We explore the effect of multiscale structure on weakly interacting diffusions through two main projects. In the first, we consider a collection of weakly interacting diffusion processes moving in a two-scale locally periodic environment. We study the large deviations principle of the empirical distribution of the particles' positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We make use of weak convergence methods providing a convenient representation for the large deviations rate function, which allow us to characterize the effective controlled mean field dynamics. In addition, we obtain equivalent representations for the large deviations rate function of the form of Dawson-Gartner which hold even in the case where the diffusion matrix depends on the empirical measure and when the particles undergo averaging in addition to the propagation of chaos. In the second, we consider a fully-coupled slow-fast system of McKean-Vlasov SDEs with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy-Problem on the Wasserstein space that are of independent interest.

Hamilton-Jacobi equations are a class of partial differential equations that arise in many areas of science and engineering. Originating from classical mechanics, they are widely used in various fields such as optimal control theory, quantitative finance, and game theory. Stochastic homogenization is a phenomenon used to study the behavior of solutions to partial differential equations in stationary ergodic media, aiming to understand how these solutions average out or 'homogenize' over large scales. This process results in effective deterministic descriptions, called effective Hamiltonians, which capture the essential behavior of the system. We consider nonconvex Hamilton-Jacobi equations in one space dimension. We provide a fully constructive proof of homogenization, which yields a formula for the effective Hamiltonian. Our proof employs sublinear correctors, functions extensively discussed in the literature. The proof involves strong induction: we first show homogenization for our base cases, then use gluing processes to generalize the solution for the strong induction. Finally, we extend the result to a wide class of functions. We study the properties of the resulting effective Hamiltonian and investigate the occurrence of flat pieces. Additionally, we develop a Python-based computational tool that performs the same homogenization steps in a computing environment, returning the effective Hamiltonian along with its graph and properties. / Mathematics

This thesis is divided into two parts: In the first one (Chapters 1 and 2), we deal with problems arising from quantitative homogenization of the random elliptic operator in divergence form $-\\nabla \\cdot a \\nabla$. In Chapter 1 we study existence and stochastic bounds for the Green function $G$ associated to $-\\nabla \\cdot a \\nabla$ in the case of systems. Without assuming any regularity on the coefficient field $a= a(x)$, we prove that for every (measurable) uniformly elliptic tensor field $a$ and for almost every point $y \\in \\mathbb^d$, there exists a unique Green\'s function centred in $y$ associated to the vectorial operator $-\\nabla \\cdot a\\nabla $ in $\\mathbb{R}^d$, $d> 2$. In addition, we prove that if we introduce a shift-invariant ensemble $\\langle\\cdot \\rangle$ over the set of uniformly elliptic tensor fields, then $\\nabla G$ and its mixed derivatives $\\nabla \\nabla G$ satisfy optimal pointwise $L^1$-bounds in probability. Chapter 2 deals with the homogenization of $-\\nabla \\cdot a \\nabla$ to $-\\nabla \\ah \\nabla$ in the sense that we study the large-scale behaviour of $a$-harmonic functions in exterior domains $\\{ |x| > r \\}$ by comparing them with functions which are $\\ah$-harmonic. More precisely, we make use of the first and second-order correctors to compare an $a$-harmonic function $u$ to the two-scale expansion of suitable $\\ah$-harmonic function $u_h$. We show that there is a direct correspondence between the rate of the sublinear growth of the correctors and the smallness of the relative homogenization error $u- u_h$. The theory of stochastic homogenization of elliptic operators admits an equivalent probabilistic counterpart, which follows from the link between parabolic equations with elliptic operators in divergence form and random walks. This allows to reformulate the problem of homogenization in terms of invariance principle for random walks. The second part of thesis (Chapters 3 and 4) focusses on this interplay between probabilistic and analytic approaches and aims at exploiting it to study invariance principles in the case of degenerate random conductance models and systems of interacting particles. In Chapter 3 we study a random conductance model where we assume that the conductances are independent, stationary and bounded from above but not uniformly away from $0$. We give a simple necessary and sufficient condition for the relaxation of the environment seen by the particle to be diffusive in the sense of every polynomial moment. As a consequence, we derive polynomial moment estimates on the corrector which imply that the discrete elliptic operator homogenises or, equivalently, that the random conductance model satisfies a quenched invariance principle. In Chapter 4 we turn to a more complicated model, namely the symmetric exclusion process. We show a diffusive upper bound on the transition probability of a tagged particle in this process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type.

