QUANTITATIVE VERSION OF THE KIPNIS-VARADHAN THEOREM AND MONTE CARLO APPROXIMATION OF HOMOGENIZED COEFFICIENTS

• 成果类型: Article
• 署名作者: Gloria, Antoine; Mourrat, Jean-Christophe
• 署名单位: Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP880
• 发表日期: 2013
• 页码: 1544-1583
• 关键词: quenched invariance-principles; reversible markov-processes; random-walks; Stochastic Homogenization; percolation; functionals; limit

摘要

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.