Off-Diagonal Heat Kernel Estimates for Symmetric Diffusions in a Degenerate Ergodic Environment

TL;DR

This paper establishes off-diagonal heat kernel estimates for symmetric diffusions in degenerate, ergodic random environments, providing insights into their behavior and scaling limits despite degeneracies.

Contribution

It introduces new upper and lower off-diagonal heat kernel estimates for diffusions with degenerate coefficients in ergodic environments, extending previous results to more general settings.

Findings

Derived upper off-diagonal heat kernel estimates for degenerate diffusions.

Proved lower estimates under mixing conditions on the environment.

Established a scaling limit for the Green's function in this setting.

Abstract

We study a symmetric diffusion process on $\mathbb{R}^d$, $d\geq 2$, in divergence form in a stationary and ergodic random environment. The coefficients are assumed to be degenerate and unbounded but satisfy a moment condition. We derive upper off-diagonal estimates on the heat kernel of this process for general speed measure. Lower off-diagonal estimates are also shown for a natural choice of speed measure, under an additional mixing assumption on the environment. Using these estimates, a scaling limit for the Green's function is proven.