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

Abstract

We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on 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.

A Parabolic Harnack Principle for Balanced Difference Equations in Random Environments

“…More recently, there is a growing interest in PHIs for settings which are not necessarily uniformly elliptic. We mention the paper [12], where heat equations arising from random walks on percolation clusters (RWPC) are studied, and the articles [1,3,7], where the PHI is proved for equations related to the random conductance model (RCM). In these works the PHI was used to prove a local limit theorem for the corresponding stochastic processes.…”

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

“…Many of the techniques take inspiration from the random conductance model (RCM) setting, cf. [1,3,7,12] for recent RCM local limit theorems in a degenerate, ergodic setting. The diffusion studied in this paper is a continuum analogue of that model, where a random walk moves on a lattice, usually Z d equipped with nearestneighbour edges.…”