Homogenization of a bond diffusion in a locally ergodic random environment

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 η(x). We assume that {η(x)}x∈Z is distributed by a 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 R with infinitesimal generator d/dX(a(X)d/dX), for a certain homogenized diffusion function a(X), independent of η. The main tools of the proof are a local ergodic result and the explicit solution of the corresponding Poisson equation.