′ > D + ν -s -σ ′ by assumption.

•Proof of (4.7.17). Finally, we put the term F in S N,D ′ :

Then we have

After multiplying by C ≤N,D (M ) + C N,α (M ) the sum over M is bounded by N -σ ′ .

Remarks on ill-posedness 4.8.1 Semilinear ill-posedness

From the traveling wave profiles for the Szegő equation, we cook up a one-parameter family of profiles from which we deduce that the bilinear estimate only holds when s ≥ 1 2 . Then, we proceed as in Remark 2.12 from [START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schröinger equation on surfaces[END_REF] to deduce that the flow map cannot be of class C 3 at the origin when 1 4 < s < 1 2 . We recall that

Theorem 4.8.1 (Semilinear ill-posedness). If there exists a local in time flow map on H s with regularity C 3 at the origin, then s ≥ 1 2 .

Proof. As a corollary of [START_REF] Burq | Bilinear eigenfunction estimates and the nonlinear Schröinger equation on surfaces[END_REF], Remark 2.12, if there exists a C 3 local in time flow map at the vicinity of the origin in the space H s , then the following bilinear estimate holds:

As usual, we use a one-parameter family of stationary solutions to evidence the instabilities of the flow-map. Namely, we consider a Gaussian distribution G, a family of traveling waves profiles for the Szegő equation K ρ (y), for ρ ∈ (0 , +∞), and set

First, let us look at the scaling of L 4 t,x -norm of f ρ as ρ goes to 0. Since K ρ is a traveling wave for the Szegő equation on the line [START_REF] Pocovnicu | Traveling waves for the cubic Szegő equation on the real line[END_REF], and in particular a traveling wave for equation (NLS-HW), one can see that for every t, there holds

Theorem 4.8.2 (Norm inflation [START_REF] Kato | Ill-posedness for the Half wave Schrödinger equation[END_REF]). Let s < 1 4 . For every bounded set B of H s and for every T > 0, the flow map cannot be extended as a continuous map from B to C([-T, T ], H s ). More precisely, there exists a sequence (t n ) n∈N of positive numbers tending to zero and a sequence (u N (t)) n≫1 of C ∞ (R 2 ) solutions of (NLS-HW) defined for t ∈ [0, t n ], and as n → ∞ ∥u n (0)∥ H s → 0 ,

Non-uniform continuity of the flow map for s = 1 4 . has also been investigated in [START_REF] Kato | Ill-posedness for the Half wave Schrödinger equation[END_REF]. Using the method of Sun and Tzvetkov [START_REF] Sun | Concerning the pathological set in the context of probabilistic well-posedness[END_REF] adapted to Schrödinger-type equations in [START_REF] Camps | Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations[END_REF], we make precise this norm inflation result in the context of probabilistic well-posedness. We fix ρ ∈ C ∞ c (R 2 ), valued in [0, 1], such that R 2 ρ(x) dx = 1 and ρ vanishes for x 2 + y 2 ≥ 1 10 4 . Due to the anisotropy, we define an approximate identity (ρ ε ) ε>0 of the form

Theorem 4.8.3 (Generic ill-posedness for (NLS-HW)). Let s < 1/2. There exists a dense set S ⊂ X s such that for every f ∈ S, the family of local solutions (u ε ) ε>0 of (NLS-HW) with initial data ρ ε * f does not converge. More precisely, there exist ε n → 0 and t n → 0 such that u εn (t n ) is well-defined and lim

Since the argument is the same as in [START_REF] Camps | Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations[END_REF], we only make a sketch of the proof adapted to the anisotropy. We construct unstable profiles with growing H s norm based on the bounded time-periodic solution V (t) = e it to the ODE

), radial such that 0 ≤ φ ≤ 1. Let γ to be chosen later, and

We define v n (0, x, y) := λ n φ(nx, n 2 y) .

