Abstract

In this talk I will present recent results on random processes moving in random environments.

In the first part of the talk, we introduce the Random Conductance Model (RCM); a random walk on an infinite lattice (usually taken to be $\mathbb{Z}^d$) whose law is determined by random weights on the (nearest neighbour) edges. In the setting of degenerate, ergodic weights and general speed measure, we present a local limit theorem for this model which tells us how the heat kernel of this process has a Gaussian scaling limit. Furthermore, we exhibit applications of said local limit theorems to the Ginzburg-Landau gradient model. This is a model for a stochastic interface separating two distinct thermodynamic phases, using an infinite system of coupled SDEs. Based on joint work with Sebastian Andres.

If time permits I will define another process – symmetric diffusion in a degenerate, ergodic medium. This is a continuum analogue of the above RCM and the techniques take inspiration from there. We show upper off-diagonal (Gaussian-like) heat kernel estimates, given in terms of the intrinsic metric of this process, and a scaling limit for the Green’s kernel.