Approximation of the effective conductivity of ergodic media by periodization

Details Approximation of the effective conductivity of ergodic media by periodization Approximation of the effective conductivity of ergodic media by periodization

2002-01-08

arxiv.org/abs/math/0201062v2

Probability Theory and Related Fields. 125, 225-258, 2003

Mathematics

Probability

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

Scientific paper

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.

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