HITTING PROBABILITIES OF A BROWNIAN FLOW WITH RADIAL DRIFT

We consider a stochastic flow phi(t) (x, omega) in R-n with initial point phi(0)(x, omega) = x, driven by a single n-dimensional Brownian motion, and with an outward radial drift of magnitude F(parallel to phi(t)(x)parallel to/parallel to phi(t)(x)parallel to), with F nonnegative, bounded and Lipschitz. We consider initial points x lying in a set of positive distance from the origin. We show that there exist constants C*, c* > 0 not depending on n, such that if F > C* n then the image of the initial set under the flow has probability 0 of hitting the origin. If 0 <= F <= c*n(3/4), and if the initial set has a nonempty interior, then the image of the set has positive probability of hitting the origin.