ergodic media by periodization.

This paper is concerned with the approximation of the effective conductivity σ(A,μ) associated to an elliptic operator ∇xA(x, η)∇x where for x ∈ Rd, d ≥ 1, A(x, η) is a bounded elliptic random symmetric d× d matrix and η takes value in an ergodic probability space (X,μ). Writing AN (x, η) the periodization of A(x, η) on the torus T d N of dimension d and side N we prove that for μ-almost all η lim N→+∞ σ(A , η) = σ(A,μ) We extend this result to non-symmetric operators ∇x(a + E(x, η))∇x corresponding to diffusions in ergodic divergence free flows (a is d × d elliptic symmetric matrix and E(x, η) an ergodic skew-symmetric matrix); and to discrete operators corresponding to random walks on Zd with ergodic jump rates. The core of our result is to show that the ergodic Weyl decomposition associated to L(X,μ) can almost surely be approximated by periodic Weyl decompositions with increasing periods, implying that semi-continuous variational formulae associated to L(X,μ) can almost surely be approximated by variational formulae minimizing on periodic potential and solenoidal functions.