A stability theorem for elliptic Harnack inequalities

TL;DR

This paper establishes a stability theorem for the elliptic Harnack inequality, showing its invariance under graph equivalence and providing a characterization of the inequality.

Contribution

It introduces a stability result for the elliptic Harnack inequality and characterizes the inequality in the context of weighted graphs.

Findings

Elliptic Harnack inequality is stable under graph equivalence.

Provides a new characterization of the elliptic Harnack inequality.

Theorem applies to harmonic functions on weighted graphs.

Abstract

We prove a stability theorem for the elliptic Harnack inequality: if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic functions with respect to one of the graphs if and only if it holds for harmonic functions with respect to the other graph. As part of the proof, we give a characterization of the elliptic Harnack inequality.