# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## Reflecting Brownian motion in a cuspHTML articles powered by AMS MathViewer

by R. Dante DeBlassie and Ellen H. Toby
Trans. Amer. Math. Soc. 339 (1993), 297-321 Request permission

## 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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