This thesis is split into two parts, the first one is concerned with some problems in stochastic homogenization and the second addresses a problem in singular SPDEs. In the part on stochastic homogenization we are interested in developing large-scale regularity theories for random linear elliptic operators by using estimates for the homogenization error to transfer regularity from the homogenized operator to the heterogeneous one at large scales. In the whole-space case this has been done by Gloria, Neukamm, and Otto through means of a homogenization-inspired Campanato iteration. Here we are specifically interested in boundary regularity and as a model setting we consider random linear elliptic operators on the half-space with either homogeneous Dirichlet or Neumann boundary data. In each case we obtain a large-scale regularity theory and the main technical difficulty turns out to be the construction of a sublinear homogenization corrector that is adapted to the boundary data. The case of Dirichlet boundary data is taken from a joint work with Julian Fischer. In an attempt to head towards a percolation setting, we have also included a chapter concerned with the large-scale behaviour of harmonic functions on a domain with random holes assuming that these are 'well-spaced'. In the second part of this thesis we would like to provide a pathwise solution theory for a singular quasilinear parabolic initial value problem with a periodic forcing. The difficulty here is that the roughness of the data limits the regularity the solution such that it is not possible to define the nonlinear terms in the equation. A well-posedness result, therefore, comes with two steps: 1) Giving meaning to the nonlinear terms and 2) Showing that with this meaning the equation has a solution operator with some continuity properties. The solution theory that we develop in this contribution is a perturbative result in the sense that we think of the solution of the initial value problem as a perturbation of the solution of an associated periodic problem, which has already been handled in a work by Otto and Weber. The analysis in this part relies entirely on estimates for the heat semigroup. The results in the second part of this thesis will be in an upcoming joint work with Felix Otto and Jonas Sauer.

Le travail de cette thèse a porté sur le développement de techniques numériques pour l'homogénéisation de matériaux présentant à une petite échelle des hétérogénéités aléatoires. Sous certaines hypothèses, la théorie mathématique de l'homogénéisation stochastique permet d'expliciter les propriétés effectives de tels matériaux. Néanmoins, en pratique, la détermination de ces propriétés demeure difficile. En effet, celle-ci requiert la résolution d'équations aux dérivées partielles stochastiques posées sur l'espace tout entier. Dans cette thèse, cette difficulté est abordée de deux manières différentes. Les méthodes classiques d'approximation conduisent à approcher les propriétés effectives par des quantités aléatoires. Réduire la variance de ces quantités est l'objectif des travaux de la Partie I. On montre ainsi comment adapter au cadre de l'homogénéisation stochastique une technique de réduction de variance déjà éprouvée dans d'autres domaines. Les travaux de la Partie II s'intéressent à des cas pour lesquels le matériau d'intérêt est considéré comme une petite perturbation aléatoire d'un matériau de référence. On montre alors numériquement et théoriquement que cette simplification de la modélisation permet effectivement une réduction très importante du coût calcul / In this thesis we investigate numerical methods for the homogenization of materials the structures of which, at fine scales, are characterized by random heterogenities. Under appropriate hypotheses, the effective properties of such materials are given by closed formulas. However, in practice the computation of these properties is a difficult task because it involves solving partial differential equations with stochastic coefficients that are additionally posed on the whole space. In this work, we address this difficulty in two different ways. The standard discretization techniques lead to random approximate effective properties. In Part I, we aim at reducing their variance, using a well-known variance reduction technique that has already been used successfully in other domains. The works of Part II focus on the case when the material can be seen as a small random perturbation of a periodic material. We then show both numerically and theoretically that, in this case, computing the effective properties is much less costly than in the general case

In this thesis we investigate numerical methods for the homogenization of materials the structures of which, at fine scales, are characterized by random heterogenities. Under appropriate hypotheses, the effective properties of such materials are given by closed formulas. However, in practice the computation of these properties is a difficult task because it involves solving partial differential equations with stochastic coefficients that are additionally posed on the whole space. In this work, we address this difficulty in two different ways. The standard discretization techniques lead to random approximate effective properties. In Part I, we aim at reducing their variance, using a well-known variance reduction technique that has already been used successfully in other domains. The works of Part II focus on the case when the material can be seen as a small random perturbation of a periodic material. We then show both numerically and theoretically that, in this case, computing the effective properties is much less costly than in the general case

