Approximation of the effective conductivity of 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 ∈ R , 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 A(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 Z with ergodic jump rates. The core of our result is to show that the ergodic Weyl decomposition associated to L 2(X,μ) can almost surely be approximated by periodic Weyl decompositions with increasing periods, implying that semi-continuous variational formulae associated to L2(X,μ) can almost surely be approximated by variational formulae minimizing on periodic potential and solenoidal functions.