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 Rd with positive continuous density of the Lévy measure; stable-like processes in Rd and in domains; and stable-like subordinate diffusions in metric measure spaces.
Highlights
The purpose of this article is to study boundary limits of ratios of positive functions which are harmonic in an arbitrary open set with respect to a Markov process with jumps
For the existence of boundary limits, we follow the approach of [11] using the boundary Harnack inequality of [12], and prove in our main results, Theorems 2 and 3, the existence of boundary limits of ratios of harmonic functions for arbitrary open sets and rather general Markov processes with jumps, as well as Martin representation of such functions
We provide a counter-example, which shows that the boundary limits (2) typically fail to exist in irregular domains when the process Xt has a non-trivial diffusion part
Summary
The purpose of this article is to study boundary limits of ratios of positive functions which are harmonic in an arbitrary open set with respect to a Markov process with jumps. Bogdan ([6]), where he proved the result for the isotropic stable Levy process (equivalently: for the fractional Laplace operator −(−Δ)α/2) and Lipschitz domains Later this was extended to more general sets ([11, 34]) and processes ([8, 13, 21,22,23,24,25,26]). For the existence of boundary limits, we follow the approach of [11] using the boundary Harnack inequality of [12], and prove in our main results, Theorems 2 and 3, the existence of boundary limits of ratios of harmonic functions for arbitrary open sets and rather general Markov processes with jumps, as well as Martin representation of such functions.
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