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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Reflecting Brownian motion in a cusp


Authors: R. Dante DeBlassie and Ellen H. Toby
Journal: Trans. Amer. Math. Soc. 339 (1993), 297-321
MSC: Primary 60J60; Secondary 60H99, 60J65
DOI: https://doi.org/10.1090/S0002-9947-1993-1149119-1
MathSciNet review: 1149119
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Abstract: Let $C$ be the cusp $\{ (x,y):x \geq 0$, $- {x^\beta } \leq y \leq {x^\beta }\}$ where $\beta > 1$. Set $\partial {C_1} = \{ (x,y):x \geq 0, y = - {x^\beta }\}$ and $\partial {C_2} = \{ (x,y):x \geq 0$, $y = {x^\beta }\}$. We study the existence and uniqueness in law of reflecting Brownian motion in $C$. The angle of reflection at $\partial {C_j}\backslash \{ 0\}$ (relative to the inward unit normal) is a constant ${\theta _j} \in \left ( { - \frac {\pi } {2},\frac {\pi } {2}} \right )$, and is positive iff the direction of reflection has a negative first component in all sufficiently small neighborhoods of $0$. When ${\theta _1} + {\theta _2} \leq 0$, existence and uniqueness in law hold. When ${\theta _1} + {\theta _2} > 0$, existence fails. We also obtain results for a large class of asymmetric cusps. We make essential use of results of Warschawski on the differentiability at the boundary of conformal maps.


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Keywords: Brownian motion with skew reflection, cusp, conformai transformation, differentiability at the boundary
Article copyright: © Copyright 1993 American Mathematical Society