This thesis is divided into two parts: In the first one (Chapters 1 and 2), we deal with problems arising from quantitative homogenization of the random elliptic operator in divergence form $-\\nabla \\cdot a \\nabla$. In Chapter 1 we study existence and stochastic bounds for the Green function $G$ associated to $-\\nabla \\cdot a \\nabla$ in the case of systems. Without assuming any regularity on the coefficient field $a= a(x)$, we prove that for every (measurable) uniformly elliptic tensor field $a$ and for almost every point $y \\in \\mathbb{R}^d$, there exists a unique Green\''s function centred in $y$ associated to the vectorial operator $-\\nabla \\cdot a\\nabla $ in $\\mathbb^d$, $d> 2$. In addition, we prove that if we introduce a shift-invariant ensemble $\\langle\\cdot \\rangle$ over the set of uniformly elliptic tensor fields, then $\\nabla G$ and its mixed derivatives $\\nabla \\nabla G$ satisfy optimal pointwise $L^1$-bounds in probability. Chapter 2 deals with the homogenization of $-\\nabla \\cdot a \\nabla$ to $-\\nabla \\ah \\nabla$ in the sense that we study the large-scale behaviour of $a$-harmonic functions in exterior domains $\\$ by comparing them with functions which are $\\ah$-harmonic. More precisely, we make use of the first and second-order correctors to compare an $a$-harmonic function $u$ to the two-scale expansion of suitable $\\ah$-harmonic function $u_h$. We show that there is a direct correspondence between the rate of the sublinear growth of the correctors and the smallness of the relative homogenization error $u- u_h$. The theory of stochastic homogenization of elliptic operators admits an equivalent probabilistic counterpart, which follows from the link between parabolic equations with elliptic operators in divergence form and random walks. This allows to reformulate the problem of homogenization in terms of invariance principle for random walks. The second part of thesis (Chapters 3 and 4) focusses on this interplay between probabilistic and analytic approaches and aims at exploiting it to study invariance principles in the case of degenerate random conductance models and systems of interacting particles. In Chapter 3 we study a random conductance model where we assume that the conductances are independent, stationary and bounded from above but not uniformly away from $0$. We give a simple necessary and sufficient condition for the relaxation of the environment seen by the particle to be diffusive in the sense of every polynomial moment. As a consequence, we derive polynomial moment estimates on the corrector which imply that the discrete elliptic operator homogenises or, equivalently, that the random conductance model satisfies a quenched invariance principle. In Chapter 4 we turn to a more complicated model, namely the symmetric exclusion process. We show a diffusive upper bound on the transition probability of a tagged particle in this process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type.

The issue of establishing the status of nonlinear elasticity theory for rubber with respect to the point of view of polymer physics is at the heart of this manuscript. Our aim is to develop mathematical methods to describe, understand, and solve this multiscale problem. At the level of the polymer chains, rubber can be described as a network whose nodes represent the cross-links between the polymer chains. This network can be considered as the realization of some stochastic process. Given the free energy of the polymer network, we'd like to derive a continuum model as the characteristic length of the polymer chains vanishes. In mathematical terms, this process can be viewed as a hydrodynamic limit or as a discrete homogenization, depending on the nature of the free energy of the network. In view of the works by Treloar, by Flory, and by Rubinstein and Colby on polymer physics, and in view of the stochastic nature of the network, stochastic discrete homogenization seems to be the right tool for the analysis. Hence, in order to complete our program we need to understand the stochastic homogenization of discrete systems. Two features make the analysis rich and challenging from a mathematical perspective: the randomness and the nonlinearity of the problem. The achievement of this manuscript is twofold: - a complete and sharp quantitative theory for the approximation of homogenized coefficients in stochastic homogenization of discrete linear elliptic equations; - the first rigorous and global picture on the status of nonlinear elasticity theory with respect to polymer physics, which partially answers the question raised by Ball in his review paper on open problems in elasticity. Although the emphasis of this manuscript is put on discrete models for rubber, and more generally on the homogenization of discrete elliptic equations, we have also extended most of the results to the case of elliptic partial differential equations --- some of the results being even more striking in that case.

