Approximation of the effective conductivity of ergodic media by periodization

• 成果类型: Article
• 署名作者: Owhadi, H
• 署名单位: Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-002-0240-4
• 发表日期: 2003
• 页码: 225-258
• 关键词: reversible markov-processes; invariance-principle; diffusion

摘要

This paper is concerned with the approximation of the effective conductivity sigma (A, mu) associated to an elliptic operator del(x)A (x, eta)del(x) where for x is an element of R-d, d greater than or equal to 1, A (x, eta) is a bounded elliptic random symmetric d x d matrix and eta takes value in an ergodic probability space (X, mu). Writing A(N) (x, eta) the periodization of A(x, eta) the torus T-N(d) of dimension d and side N we prove that for mu-almost all eta lim sigma(A(N), eta) = sigma(A, mu) N-->+infinity We extend this result to non-symmetric operators del(x) (a + E(x, eta))del(x) corresponding to diffusions in ergodic divergence free flows (a is d x d elliptic symmetric matrix and E (x, eta) an ergodic skew-symmetric matrix); and to discrete operators corresponding to random walks on Z(d) with ergodic jump rates. The core of our result is to show that the ergodic Weyl decomposition associated to L-2(X, mu) can almost surely be approximated by periodic Weyl decompositions with increasing periods, implying that semi-continuous variational formulae associated to L-2(X, mu) can almost surely be approximated by variational formulae minimizing on periodic potential and solenoidal functions.