Limiting Behaviour for Heat Kernels of Random Processes in Random Environments

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.

• Speaker: Peter Taylor (Statslab)
• Tuesday 16 February 2021, 14:00–15:00
• Venue: Zoom.
• Series: Probability; organiser: Perla Sousi.