An elliptic Harnack inequality for difference equations with random balanced coefficients

Abstract

We prove an elliptic Harnack inequality at large scale on the lattice ${\mathbb{Z}}^{\mathit{d}}$ for nonnegative solutions of a difference equation with balanced i.i.d. coefficients which are not necessarily elliptic. We also identify the optimal constant in the Harnack inequality. Our proof relies on a quantitative homogenization result of the corresponding invariance principle to Brownian motion and on percolation estimates. As a corollary of our main theorem, we derive an almost optimal Hölder estimate.