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August 2013 Quantitative version of the Kipnis–Varadhan theorem and Monte Carlo approximation of homogenized coefficients
Antoine Gloria, Jean-Christophe Mourrat
Ann. Appl. Probab. 23(4): 1544-1583 (August 2013). DOI: 10.1214/12-AAP880


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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Antoine Gloria. Jean-Christophe Mourrat. "Quantitative version of the Kipnis–Varadhan theorem and Monte Carlo approximation of homogenized coefficients." Ann. Appl. Probab. 23 (4) 1544 - 1583, August 2013.


Published: August 2013
First available in Project Euclid: 21 June 2013

zbMATH: 1276.35026
MathSciNet: MR3098442
Digital Object Identifier: 10.1214/12-AAP880

Primary: 35B27 , 60G50 , 60H25 , 60H35 , 60K37 , 65C05

Keywords: effective coefficients , Monte Carlo method , quantitative estimates , random environment , Random walk , Stochastic homogenization

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.23 • No. 4 • August 2013
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