Martin kernels for Markov processes with jumps

2015/09/18 by Tomasz Juszczyszyn, Juszczyszyn, Tomasz, Mateusz Kwaśnicki +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1509.05677

openalex publication_date 2015/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Boundary Harnack inequalities for regional fractional Laplacian

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