Quantitative version of the kipnis-varadhan theorem and monte carlo approximation of homogenized coefficients

Antoine Gloria, Jean Christophe Mourrat

Research output: Contribution to journalArticlepeer-review

Abstract

This article is devoted to the analysis of aMonte 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.

Original languageEnglish (US)
Pages (from-to)1544-1583
Number of pages40
JournalAnnals of Applied Probability
Volume23
Issue number4
DOIs
StatePublished - Aug 2013

Keywords

  • Effective coefficients
  • Monte Carlo method
  • Quantitative estimates
  • Random environment
  • Random walk
  • Stochastic homogenization

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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