Martin kernels for Markov processes with jumps

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.