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Transactions of the American Mathematical Society

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Potential theory of subordinate killed Brownian motion


Authors: Panki Kim, Renming Song and Zoran Vondraček
Journal: Trans. Amer. Math. Soc. 371 (2019), 3917-3969
MSC (2010): Primary 60J45; Secondary 60J50, 60J75
DOI: https://doi.org/10.1090/tran/7358
Published electronically: July 6, 2018
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Abstract: Let $ W^D$ be a killed Brownian motion in a domain $ D\subset \mathbb{R}^d$ and $ S$ an independent subordinator with Laplace exponent $ \phi $. The process $ Y^D$ defined by $ Y^D_t=W^D_{S_t}$ is called a subordinate killed Brownian motion. It is a Hunt process with infinitesimal generator $ -\phi (-\Delta \vert _D)$, where $ \Delta \vert _D$ is the Dirichlet Laplacian. In this paper we study the potential theory of $ Y^D$ under a weak scaling condition on the derivative of $ \phi $. We first show that non-negative harmonic functions of $ Y^D$ satisfy the scale invariant Harnack inequality. Subsequently we prove two types of scale invariant boundary Harnack principles with explicit decay rates for non-negative harmonic functions of $ Y^D$. The first boundary Harnack principle deals with a $ C^{1,1}$ domain $ D$ and non-negative functions which are harmonic near the boundary of $ D$, while the second one is for a more general domain $ D$ and non-negative functions which are harmonic near the boundary of an interior open subset of $ D$. The obtained decay rates are not the same, reflecting different boundary and interior behaviors of $ Y^D$.


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Additional Information

Panki Kim
Affiliation: Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Building 27, 1 Gwanak-ro, Gwanak-gu Seoul 151-747, Republic of Korea
Email: pkim@snu.ac.kr

Renming Song
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801; and School of Mathematical Sciences, Nankai University, Tianjin 300071, People’s Republic of China
Email: rsong@math.uiuc.edu

Zoran Vondraček
Affiliation: Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia; and Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: vondra@math.hr

DOI: https://doi.org/10.1090/tran/7358
Keywords: Subordinate killed Brownian motion, subordinate Brownian motion, harmonic functions, Harnack inequality, boundary Harnack principle
Received by editor(s): December 6, 2016
Received by editor(s) in revised form: July 21, 2017, and October 17, 2017
Published electronically: July 6, 2018
Additional Notes: This work was supported by National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIP) (No. 2016R1E1A1A01941893).
The second author was supported in part by a grant from the Simons Foundation (No. 429343).
The third author was supported in part by the Croatian Science Foundation under the project 3526.
Article copyright: © Copyright 2018 American Mathematical Society