Approximation of the effective conductivity of ergodic media by periodization
| dc.creator | Owhadi, Houman | |
| dc.date | 2002-01-08 | |
| dc.date | 2004-03-30 | |
| dc.date.accessioned | 2026-07-07T04:45:45Z | |
| dc.date.available | 2026-07-07T04:45:45Z | |
| dc.description | This paper is concerned with the approximation of the effective conductivity $σ(A,μ)$ associated to an elliptic operator $\nabla_x A(x,η) \nabla_x$ where for $x\in \R^d$, $d\geq 1$, $A(x,η)$ is a bounded elliptic random symmetric $d\times d$ matrix and $η$ takes value in an ergodic probability space $(X,μ)$. Writing $A^N(x,η)$ 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 $η$ $$ \lim_{N\to +\infty}σ(A^N,η)=σ(A,μ) $$ We extend this result to non-symmetric operators $\nabla_x (a+E(x,η)) \nabla_x$ corresponding to diffusions in ergodic divergence free flows ($a$ is $d\times d$ elliptic symmetric matrix and $E(x,η)$ 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 $Ł^2(X,μ)$ can almost surely be approximated by periodic Weyl decompositions with increasing periods, implying that semi-continuous variational formulae associated to $Ł^2(X,μ)$ can almost surely be approximated by variational formulae minimizing on periodic potential and solenoidal functions. | |
| dc.description | published version. The approximation result is given for general non linear semi-continuous variational formulae | |
| dc.identifier | https://arxiv.org/abs/math/0201062 | |
| dc.identifier | http://arxiv.org/abs/math/0201062 | |
| dc.identifier | Probability Theory and Related Fields. 125, 225-258, 2003 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/63069 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Primary 74Q20, 37A15; secondary 37A25 | |
| dc.title | Approximation of the effective conductivity of ergodic media by periodization | |
| dc.type | text |