- MARTIN BARLOW, University of British Columbia, Vancouver, BC

*Random walks in symmetric random environments*[PDF] -
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

where\tag1 ¶ *u*¶*t*=

å

*ij*¶ ¶*x*_{i}*a*_{ij}(*x*/e)¶ ¶*x*_{j}*u*_{e}(*t*,*x*),(1) *x*Î**R**^{d},*u*_{e}(0,*x*) =*v*_{0}(*x*), and*a*(*x*) = (*a*_{ij}(*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*=**R**^{d}-*Z*on which diffusion can occur.I will discuss a discrete version of this problem. Here

**R**^{d}is replaced by the lattice e**Z**^{d}, and the set*C*by the unique unbounded connected component of a supercritical percolation process on e**Z**^{d}. 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.