# Quantitative version of the Kipnis-Varadhan theorem and Monte-Carlo approximation of homogenized coefficients

1 SIMPAF - SImulations and Modeling for PArticles and Fluids
LPP - Laboratoire Paul Painlevé - UMR 8524, Inria Lille - Nord Europe
Abstract : This article is devoted to the analysis of a Monte-Carlo method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We consider the case of independent and identically distributed coefficients, and adopt the point of view of the random walk in a random environment. Given some final time $t>0$, a natural approximation of the homogenized coefficients is given by the empirical average of the final squared positions rescaled by $t$ of $n$ independent random walks in $n$ independent environments. Relying on a quantitative version of the Kipnis-Varadhan theorem combined with estimates of spectral exponents obtained by an original combination of pde arguments and spectral theory, we first give a sharp estimate of the error between the homogenized coefficients and the expectation of the rescaled final position of the random walk in terms of $t$. We then complete the error analysis by quantifying the fluctuations of the empirical average in terms of $n$ and $t$, and prove a large-deviation estimate, as well as a central limit theorem. Our estimates are optimal, up to a logarithmic correction in dimension $2$.
Type de document :
Article dans une revue
Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2013, pp.31

Littérature citée [21 références]

https://hal.inria.fr/inria-00579424
Contributeur : Antoine Gloria <>
Soumis le : jeudi 24 mai 2012 - 12:25:38
Dernière modification le : jeudi 11 janvier 2018 - 06:22:13
Document(s) archivé(s) le : samedi 25 août 2012 - 02:32:42

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• HAL Id : inria-00579424, version 4

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Antoine Gloria, Jean-Christophe Mourrat. Quantitative version of the Kipnis-Varadhan theorem and Monte-Carlo approximation of homogenized coefficients. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2013, pp.31. 〈inria-00579424v4〉

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