Cette thèse est consacrée à l'étude mathématique des effets de désordre dans divers systèmes physiques. On commence par trois problèmes d'homogénéisation stochastique en lien avec des questions statiques de physique classique. Premièrement, en vue de la déduction rigoureuse de l'élasticité non linéaire à partir de la physique statistique de réseaux de chaînes de polymères, on établit l'existence de propriétés effectives pour des matériaux hyperélastiques hétérogènes aléatoires sous des hypothèses générales de croissance. Deuxièmement, dans un cadre linéarisé simplifié, on étudie les formules de Clausius-Mossotti pour les propriétés effectives d'alliages binaires dilués: on donne la première preuve générale et rigoureuse de ces formules, ainsi qu'une extension aux ordres supérieurs. Troisièmement, encore pour des systèmes linéarisés, on propose d'étudier les déviations par rapport aux propriétés effectives et on établit la première théorie générale des fluctuations en homogénéisation stochastique. Dans la seconde partie de cette thèse, on se focalise sur la compétition entre désordre et interactions, et on étudie plus particulièrement la dynamique des vortex de Ginzburg-Landau dans des supraconducteurs 2D de type II en présence d'impuretés. Bien que la compréhension mathématique des propriétés vitreuses complexes de ces systèmes semble hors de portée, on établit rigoureusement la limite de champ moyen pour la dynamique d'un grand nombre de vortex, et on étudie l'homogénéisation de ces équations limites et leurs propriétés. / This thesis is devoted to the mathematical study of effects of disorder in various physical systems. We start with three stochastic homogenization problems in connection with static classical physics questions. First, motivated by the rigorous derivation of nonlinear elasticity from the statistical physics of polymer-chain networks, we establish the existence of effective properties for randomly heterogeneous hyperelastic materials under general growth assumptions. Second, in the simplest linearized setting, we investigate the so-called Clausius-Mossotti formulas for the effective properties of dilute two-phase dispersed media: we provide the first general and rigorous proof of these formulas, as well as an extension to higher orders. Third, again for linearized models, we propose to study deviations with respect to effective properties and we establish the first general theory of fluctuations in stochastic homogenization. In the second part of this thesis, the focus is on the interplay between disorder and interactions, and more precisely we study the dynamics of Ginzburg-Landau vortices in 2D type-II superconductors in the presence of several impurities. Although a complete mathematical understanding of the complex glassy properties of such systems seems out of reach, we rigorously establish the mean-field dynamics of a large number of vortices, and we investigate the homogenization of the fluid-like mean-field equations and their stick-slip properties.

Cette thèse porte sur l’homogénéisation quantitative d’équations aux dérivées partielles paraboliques, et de problèmes elliptiques discrets. Dans l’introduction, nous voyons comment de tels problèmes, même lorsque les coefficients sont déterministes, résultent d’un modèle aléatoire. Nous donnons ensuite une notion de ce qu’est l’homogénéisation : que se passe-t-il lorsque les coefficients eux-mêmes sont aléatoires, est-il possible de considérer qu’un environnement présentant des inhomogénéités sur de très petites échelles, se comporte d’une manière proche d’un environnement fictif qui serait homogène ?Nous donnons ensuite une interprétation de cette question en terme de marche aléatoire en conductances aléatoires, puis donnons une idée des outils utilisés dans les preuves des deux chapitres suivants. Dans le chapitre II, nous démontrons un résultat d’homogénéisation quantitative pour une équation parabolique – l’équation de la chaleur par exemple – dans un environnement admettant des coefficients aléatoires et dépendant du temps. La méthode utilisée consiste à considérer les solutions d’un tel problème comme optimiseurs de fonctionnelles qui seront définies au préalable, puis d’utiliser la propriété cruciale de sous-additivité de ces quantités, afin d’en déduire une convergence puis un résultat de concentration, qui permettra d’en déduire une vitesse de convergence des solutions vers la solution du problème homogénéisé, Dans le chapitre III, nous adaptons ces méthodes pour un problème elliptique sur le graphe Zd. / This thesis deals with quantitative stochastic homogenization of parabolic partial differential equations, and discrete elliptic problems. In the introduction, we see how can such problems come from random models, even when the coefficients are deterministic. Then, we introduce homogenization : what happen if the coefficients themselves are random ? Could we consider that an environment with microscopical random heterogeneities behaves, at big scale, as a fictious deterministic homogeneous environment ? Then, we give a random walk in random environment interpretation and the sketch of the proofs in the two following chapters. In chapter II, we prove a quantitative homogenization result for parabolic PDEs, such as heat equation, in environment admitting time and space dependent coefficients. The method of the proof consists in considering solutions of such problems as minimizers of variational problems. The first step is to express solutions as minimizers, and then to use the capital property of subadditivity of the corresponding quantities, in order to deduce convergence and concentration result. From that, we deduce a rate of convergence of the actual solutions to the homogenized solution. In chapter III, we adapt these methods to a discrete elliptic problem on the lattice Zd.