Reflecting Brownian motion in a cusp

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.