Pathwise uniqueness for a SDE with non-Lipschitz coefficients

We consider the ordinary stochastic differential equation on the closed unit ball E in . While it is easy to prove existence and distribution uniqueness for solutions of this SDE for each c[greater-or-equal, slanted]0, pathwise uniqueness can be proved by standard methods only in dimension n=1 and in dimensions n[greater-or-equal, slanted]2 if c=0 or if c[greater-or-equal, slanted]2 and the initial condition is in the interior of E. We sharpen these results by proving pathwise uniqueness for c[greater-or-equal, slanted]1. More precisely, we show that for X1,X2 solutions relative to the same Brownian motion, the function is almost surely nonincreasing. Whether or not pathwise uniqueness holds in dimensions n[greater-or-equal, slanted]2 for 0<c<1 is still open.