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Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics

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Abstract

We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice \(\mathbb Z^d\) with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions \(d>2\)) and prove new optimal estimates for regularized versions of the corrector (in dimensions \(d\ge 2\)). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.

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Acknowledgments

Felix Otto wants to acknowledge the hospitality of the University of Paris-Sud (Orsay)—most of the presented material was covered by a Hadamard lecture Felix Otto gave in 2012 at that institution, which in turn was an extended version of a minitutorial at the SIAM PDE conference of 2011.

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Authors and Affiliations

  1. Université Libre de Bruxelles (ULB), Brussels, Belgium

    A. Gloria

  2. Project-Team MEPHYSTO, Inria Lille - Nord Europe, Villeneuve d’Ascq, France

    A. Gloria

  3. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    S. Neukamm

  4. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 , Leipzig, Germany

    F. Otto

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  1. A. Gloria
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  2. S. Neukamm
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Correspondence to A. Gloria.

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Gloria, A., Neukamm, S. & Otto, F. Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. Invent. math. 199, 455–515 (2015). https://doi.org/10.1007/s00222-014-0518-z

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  • DOI: https://doi.org/10.1007/s00222-014-0518-z

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