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The goal of the current paper is to provide information about the limiting probability of the granular media equation when there are several stable states. Indeed, it has been proved in our previous works \cite{AOP,SPA} that there is convergence. However, very few is known about the limiting probability, even with a small diffusion coefficient. The techniques that we use here are related to the ones about the exit-problem of the associated McKean-Vlasov diffusion.

International audience; The present paper is devoted to the study of a McKean-Vlasov diffusion of the type self-stabilizing. We obtain this model by taking the hydrodynamical limit of a mean-field system of particles. It also corresponds to the probabilistic interpretation of the granular media equation. The main question that we study is the exit-time. We take a confining potential with two wells : a < 0 and b > 0. We start with a deterministic condition x0 > 0 and we show that the first time that this diffusion leaves an interval of the form ]d; +∞[ (d verifying some assumptions) satisfies a Kramers'type law. In other words, this time is exponentially equivalent to exp 2 σ 2 H as the coefficient diffusion σ goes to 0, H being the exit cost. Incidentally, we also prove that the solution of the granular media equation is trapped (for the 2-Wasserstein distance) in a ball around δ b during a time at least exponentially equivalent to exp 2 σ 2 H .

The current work deals with the granular media equation whose probabilistic interpretation is the McKean–Vlasov diffusion. It is well known that the Laplacian provides a regularization of the solution. Indeed, for any t > 0, the solution is absolutely continuous with respect to the Lebesgue measure. It has also been proved that all the moments are bounded for positive t. However, the finiteness of the entropy of the solution is a new result which will be presented here.

We consider the one-dimensional free Fokker–Planck equation ¶ \[\frac{\partial \mu_{t}}{\partial t}=\frac{\partial }{\partial x}[\mu_{t}\cdot (\frac{1}{2}V'-H\mu_{t})],\] where $H$ denotes the Hilbert transform and $V$ is a particular double-well quartic potential, namely $V(x)=\frac{1}{4}x^{4}+\frac{c}{2}x^{2}$, with $c\ge -2$. We prove that the solution $(\mu_{t})_{t\ge 0}$ of this PDE converges in Wasserstein distance of any order $p\ge 1$ to the equilibrium measure $\mu_{V}$ as $t$ goes to infinity. This provides a first result of convergence for this equation in a non-convex setting. The proof involves free probability and complex analysis techniques.

International audience; We are interested in a nonlinear partial differential equation: the gran-ular media one. Thanks to some of our previous results [Tug14a, Tug14b], we know that under easily checked assumptions, there is a unique steady state. We point out that we consider a case in which the confining potential is not globally convex. According to recent articles [Tug13a, Tug13b], we know that there is weak convergence towards this steady state. However , we do not know anything about the rate of convergence. In this paper, we make a first step to this direction by proving a deterministic Kramers'type law concerning the first time that the solution of the gran-ular media equation leaves a local well. In other words, we show that the solution of the granular media equation is trapped around a local minimum during a time exponentially equivalent to exp 2 σ 2 H , H being the so-called exit-cost.

This thesis is devoted to the Fokker-Planck équation partial_t f =∆f + div(E f).It is divided into two parts. The rst part deals with the linear problem. In this part we consider a vector E(x) depending only on x. It is composed of chapters 3, 4 and 5. In chapter 3 we prove that the linear operator Lf :=∆f + div(Ef ) is an in nitesimal generator of a strong continuous semigroup (SL(t))_{t≥0}. We establish also that (SL(t))_{t≥0} is positive and ultracontractive. In chapter 4 we show how an adequate decomposition of the linear operator L allows us to deduce interesting properties for the semigroup (SL(t))_{t≥0}. Indeed using this decomposition we prove that (SL(t))_{t≥0} is a bounded semigroup. In the last chapter of this part we establish that the linear Fokker-Planck admits a unique steady state. Moreover this stationary solution is asymptotically stable.In the nonlinear part we consider a vector eld of the form E(x, f ) := x +nabla (a *f ), where a and f are regular functions. It is composed of two chapters. In chapter 6 we establish that fora in W^{2,infini}_locthe nonlinear problem has a unique local solution in L^2_{K_alpha}(R^d); . To end this part we prove in chapter 7 that the nonlinear problem has a unique global solution in L^2_k(R^d). This solution depends continuously on the data.; Ce mémoire de thèse est consacré à l’équation de Fokker-Planckpartial_ f=∆f+div(Ef).Il est subdivisé en deux parties :une partie linéaire et une partie non linéaire. Dans la partie linéaire on considère un champ de vecteur E(x) dépendant seulement de x. Cette partie est constituée des chapitres 3, 4 et 5. Dans le chapitre 3 on montre que l’opérateur linéaire Lf :=∆ f + div(E f ) est le générateur d’un semi-groupe fortement continu (SL(t))_{t≥0} dans tous les espaces L^p. On y établit également que le semi-groupe (SL(t))_{t≥0} est positif et ultracontractif. Dans le chapitre 4 nous montrons comment est qu’une décomposition adéquate de l’opérateur L permet d’établir certaines propriétés du semi-groupe (SL(t))_{t≥0} notamment sa bornitude. Le chapitre 5 est consacré à l’existence d’un état d’équilibre. De plus on y montre que cet état d’équi- libre est asymptotiquement stable. Dans la partie non linéaire on considère un champ de vecteur de la forme E(x,f) := x+nabla (a*f) ou a et f sont des fonctions assez régulières et * est l’opérateur de convolution. Cette parties est contituée des chapitre 6 et 7. Dans le chapitre 6 nous établissons que poura appartenant à W^{2,infini}_locl’équation de Fokker-Planck non linéaire admet une unique solution locale dans l’espace L^2_{K_alpha} (R^d). Dans le dernier chapitre nous montrons que le problème non linéaire admet une solution globale. De plus cette solution dépend continument des données.

