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Homogénéisation stochastique quantitative

Abstract : 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.
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Submitted on : Tuesday, November 6, 2018 - 2:53:05 PM
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Alexandre Bordas. Homogénéisation stochastique quantitative. Probabilités [math.PR]. Université de Lyon, 2018. Français. ⟨NNT : 2018LYSEN053⟩. ⟨tel-01913702⟩

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