Boundary Harnack inequality for Markov processes with jumps

By Krzysztof Bogdan, Takashi Kumagai, Mateusz Kwaśnicki. 2012

78 citations recorded by the source corpus.

We prove a boundary Harnack inequality for jump-type Markov processes on metric measure state spaces, under comparability estimates of the jump kernel and Urysohn-type property of the domain of the generator of the process. The result holds for positive harmonic functions in arbitrary open sets. It applies, e.g., to many subordinate Brownian motions, L\'evy processes with and without continuous part, stable-like and censored stable processes, jump processes on fractals, and rather general Schr\"odinger, drift and jump perturbations of such processes.

Subfields: Computational Theory and Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eabacad7fe9505b6f6f1c95263a88d0702ea15ce

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Citations in the corpus

Global uniform boundary Harnack principle with explicit decay rate and its application (2012, background): “13 Very recently, the boundary Harnack principle for (discontinuous) Markov processes (not necessarily Lévy processes) on metric measure state spaces is discussed in [6].”
Boundary regularity for fully nonlinear integro-differential equations (2014, methodology): “Linear equations Lu = f with L in the class L∗ have already been studied, specially by Sztonyk and Bogdan; see for instance [49, 6, 41, 7, 8, 50].”
Ten equivalent definitions of the fractional laplace operator (2015, background): “Various results are also known for the restriction of L to a domain: detailed properties of L in a domain with Dirichlet condition can be found in [8, 9, 10, 11, 33, 38]; see also [6] for the study of L in a domain with certain reflection.”