Regularity of harmonic functions for some Markov chains with unbounded range

TL;DR

This paper proves that harmonic functions for certain continuous-time Markov chains with unbounded jumps on integer lattices are Hölder continuous, extending regularity results to chains with potentially large jumps.

Contribution

It establishes Hölder continuity of harmonic functions for a class of Markov chains with unbounded range, a novel regularity result in this context.

Findings

Harmonic functions are Hölder continuous under specified conditions.

The results extend regularity theory to Markov chains with unbounded jumps.

Provides conditions ensuring regularity for discrete-space jump processes.

Abstract

We consider a class of continuous time Markov chains on $\Z^d$. These chains are the discrete space analogue of Markov processes with jumps. Under some conditions, we show that harmonic functions associated with these Markov chains are H\"{o}lder continuous.