Uniqueness for reflecting Brownian motion in lip domains

Abstract

A lip domain is a Lipschitz domain where the Lipschitz constant is strictly less than one. We prove strong existence and pathwise uniqueness for the solution X = {X [subscript] t, t [is less than or equal to] 0} to the Skorokhod equation dX [subscript] t = dW [subscript] t + n(X [subscript] t)dL [subscript] t, in planar lip domains, where W = {W [subscript] t, t [is greater than or equal to] 0} is a Brownian motion, n is the inward pointing unit normal vector, and L = {L [subscript] t, t [is greater than or equal to] 0} is a local time on the boundary which satisfies some additional regularity conditions. Counterexamples are given for some Lipschitz (but not lip) three dimensional domains.