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 $\nabla_x A(x,\eta) \nabla_x$ where for $x\in \R^d$, $d\geq 1$, $A(x,\eta)$ is a bounded elliptic random symmetric $d\times d$ matrix and $\eta$ takes value in an ergodic probability space $(X,\mu)$. Writing $A^N(x,\eta)$ the periodization of $A(x,\eta)$ on the torus $T^d_N$ of dimension $d$ and side $N$ we prove that for $\mu$-almost all $\eta$ $$\lim_{N\to +\infty}\sigma(A^N,\eta)=\sigma(A,\mu)$$ We extend this result to non-symmetric operators $\nabla_x (a+E(x,\eta)) \nabla_x$ corresponding to diffusions in ergodic divergence free flows ($a$ is $d\times d$ 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.