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

TL;DR

This paper investigates the conditions under which a symmetric random walk in a non-homogeneous medium converges to a Brownian motion, analyzing explicit formulas for the diffusion coefficient and providing bounds for higher dimensions.

Contribution

It clarifies the homogenization conditions and diffusion coefficients for symmetric random walks in non-homogeneous media, extending understanding beyond the one-dimensional case.

Findings

Explicit expression for Green's function in 1D case.

Formulas for effective diffusion matrix in higher dimensions.

Bounds on the diffusion coefficient for d>1.

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 $d>1$.