Approximation of the effective conductivity of ergodic media

This paper is concerned with the approximation of the effective conductivity σ(A) associated to an elliptic operator ∇xA(x, η)∇x where for x ∈ R d, d ≥ 1, A(x, η) is a bounded elliptic random symmetric d × d matrix and η takes value in an ergodic probability space. 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 η-a.s. lim N→+∞ σ(A (x, η)) = σ(A) 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.