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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Boundary Harnack inequality for Markov processes with jumps


Authors: Krzysztof Bogdan, Takashi Kumagai and Mateusz Kwaśnicki
Journal: Trans. Amer. Math. Soc. 367 (2015), 477-517
MSC (2010): Primary 60J50; Secondary 60J75, 31B05
DOI: https://doi.org/10.1090/S0002-9947-2014-06127-8
Published electronically: July 24, 2014
MathSciNet review: 3271268
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Abstract: 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, Lévy processes with and without continuous part, stable-like and censored stable processes, jump processes on fractals, and rather general Schrödinger, drift and jump perturbations of such processes.


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Additional Information

Krzysztof Bogdan
Affiliation: Institute of Mathematics, Polish Academy of Science, ul. Śniadeckich 8, 00-958 Warsaw, Poland – and – Institute of Mathematics and Computer Science, Wrocław University of Technology, ul. Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Email: krzysztof.bogdan@pwr.wroc.pl

Takashi Kumagai
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Email: kumagai@kurims.kyoto-u.ac.jp

Mateusz Kwaśnicki
Affiliation: Institute of Mathematics, Polish Academy of Science, ul. Śniadeckich 8, 00-958 Warsaw, Poland — and — Institute of Mathematics and Computer Science, Wrocław University of Technology, ul. Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Email: mateusz.kwasnicki@pwr.wroc.pl

DOI: https://doi.org/10.1090/S0002-9947-2014-06127-8
Keywords: Boundary Harnack inequality, jump Markov process
Received by editor(s): July 16, 2012
Received by editor(s) in revised form: March 2, 2013, and March 9, 2013
Published electronically: July 24, 2014
Additional Notes: The first author was supported in part by grant N N201 397137.
The second author was supported by the Grant-in-Aid for Challenging Exploratory Research 24654033.
The third author was supported by the Foundation for Polish Science and by the Polish National Science Centre grant no. 2011/03/D/ST1/00311.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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