## Limiting Behaviour for Heat Kernels of Random Processes in Random Environments

### With Peter Taylor (Statslab)

# 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.