Brownian Motion Limit of Random Walks in Symetric Non-Homogeneous Media

Abstract

The phenomenon of macroscopic homogenization is illustrated with a simple example of diffusion. We examine the conditions under which a $d$--dimensional simple random walk in a symmetric random media converges to a Brownian motion. For $d=1$, both the macroscopic homogeneity condition and the diffusion coefficient can be read from an explicit expression for the Green's function. Except for this case, the two available formulas for the effective diffusion matrix $\kappa $ do not explicit show how macroscopic homogenization takes place. Using an electrostatic analogy due to Anshelevich, Khanin and Sinai \cite{AKS}, we discuss upper and lower bounds on the diffusion coefficient $\kappa $ for $d>1$.