Abstract
Let C C be the cusp { ( x , y ) : x ≥ 0 \{ (x,y):x \geq 0 , − x β ≤ y ≤ x β } - {x^\beta } \leq y \leq {x^\beta }\} where β > 1 \beta > 1 . Set ∂ C 1 = { ( x , y ) : x ≥ 0 , y = − x β } \partial {C_1} = \{ (x,y):x \geq 0, y = - {x^\beta }\} and ∂ C 2 = { ( x , y ) : x ≥ 0 \partial {C_2} = \{ (x,y):x \geq 0 , y = x β } y = {x^\beta }\} . We study the existence and uniqueness in law of reflecting Brownian motion in C C . The angle of reflection at ∂ C j ∖ { 0 } \partial {C_j}\backslash \{ 0\} (relative to the inward unit normal) is a constant θ j ∈ ( − π 2 , π 2 ) {\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 0 . When θ 1 + θ 2 ≤ 0 {\theta _1} + {\theta _2} \leq 0 , existence and uniqueness in law hold. When θ 1 + θ 2 > 0 {\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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