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

DOI:
https://doi.org/10.1090/S0002-9947-04-03568-8

Published electronically:
September 2, 2004

MathSciNet review:
2110439

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 units along the unit normal to the curve when the traveler is units away from the origin. The function is called the growth radius. Such domains can be highly nonconvex and asymmetric. We give a detailed account of the case , . When , a twisted domain can reasonably be interpreted as a ``twisted cone.''

**1.**Bañuelos, R. and Davis, B. (1989). Heat kernel, eigenfunctions, and conditioned Brownian motion in planar domains, Journal of Functional Analysis**84**188-200. MR**91a:60203****2.**Bañuelos, R., DeBlassie, R.D. and Smits, R. (2001). The first exit time of planar Brownian motion from the interior of a parabola, Annals of Probability**29**882-901. MR**2002h:60165****3.**van den Berg, M. (2003). Subexponential behavior of the Dirichlet heat kernel, Journal of Functional Analysis**198**28-42. MR**2003m:60211****4.**Li, W. (2003). The first exit time of Brownian motion from an unbounded convex domain, Annals of Probability**31**1078-1096.MR**2004c:60126****5.**Lifshits, M. and Shi, Z. (2002), The first exit time of Brownian motion from a parabolic domain, Bernoulli**8**745-765. MR**2004d:60213****6.**Spitzer, F. (1958). Some theorems concerning two-dimensional Brownian motion, Transactions of the American Mathematical Society**87**187-197. MR**21:3051****7.**Warschawski, S.E. (1942). On conformal mapping of infinite strips, Transactions of the American Mathematical Society**51**280-335. MR**4:9b**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
60J65,
60J50,
60F10

Retrieve articles in all journals with MSC (2000): 60J65, 60J50, 60F10

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:
https://doi.org/10.1090/S0002-9947-04-03568-8

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