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 Lévy processes in R^d with positive continuous density of the Lévy measure; stable-like processes in R^d and in domains; and stable-like subordinate diffusions in metric measure spaces.

arXiv comment20 pages

References in corpus (4)

• Estimates and structure of $\alpha$-harmonic functions
• On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes
• Boundary Harnack inequality for Markov processes with jumps
• Boundary Harnack inequalities for regional fractional Laplacian

Cited by in corpus (3)

• Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes
• Accessibility, Martin boundary and minimal thinness for Feller processes in metric measure spaces
• Martin boundary of unbounded sets for purely discontinuous Feller processes