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.

Evolutionäre ratenunabhängige Systeme (ERIS) beschreiben beispielsweise stark heterogene, elastoplastische Materialien. Grundlage dieser Arbeit wird ein diskretes Gitter mit Gitterweite größer Null, beschrieben durch ein ERIS, sein, dessen Kanten als elastoplastische Federn angenommen werden. Für konstante Federparameter wird analytisch bewiesen, dass die Lösungen des diskreten ERIS gegen die Lösungen eines kontinuierlichen Systems konvergieren, wenn die Gitterweite gegen Null konvergiert. Für zufällige Federparameter wird das Verhalten von ein- und zweidimensionalen Federsystemen simuliert. Das zugehörige Grenzwertsystem weist jedoch numerische Schwierigkeiten auf, weshalb zur Lösung dessen die Methode der Approximation durch Periodisierung herangezogen wird, welche wiederum zum Lösen eines diskreten, stochastischen Systems führt. Dieses wird durch einen stochastischen Spannungstensor charakterisiert. Für diesen sind jedoch lediglich analytische Resultate über das Verhalten und dessen Varianz in Abhängigkeit von der Zeit und Gittergröße bekannt, wenn keine der Federn eine plastische Verformung annimmt. Durch numerische Implementierung solcher Federsysteme mit Hilfe des TNNMG-Algorithmus werden auch Verhaltensweisen des Spannungstensors im plastischen Bereich für ein- und zweidimensionale Federsysteme untersucht. So wird in Simulationen deutlich, dass ein starker Zusammenhang mit dem Anteil an verformten Federn besteht. Neben den bereits bekannten Resultaten wird anhand von Simulationen vermutet, dass bei vorgegebener, linearer makroskopischer Verschiebung sich der Verlauf des Operators im plastischen Bereich einem linearen Anstieg und dessen Varianz einem quadratischen Anstieg in der Zeit asymptotisch nähert. / Evolutionary rate-independent systems (ERIS) describe for example strongly heterogeneous elastoplastic materials. In this thesis a discrete network of elastoplastic springs with length greater than zero, described by an ERIS, is considered. For constant spring parameters it is analytically proved that the solution of the discrete ERIS converges to the solution of a continuous system as the length tends to zero. For random spring parameters, the behaviour of one- and two-dimensional spring network is simulated. However, the associated limit system presents some numerical difficulties, so the method of approximation by periodization is used to solve it. This, in turn, leads to solving a discrete stochastic system, which is characterised by a stochastic stress tensor, for which, however, only analytical results about the behaviour as a function of time and grid size and its variance are known, if none of the springs assume a plastic deformation. By implementing such systems with the help of the TNNMG algorithm, it is also possible to investigate the behaviour of the stress tensor in the plastic range for one- and two-dimensional spring networks. Thus, in simulations it becomes clear that there is a strong correlation with the proportion of deformed springs. In addition to the already known results, it is assumed on the basis of simulations that for an predetermined linear macroscopic displacement the course of the operator in the plastic range approaches a linear increase and its variance approaches a quadratic increase in time asymptotically.

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.