A parabolic Harnack principle for balanced difference equations in random environments

TL;DR

This paper establishes a parabolic Harnack inequality for non-negative solutions to discrete heat equations in balanced, i.i.d. environments, even without ellipticity, and identifies the optimal Harnack constant.

Contribution

It introduces a sharp growth condition necessary for the PHI and proves a parabolic oscillation inequality and a weak homogenization result in this setting.

Findings

Proved a parabolic Harnack inequality for non-elliptic environments.

Identified the optimal Harnack constant for the inequality.

Established a parabolic oscillation inequality and a weak homogenization result.

Abstract

We consider difference equations in balanced, i.i.d. environments which are not necessary elliptic. In this setting we prove a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather mild) growth condition, and we identify the optimal Harnack constant for the PHI. We show by way of an example that a growth condition is necessary and that our growth condition is sharp. Along the way we also prove a parabolic oscillation inequality and a (weak) quantitative homogenization result, which we believe to be of independent interest.