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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Spectral analysis of Brownian motion with jump boundary


Authors: Yuk J. Leung, Wenbo V. Li and Rakesh
Journal: Proc. Amer. Math. Soc. 136 (2008), 4427-4436
MSC (2000): Primary 60J25; Secondary 30D10, 47D07, 60J35.
Published electronically: June 20, 2008
MathSciNet review: 2431059
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Abstract: Consider a family of probability measures, indexed by $ \partial D$, on a bounded open region $ D\subset \mathbb{R}^d$ with a smooth boundary. For any starting point inside $ D$, we run a standard $ d$-dimensional Brownian motion in $ \mathbb{R}^d $ until it first exits $ D$, at which time it jumps to a point inside the domain $ D$ according to the jump measure at the exit point and starts a new Brownian motion. The same evolution is repeated independently each time the process reaches the boundary. We study the exponential rate at which the transition distribution of the process converges to its invariant measure, in terms of the spectral gap of the generator. In particular, we prove two conjectures of I. Ben-Ari and R. Pinsky for an interval (see J. Funct. Anal. 251 (2007), 122-140, and preprint (2007)) by studying when a combination of the sine and cosine transforms of probability measures on an interval has only real zeros.


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Additional Information

Yuk J. Leung
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
Email: yleung@math.udel.edu

Wenbo V. Li
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
Email: wli@math.udel.edu

Rakesh
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
Email: rakesh@math.udel.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09451-3
PII: S 0002-9939(08)09451-3
Received by editor(s): September 25, 2007
Received by editor(s) in revised form: November 13, 2007
Published electronically: June 20, 2008
Additional Notes: The second author was supported in part by NSF Grant DMS-0505805.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.