On pathwise uniqueness for reflecting Brownian motion in \(C^{1+\gamma}\) domains (Q2519684)

Let \(X_t\) be a solution to the Skorokhod equation \[ X_t=X_0+W_t+\int_0^t{\mathbf n}(X_s)dL_s \] in a domain \(D\), where \(W_t\) is a \(d\)-dimensional Brownian motion, \({\mathbf n}\) is the inward pointing unit normal vector field on \(\partial D\) and \(L_t\) is the local time on \(\partial D\). When \(D\) is \(C^{1+\gamma}\), \(\gamma>1/2\), and \(d\geq 3\), pathwise uniqueness holds true. The last section of the paper displays a counterexample about a nearby stochastic differential equation with \(d=3\) and \(\gamma\in (0,1/2)\) where pathwise uniqueness does not hold any more.