Approximation of the effective conductivity of ergodic media by periodization

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.