Hitting probabilities of a Brownian flow with radial drift

Abstract

We consider a stochastic flow $\phi_{t}(x,\omega )$ in $\mathbb{R}^{n}$ with initial point $\phi_{0}(x,\omega )=x$, driven by a single $n$-dimensional Brownian motion, and with an outward radial drift of magnitude $\frac{F(\|\phi_{t}(x)\|)}{\|\phi_{t}(x)\|}$, 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 $0\leq F\leq c^{*}n^{3/4}$, and if the initial set has a nonempty interior, then the image of the set has positive probability of hitting the origin.