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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Green function estimates for subordinate Brownian motions: Stable and beyond


Authors: Panki Kim and Ante Mimica
Journal: Trans. Amer. Math. Soc. 366 (2014), 4383-4422
MSC (2010): Primary 60J45; Secondary 60J75, 60G51
Published electronically: January 16, 2014
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Abstract: A subordinate Brownian motion $ X$ is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $ \phi $ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for $ X$ on arbitrary open sets. Then we give an explicit form of sharp two-sided estimates of the Green functions of these subordinate Brownian motions in any bounded $ C^{1,1}$ open set. As a consequence, we prove the boundary Harnack inequality for $ X$ on any $ C^{1,1}$ open set with explicit decay rate. Unlike previous work of Kim, Song and Vondraček, our results cover geometric stable processes and relativistic geometric stable process, i.e. the cases when the subordinator has the Laplace exponent

$\displaystyle \phi (\lambda )=\log (1+\lambda ^{\alpha /2})\ \ \ \ (0<\alpha \leq 2, d > \alpha )$

and

$\displaystyle \phi (\lambda )=\log (1+(\lambda +m^{2/\alpha })^{\alpha /2}-m)\ \ \ \ (0<\alpha <2,\, m>0, d >2)\,. $


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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

Ante Mimica
Affiliation: Department of Mathematics, University of Zagreb, Bijenicka Cesta 30, 10000 Zagreb, Croatia
Email: amimica@math.hr

DOI: http://dx.doi.org/10.1090/S0002-9947-2014-06017-0
PII: S 0002-9947(2014)06017-0
Keywords: Geometric stable process, Green function, Harnack inequality, Poisson kernel, harmonic function, potential, subordinator, subordinate Brownian motion
Received by editor(s): August 21, 2012
Received by editor(s) in revised form: November 4, 2012, and November 10, 2012
Published electronically: January 16, 2014
Additional Notes: The research of the first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology(2011-0011199)
The research of the second author was supported in part by the German Science Foundation DFG via IGK “Stochastics and real world models” and SFB 701.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.