Hitting probabilities of a Brownian flow with radial drift

We consider a stochastic flow .t(x,.) in Rn with initial point .0(x,.)=x, driven by a single n-dimensional Brownian motion, and with an outward radial drift of magnitude F(..t(x).)..t(x)., 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.n3/4, and if the initial set has a nonempty interior, then the image of the set has positive probability of hitting the origin.