Non-uniformly parabolic equations and applications to the random conductance model

TL;DR

This paper investigates local regularity of non-uniformly parabolic difference operators linked to the random conductance model, improving oscillation decay results and establishing a local limit theorem for random walks in complex environments.

Contribution

It introduces new oscillation decay estimates under minimal assumptions and applies these to prove a local limit theorem for random walks in degenerate, unbounded conductance environments.

Findings

Improved oscillation decay under weaker conditions

Established a local limit theorem for complex environments

Extended understanding of regularity in random conductance models

Abstract

We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on $\mathbb Z^d$. In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment.