Annals of Applied Probability

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

Antoine Gloria and Jean-Christophe Mourrat

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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 re-scaled 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 re-scaled 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 $2$.

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Ann. Appl. Probab., Volume 23, Number 4 (2013), 1544-1583.

First available in Project Euclid: 21 June 2013

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Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 60K37: Processes in random environments 60H25: Random operators and equations [See also 47B80] 65C05: Monte Carlo methods 60H35: Computational methods for stochastic equations [See also 65C30] 60G50: Sums of independent random variables; random walks

Random walk random environment stochastic homogenization effective coefficients Monte Carlo method quantitative estimates


Gloria, Antoine; Mourrat, Jean-Christophe. Quantitative version of the Kipnis–Varadhan theorem and Monte Carlo approximation of homogenized coefficients. Ann. Appl. Probab. 23 (2013), no. 4, 1544--1583. doi:10.1214/12-AAP880.

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