Translator Disclaimer
May 2022 An elliptic Harnack inequality for difference equations with random balanced coefficients
Noam Berger, Moran Cohen, Jean-Dominique Deuschel, Xiaoqin Guo
Author Affiliations +
Ann. Probab. 50(3): 835-873 (May 2022). DOI: 10.1214/21-AOP1544

Abstract

We prove an elliptic Harnack inequality at large scale on the lattice Zd 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.

Acknowledgments

We thank David Criens for sharing with us his thesis [15] which helped us fix some errors in earlier versions.

We thank two anonymous referees for their thoughtful comments and suggestions which improved the presentation of our work.

Citation

Download Citation

Noam Berger. Moran Cohen. Jean-Dominique Deuschel. Xiaoqin Guo. "An elliptic Harnack inequality for difference equations with random balanced coefficients." Ann. Probab. 50 (3) 835 - 873, May 2022. https://doi.org/10.1214/21-AOP1544

Information

Received: 1 December 2019; Revised: 1 July 2021; Published: May 2022
First available in Project Euclid: 27 April 2022

Digital Object Identifier: 10.1214/21-AOP1544

Subjects:
Primary: 60K37
Secondary: 35B27

Keywords: balanced environment , elliptic Harnack inequality , nondivergence form operators , nonellipticity , percolation , Random walks in random environments

Rights: Copyright © 2022 Institute of Mathematical Statistics

JOURNAL ARTICLE
39 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.50 • No. 3 • May 2022
Back to Top