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Bessel bridge representation for the heat kernel in hyperbolic space


Authors: Xue Cheng and Tai-Ho Wang
Journal: Proc. Amer. Math. Soc. 146 (2018), 1781-1792
MSC (2010): Primary 60G99, 60J35, 60J60
DOI: https://doi.org/10.1090/proc/13952
Published electronically: January 16, 2018
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Abstract: This article shows a Bessel bridge representation for the transition density of Brownian motion on the Poincaré space. This transition density is also referred to as the heat kernel on the hyperbolic space in differential geometry literature. The representation recovers the well-known closed form expression for the heat kernel on hyperbolic space in dimension three. However, the newly derived bridge representation is different from the McKean kernel in dimension two and from the Gruet's formula in higher dimensions. The methodology is also applicable to the derivation of an analogous Bessel bridge representation for the heat kernel on a Cartan-Hadamard radially symmetric space and for the transition density of the hyperbolic Bessel process.


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Additional Information

Xue Cheng
Affiliation: LMEQF, Department of Financial Mathematics, School of Mathematical Sciences, Peking University, Beijing, People’s Republic of China
Email: chengxue@pku.edu.cn

Tai-Ho Wang
Affiliation: Department of Mathematics, Baruch College, The City University of New York, 1 Bernard Baruch Way, New York, New York 10010
Email: tai-ho.wang@baruch.cuny.edu

DOI: https://doi.org/10.1090/proc/13952
Received by editor(s): February 2, 2017
Published electronically: January 16, 2018
Additional Notes: The authors are partially supported by the Natural Science Foundation of China, grant 11601018. The first author is also partially supported by the Natural Science Foundation of China grant 11471051.
Communicated by: Zhen-Qing Chen
Article copyright: © Copyright 2018 American Mathematical Society

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