This thesis is devoted to the Fokker-Planck équation partial_t f =∆f + div(E f).It is divided into two parts. The rst part deals with the linear problem. In this part we consider a vector E(x) depending only on x. It is composed of chapters 3, 4 and 5. In chapter 3 we prove that the linear operator Lf :=∆f + div(Ef ) is an in nitesimal generator of a strong continuous semigroup (SL(t))_{t≥0}. We establish also that (SL(t))_{t≥0} is positive and ultracontractive. In chapter 4 we show how an adequate decomposition of the linear operator L allows us to deduce interesting properties for the semigroup (SL(t))_{t≥0}. Indeed using this decomposition we prove that (SL(t))_{t≥0} is a bounded semigroup. In the last chapter of this part we establish that the linear Fokker-Planck admits a unique steady state. Moreover this stationary solution is asymptotically stable.In the nonlinear part we consider a vector eld of the form E(x, f ) := x +nabla (a *f ), where a and f are regular functions. It is composed of two chapters. In chapter 6 we establish that fora in W^{2,infini}_locthe nonlinear problem has a unique local solution in L^2_{K_alpha}(R^d); . To end this part we prove in chapter 7 that the nonlinear problem has a unique global solution in L^2_k(R^d). This solution depends continuously on the data.; Ce mémoire de thèse est consacré à l’équation de Fokker-Planckpartial_ f=∆f+div(Ef).Il est subdivisé en deux parties :une partie linéaire et une partie non linéaire. Dans la partie linéaire on considère un champ de vecteur E(x) dépendant seulement de x. Cette partie est constituée des chapitres 3, 4 et 5. Dans le chapitre 3 on montre que l’opérateur linéaire Lf :=∆ f + div(E f ) est le générateur d’un semi-groupe fortement continu (SL(t))_{t≥0} dans tous les espaces L^p. On y établit également que le semi-groupe (SL(t))_{t≥0} est positif et ultracontractif. Dans le chapitre 4 nous montrons comment est qu’une décomposition adéquate de l’opérateur L permet d’établir certaines propriétés du semi-groupe (SL(t))_{t≥0} notamment sa bornitude. Le chapitre 5 est consacré à l’existence d’un état d’équilibre. De plus on y montre que cet état d’équi- libre est asymptotiquement stable. Dans la partie non linéaire on considère un champ de vecteur de la forme E(x,f) := x+nabla (a*f) ou a et f sont des fonctions assez régulières et * est l’opérateur de convolution. Cette parties est contituée des chapitre 6 et 7. Dans le chapitre 6 nous établissons que poura appartenant à W^{2,infini}_locl’équation de Fokker-Planck non linéaire admet une unique solution locale dans l’espace L^2_{K_alpha} (R^d). Dans le dernier chapitre nous montrons que le problème non linéaire admet une solution globale. De plus cette solution dépend continument des données.

International audience; We consider a diffusion in which the own law of the process appears in the drift, that is a non-linearity in the sense of McKean. This kind of diffusion is obtained as the hydrodynamical limit of a mean-field system of interacting particles. The question that we deal with is the exit-time of such a diffusion when it evolves in a double-wells landscape. This has already been solved for the convex case but the previous methods relie completely on the convexity of the external force. Here, we provide a Kramers'type law for self-stabilizing process directed by a non-convex confining potential.

In this paper, we study the long-time behaviour of solutions to the Vlasov-Fokker-Planck equation where the confining potential is non-convex. This is a nonlocal nonlinear partial differential equation describing the time evolution of the probability distribution of a particle moving under the influence of a non-convex potential, an interaction potential, a friction force and a stochastic force. Using the free-energy approach, we show that under suitable assumptions solutions of the Vlasov-Fokker-Planck equation converge to an invariant probability.

