Martin kernels for Markov processes with jumps

TL;DR

This paper establishes boundary limits and uniqueness of Martin kernels for a broad class of jump Markov processes in irregular domains, extending classical potential theory results to new stochastic settings.

Contribution

It proves the existence and uniqueness of Martin kernels for jump processes in irregular domains within metric measure spaces, generalizing classical harmonic analysis.

Findings

Martin boundary coincides with the topological boundary.

Martin representation theorem for harmonic functions.

Applicability to stable Lévy processes and stable-like processes.

Abstract

We prove existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular domains, in the context of general metric measure spaces. As a corollary, we prove uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable L\'evy processes in R^d with positive continuous density of the L\'evy measure; stable-like processes in R^d and in domains; and stable-like subordinate diffusions in metric measure spaces.