Lower Gaussian heat kernel bounds for the Random Conductance Model in a degenerate ergodic environment

TL;DR

This paper establishes Gaussian lower bounds for the heat kernel in the random conductance model with unbounded, ergodic conductances, under certain moment and correlation conditions, advancing understanding of diffusion in random media.

Contribution

It provides the first Gaussian lower bounds for the heat kernel in the degenerate, ergodic setting with unbounded conductances, using chaining techniques.

Findings

Proved Gaussian lower bounds for the heat kernel.

Derived bounds on the Green's function.

Extended heat kernel analysis to degenerate environments.

Abstract

We study the random conductance model on $\mathbb{Z}^d$ with ergodic, unbounded conductances. We prove a Gaussian lower bound on the heat kernel given a polynomial moment condition and some additional assumptions on the correlations of the conductances. The proof is based on the well-established chaining technique. We also obtain bounds on the Green's function.