This thesis lies within the field of probability and statistics, and more precisely of random matrix theory. In the first part, we study the large deviations of the spectral measure of covariance matrices XX*, where X is a rectangular random matrix with i.i.d. coefficients having a probability tail like exp(-at^α), α∈]0,2[. We establish a large deviation principle similar to Bordenave and Caputo's one, with speed n^{1+α/2} and explicit rate function involving rectangular free convolution. The proof relies on a quantification result of asymptotic freeness in the information-plus-noise model. The second part of this thesis is devoted to the study of the long-time behaviour of the solution to free Fokker-Planck equation in the setting of the quartic potential V(x) = 1/4 x^4 + c/2 x² with c≥-2. We prove that when t→+∞, the solution µ_t to this partial differential equation converge in Wasserstein distance towards the equilibrium measure associated to the potential V. This result provides a first example of long-time convergence for the solution of granular media equation with a non-convex potential and a logarithmic interaction. Its proof involves in particular free probability techniques.; Cette thèse s'inscrit dans le domaine des probabilités et des statistiques, et plus précisément des matrices aléatoires. Dans la première partie, on étudie les grandes déviations de la mesure spectrale de matrices de covariance XX*, où X est une matrice aléatoire rectangulaire à coefficients i.i.d. ayant une queue de probabilité en exp(-at^α), α∈]0,2[. On établit un principe de grandes déviations analogue à celui de Bordenave et Caputo, de vitesse n^{1+α/2} et de fonction de taux explicite faisant intervenir la convolution libre rectangulaire. La démonstration repose sur un résultat de quantification de la liberté asymptotique dans le modèle information-plus-bruit. La seconde partie de cette thèse est consacrée à l'étude du comportement en temps long de la solution de l'équation de Fokker-Planck libre en présence du potentiel quartique V(x) = 1/4 x^4 + c/2 x² avec c≥-2. On montre que quand t→+∞, la solution µ_t de cette équation aux dérivées partielles converge en distance de Wasserstein vers la mesure d'équilibre associée au potentiel V. Ce résultat fournit un premier exemple de convergence en temps long de la solution de l'équation des milieux granulaires en présence d'un potentiel non convexe et d'une interaction logarithmique. Sa démonstration utilise notamment des techniques de probabilités libres.

We investigate the exit problem for a diffusion which drift is not time-homogeneous. More precisely, we study this problem for a McKean-Vlasov diffusion, that corresponds to the probabilistic interpretation of the granular media equation. This problem has already been solved in previous articles when the confining potential is uniformly strictly convex. Two different methods have been used. However, these two methods do not extend to the non-convex case. Consequently, here, we proceed in another way: by making a coupling with another McKean-Vlasov diffusion with a uniformly strictly convex confining potential. We present the result in a simple case, the one in which the interacting potential is linear. However, the result can be extended in a more general setting.

We investigate the convergence of McKean-Vlasov diffusions in a nonconvex landscape. These processes are linked to nonlinear partial differential equations. According to our previous results, there are at least three stationary measures under simple assumptions. Hence, the convergence problem is not classical like in the convex case. By using the method in Benedetto et al. [J. Statist. Phys. 91 (1998) 1261-1271] about the monotonicity of the free-energy, and combining this with a complete description of the set of the stationary measures, we prove the global convergence of the self-stabilizing processes., Published in at http://dx.doi.org/10.1214/12-AOP749 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The exit time and the exit location of a non-Markovian diffusion is analyzed. More particularly, we focus on the so-called self-stabilizing process. The question has been studied by Herrmann, Imkeller and Peithmann (in 2008) with results similar to those by Freidlin and Wentzell. We aim to provide the same results by a more intuitive approach and without reconstructing the proofs of Freidlin and Wentzell. Our arguments are as follows. In one hand, we establish a strong version of the propagation of chaos which allows to link the exit time of the McKean-Vlasov diffusion and the one of a particle in a mean-field system. In the other hand, we apply the Freidlin-Wentzell theory to the associated mean field system, which is a Markovian diffusion.

The exit time and the exit location of a non-Markovian diffusion is analyzed. More particularly, we focus on the so-called self-stabilizing process. The question has been studied by Herrmann, Imkeller and Peithmann in [Herrmann, Imkeller, Peithmann|2008]. They proved some results similar to the ones of Freidlin and Wentzell. We aim to provide the same results by an approach more intuitive and without reconstructing the proofs of Freidlin and Wentzell. Our arguments are as follows. In one hand, we establish a strong version of the propagation of chaos which permits to link the exit time of the McKean-Vlasov diffusion and the one of a particle in a mean-field system. In the other hand, we apply the Freidlin-Wentzell theory to the associated mean-field system ; which is a Markovian diffusion.

Self-stabilizing processes are inhomogeneous diffusions in which the law of the process intervenes in the drift. If the external force is the gradient of a convex potential, it has been proved that the process converges toward the unique invariant probability as the time goes to infinity. However, in a previous article, we established that the diffusion may admit several invariant probabilities, provided that the external force derives from a non-convex potential. We here provide results about the limiting values of the family $\left\{\mu_t\,;\,t\geq0\right\}$, $\mu_t$ being the law of the diffusion. Moreover, we establish the weak convergence under an additional hypothesis.