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CAIMS Minisymposia

CMS Jeffery-Williams Prize

MARTIN BARLOW, University of British Columbia, Vancouver, BC
Random walks in symmetric random environments

This talk will describe recent progress in the study of symmetric (or time reversible) random walks in random environments. There is a close connection with the homogenization of PDE. Consider the initial value problem

\tag1  u


aij (x/e)  

ue (t,x),
where x Î Rd, ue (0,x) = v0 (x), and a(x) = ( aij(x) ) is symmetric. This equation describes diffusion in an irregular medium with fluctuations at length scale e. The theory of homogenization is most developed in the case when a(·) is uniformly elliptic and periodic. Significant progress has been made in the relaxation of the hypothesis of periodicity, for example by making a a stationary random field. However, if one allows a to be zero, the set Z = {x : a(x)=0} acts as a barrier to diffusion, and one needs to consider carefully the structure of the set C = Rd - Z on which diffusion can occur.

I will discuss a discrete version of this problem. Here Rd is replaced by the lattice eZd, and the set C by the unique unbounded connected component of a supercritical percolation process on eZd. I will discuss Gaussian bounds, homogenization, Harnack inequalities and Green's functions in this setting. The differential inequalities that Nash introduced in his 1958 paper are particularly well suited to this problem.