Unité Mixte de Recherche du C.N.R.S. N°7534

Abstract : We consider a nearest neighbors random walk on $\mathbb Z$.The jump rate from site $x$ to site $x+1$ is equal to the jump rate from $x+1$ to $x$ and is a bounded, strictly positive random variable $\eta(x)$. We assume that $\{\eta(x)\}_{x\in\mathbb Z}$ is distributed by a \emph{locally ergodic} probability measure. We prove that, under diffusive scaling of space and time, the random walk converges in distribution to the diffusion process on $\mathbb R$ with infinitesimal generator $\frac{d}{dX}(a(X) \frac{d}{dX})$, for a certain homogenized diffusion function $a(X)$, independent of $\eta$. The main tools of the proof are a local ergodic result and the explicit solution of the corresponding Poisson equation.

 HOMOGENIZATION OF A BOND DIFFUSION IN A LOCALLY ERGODIC RANDOM ENVIRONNEMENT.