A Parabolic Harnack Principle for Balanced Difference Equations in Random Environments

Noam Berger, David Criens

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)899-947
Number of pages49
JournalArchive for Rational Mechanics and Analysis
Volume245
Issue number2
DOIs
StatePublished - Aug 2022

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