AN ELLIPTIC HARNACK INEQUALITY FOR DIFFERENCE EQUATIONS WITH RANDOM BALANCED COEFFICIENTS

Noam Berger; Moran Cohen; Jean Dominique Deuschel; Xiaoqin Guo

doi:10.1214/21-AOP1544

AN ELLIPTIC HARNACK INEQUALITY FOR DIFFERENCE EQUATIONS WITH RANDOM BALANCED COEFFICIENTS

Noam Berger, Moran Cohen, Jean Dominique Deuschel, Xiaoqin Guo

Associate Professorship of Stochastic Processes

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

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

Original language	English
Pages (from-to)	835-873
Number of pages	39
Journal	Annals of Probability
Volume	50
Issue number	3
DOIs	https://doi.org/10.1214/21-AOP1544
State	Published - May 2022

Keywords

Balanced environment
Elliptic harnack inequality
Non-ellipticity
Nondivergence form operators
Percolation
Random walks in random environments

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title = "AN ELLIPTIC HARNACK INEQUALITY FOR DIFFERENCE EQUATIONS WITH RANDOM BALANCED COEFFICIENTS",

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 H{\"o}lder estimate.",

keywords = "Balanced environment, Elliptic harnack inequality, Non-ellipticity, Nondivergence form operators, Percolation, Random walks in random environments",

author = "Noam Berger and Moran Cohen and Deuschel, {Jean Dominique} and Xiaoqin Guo",

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AU - Berger, Noam

AU - Cohen, Moran

AU - Deuschel, Jean Dominique

AU - Guo, Xiaoqin

N1 - Publisher Copyright: © Institute of Mathematical Statistics, 2022

PY - 2022/5

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N2 - 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.

AB - 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.

KW - Balanced environment

KW - Elliptic harnack inequality

KW - Non-ellipticity

KW - Nondivergence form operators

KW - Percolation

KW - Random walks in random environments

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JO - Annals of Probability

JF - Annals of Probability

IS - 3

ER -

AN ELLIPTIC HARNACK INEQUALITY FOR DIFFERENCE EQUATIONS WITH RANDOM BALANCED COEFFICIENTS

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