Martin Kernels for Markov Processes with Jumps |
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Authors: | Mateusz Kwaśnicki Tomasz Juszczyszyn |
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Institution: | 1.Faculty of Pure and Applied Mathematics,Wroclaw University of Science and Technology,Wroclaw,Poland |
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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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