Homogenization of a Bond Diffusion in a Locally Ergodic Random Environment

Abstract

We consider a nearest neighbors random walk on 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 (x). We assume that f(x)g x2Z is distributed by a locally ergodic probability measure. We prove that, under diusive scaling of space and time, the random walk converges in distribution to the diusion process on R with (a(X) ), for a certain homogenized diusion function a(X), independent of . The main tools of the proof are a local ergodic result and the explicit solution of the corresponding Poisson equation. Key words: random walk in random environment, homogenization, invariance principle 2000 MSC: Primary 60K37, Secondary 60F17, 82D30