Uniqueness for the Skorokhod Equation with Normal Reflection in Lipschitz Domains

Abstract

We consider the Skorokhod equation $$dX_t=dW_t+(1/2)\nu(X_t), dL_t$$ in a domain $D$, where $W_t$ is Brownian motion in $R^d$, $\nu$ is the inward pointing normal vector on the boundary of $D$, and $L_t$ is the local time on the boundary. The solution to this equation is reflecting Brownian motion in $D$. In this paper we show that in Lipschitz domains the solution to the Skorokhod equation is unique in law.