HITTING PROBABILITIES OF A BROWNIAN FLOW WITH RADIAL DRIFT

We consider a stochastic flow ϕt (x,ω) in ℝⁿ with initial point ϕ₀(x,ω) = x, driven by a single n-dimensional Brownian motion, and with an outward radial drift of magnitude F ( ‖ ϕ t ( x ) ‖ ) ‖ ϕ 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∗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.