Title: Motion in random environment: conjectures, counter examples, and "obvious" theorems
Speaker: Professor Ofer Zeitouni
Speaker Info: University of Minnesota
Brief Description:
Special Note:
Abstract:
The model of random walks in random environments generates several simple to formulate conjectures that are notoriously hard to resolve. I will describe the model and some of these conjectures. While those are undoubtedly true, I will present a couple of examples where counter-intuitive behavior occurs. I will then show, for the related model of diffusion in random environment with weak disorder, that homogenization does occur. In PDE terms, this amounts to showing the "obvious" result that, for dimension 3 or larger, solutions of the PDE $$ u_t = a_{ij}(x/\epsilon,\omega) u_{x_i,x_j} + b_i(x/\epsilon,\omega) u_{x_i}/\epsilon + g $$ with diffusion a and drift b that are small ergodic perturbations of the identity and of zero, respectively, and satisfy a statistical isotropy condition, converge uniformly on compacts to solutions of the heat equation $$ u_t = \sigma2\Delta (u)+g . $$Date: Wednesday, June 1, 2005