A parabolic Harnack principle for balanced difference equations in random environments

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.

arXiv comment35 pages, 3 figures ; Some references where updated compared to previous version

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• Optimal convergence rates in stochastic homogenization in a balanced random environment