Approximation of the effective conductivity of ergodic media by periodization

Abstract

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 $x\in {\mathbb{R}}^d$, $d\ge 1$, $A(x,\eta )$ 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 $T_N^d$ of dimension $d$ and side $N$ we prove that for $\mu$-almost all $\eta$ $$\underset{N\to +\infty }{\lim }\sigma (A^N,\eta )=\sigma (A,\mu )$$ 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 ${\mathbb{Z}}^d$ with ergodic jump rates. The core of our result is to show that the ergodic Weyl decomposition associated to ${Ł}^2(X,\mu )$ can almost surely be approximated by periodic Weyl decompositions with increasing periods, implying that semi-continuous variational formulae associated to ${Ł}^2(X,\mu )$ can almost surely be approximated by variational formulae minimizing on periodic potential and solenoidal functions.

published version. The approximation result is given for general non linear semi-continuous variational formulae