We recall that

We now fix the parameters

Lemma 2.1 in [START_REF] Camps | Pathological set of initial data for scaling-supercritical nonlinear Schrödinger equations[END_REF] becomes as follows.

Lemma 4.8.4 (Growth of the ODE profile). Let 0 ≤ s < 1 2 .

1. For every ε ≤ ε n , there holds

2. For every k, l ∈ N, there exists C > 0 such that for every n ∈ N, t ∈ R and ε > 0,

3. Moreover, with the same notation, we have

Let us denote by t → Φ(t)(f ) the local maximal solution to (NLS-HW) with initial data f in H m when m > 1 2 . In order to define a dense subset S of the pathological set

we apply the "tanghuru" construction. The k-th bubble v 0,k is centered at any point y k ∈ R 2 , with scaling parameter n k = e a k for some 1 ≪ a:

Then for k, l ≥ k 0 , we have that v 0,k and v 0,l have disjoint supports with radius of respective orders

Applying the convolution by the function ρ εn k which has a support of size ε -1

Definition 4.8.5 (Dense subset of the pathological set). We denote by S the set of initial data f which can be decomposed under the form

Introduction

We consider the cubic Schrödinger equation on a compact surface without boundary (M, g), which reads

The dynamics associated with (NLS) formally conserves the energy and the mass of the solution

and

The Gibbs measure, which is formally associated with the Hamiltonian structure of (NLS), reads dµ(u) = exp (-H(u)) du .

From the Liouville theorem and the conservation of the Hamiltonian, we expect the Gibbs measure to be invariant under the dynamics of (NLS). In particular, by using the so-called Bourgain globalization argument from [START_REF] Bourgain | Periodic nonlinear Schrödinger equation and invariant measures[END_REF], we can deduce global existence of solutions and Poincaré recurrence property from a local Cauchy theory on the support of the Gibbs measure. In order to make sense of this measure, and to realize a typical function on the support of this measure, we see it as a density measure with respect to the Gaussian free field:

The Gaussian free field dρ is induced by random series

where φ n n≥0 form an orthonormal basis of L 2 (M ) made eigenfunctions of -∆ g , and {λ 2 n } n≥0 are the associated eigenvalues, possibly with multiplicity

We deduce from (5.3.1), the chain rule and (5.3.2) that v n,k is solution in C ∞ (R) to the equation

(5.3.4) Indeed, sin(θ)∂ θ = -∂ y and it follows from (5.3.1) and from the relation

Then, we obtain (5.3.4) by dividing the above equation by cosh(y) and by noting that [∂ 2 y , cosh(y)] = cosh(y) ′′ + 2 cosh(y) ′ ∂ y = cosh(y) + 2 sinh(y)∂ y .

We introduce the semiclassical parameter h ∼ n -1 and we multiply the above equation by h 2 to get Hence, Char(P h,α ) concentrates near the equator as δ goes to 0. By the use of semiclassical functional calculus, we will deduce from this that v n,k , which is solution to P h,α f = 0 in L 2 (R), also concentrates near the equator as δ goes to 0. Denote by P 0,h,α = p 0 (y, hD y , α). Let us expand the nonlinearity as follows : Remark 5.4.1. We will see that the term II is responsible for the divergence of the first Picard iteration clamied in (5.1.2).We stress out that this term vanishes in the case of the tori since the eigenfunctions of ∆ T 2 are the plain waves e n (x) = e in•x , and satisfy |e n (x)| = 1 for all x ∈ T 2 .

To prove Theorem 5.1.2 we proceed as follows. In Lemma 5.4.2 we use independence to reduce the proof of Theorem 5.1.2 to estimates for the terms II and III. Then, we prove in Lemma 5.4.3 that III is in L 2 (S 2 ) a.s. and uniformly in N . On the other hand, we prove in Proposition 5.4.4 that the quadratic moment of the L 2 (S 2 )-norm of the Duhamel integral for II is bounded from below by ∼ ln(N ) for large N . .

(5.4.2)