Martin Kernels for Markov Processes with Jumps

Abstract

We prove the existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular (possibly disconnected) domains of harmonicity, in the context of general metric measure spaces. As a corollary, we prove the 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.

Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

“…In this section we apply the boundary Harnack inequality of [9] to unimodal Lévy processes and prove Theorem 1.9. We also apply the recent result of [20] to prove Theorem 1.11. Finally, we find estimates of the Green function of the half-space.…”

“…Application of the results of [9] and [20] requires verification of a number of conditions imposed on the process X t in [9]. Below we briefly discuss these conditions.…”

“…The following theorem is a reformulation of the main result of [9], adapted to the setting of isotropic unimodal Lévy processes. We use a version stated in [20]. We use the above notation instead of that introduced in [9].…”

“…If we add a mild regularity condition on ν, we obtain the existence of boundary limits of ratios of harmonic functions. This is an application of the main result of [20] and [30]; we use the version stated in [20]. Theorem 1.11.…”

“…(a) As a consequence of Theorem 1.11, non-negative functions harmonic with respect to X t in D admit Martin representation, with Martin boundary identified with a subset of the Euclidean boundary ∂D. We refer to [20] for more details.…”