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.
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Kwaśnicki, M., Juszczyszyn, T. Martin Kernels for Markov Processes with Jumps. Potential Anal 47, 313–335 (2017). https://doi.org/10.1007/s11118-017-9616-z
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DOI: https://doi.org/10.1007/s11118-017-9616-z
Keywords
- Markov process
- Jump process
- Killed process
- Boundary Harnack inequality
- Boundary limit
- Martin kernel
- Martin boundary
- Martin representation