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

 

Brownian motion in twisted domains


Authors: Dante DeBlassie and Robert Smits
Journal: Trans. Amer. Math. Soc. 357 (2005), 1245-1274
MSC (2000): Primary 60J65, 60J50, 60F10
Published electronically: September 2, 2004
MathSciNet review: 2110439
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Abstract: The tail behavior of a Brownian motion's exit time from an unbounded domain depends upon the growth of the ``inner radius'' of the domain. In this article we quantify this idea by introducing the notion of a twisted domain in the plane. Roughly speaking, such a domain is generated by a planar curve as follows. As a traveler proceeds out along the curve, the boundary curves of the domain are obtained by moving out $\pm g(r)$ units along the unit normal to the curve when the traveler is $r$ units away from the origin. The function $g$ is called the growth radius. Such domains can be highly nonconvex and asymmetric. We give a detailed account of the case $g(r) = \gamma r^p$, $0<p\le 1$. When $p=1$, a twisted domain can reasonably be interpreted as a ``twisted cone.''


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Additional Information

Dante DeBlassie
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: deblass@math.tamu.edu

Robert Smits
Affiliation: Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001, Department 3MB, Las Cruces, New Mexico 88003-8001
Email: rsmits@nmsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03568-8
PII: S 0002-9947(04)03568-8
Keywords: Exit times, Brownian motion, twisted domains
Received by editor(s): November 5, 2002
Received by editor(s) in revised form: November 3, 2003
Published electronically: September 2, 2004
Article copyright: © Copyright 2004 American Mathematical Society