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.

Quantitative homogenization in a balanced random environment

“…When the balanced environment is allowed to be non-elliptic and genuinelydimensional, (weak) quantitative results and Harnack inequalities for non-divergence form difference operators are obtained by Berger, Cohen, Deuschel, Guo [9], and Berger, Criens [11] for -harmonic and -caloric functions, respectively.…”