Scaling invariant Harnack inequalities in a general setting

TL;DR

This paper establishes simple, scaling-invariant Harnack inequalities for positive harmonic functions associated with general Markov processes, leading to regularity results like continuity and Hölder continuity, especially for Lévy processes.

Contribution

It provides a general framework for Harnack inequalities based solely on exit measures, applicable to a wide class of Markov and Lévy processes, with criteria adaptable to specific settings.

Findings

Harnack inequalities are valid under simple criteria based on exit measures.

Harmonic functions exhibit continuity and Hölder continuity under these inequalities.

Results apply broadly to Lévy and similar processes.

Abstract

In a setting, where only "exit measures" are given, as they are associated with an arbitrary right continuous strong Markov process on a separable metric space, we provide simple criteria for the validity of Harnack inequalities for positive harmonic functions. These inequalities are scaling invariant with respect to a metric on the state space which, having an associated Green function, may be adapted to the special situation. In many cases, this also implies continuity of harmonic functions and H\"older continuity of bounded harmonic functions. The results apply to large classes of L\'evy (and similar) processes.