Quantitative stochastic homogenization of random media: degenerate environments and stochastic interface model

Abstract : This thesis is devoted to the study of stochastic homogenization, which aims at studying the behavior of partial differential equations with highly heterogeneous, but statistically homogeneous, random coefficients. It is divided into three parts. The first part corresponds to Chapters 2 and 3 and tries to extend the theory of quantitative stochastic homogenization, developed under an assumption of uniform ellipticity, to the degenerate setting of supercritical Bernoulli bond percolation. In Chapter 2, we prove a quantitative homogenization theorem as well as a large scale regularity theory and Liouville results for harmonic functions on the infinite cluster. In Chapter 3, we obtain optimal spatial estimates in all dimension for the corrector on the infinite cluster. In Chapter 4, we study another type of degenerate environment involving differential forms and prove, in this setting, a quantitative homogenization theorem. In Chapter 5, we apply ideas from homogenization to a model of statistical physics: the discrete Ginzburg-Landau model. In this chapter, we revisit the beginning of the theory of stochastic homogenization and combine it with arguments from the theory of optimal transport to derive a quantitative rate of convergence for the finite-volume surface tension of the model.
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Submitted on : Monday, July 22, 2019 - 2:22:19 PM
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Paul Dario. Quantitative stochastic homogenization of random media: degenerate environments and stochastic interface model. Probability [math.PR]. Université Paris-Dauphine, 2019. English. ⟨tel-02190415⟩

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