Hitting Probabilities of a Brownian flow with Radial Drift

Abstract

We consider a stochastic flow ${\varphi }_t(x,\omega )$ in ${\mathbb{R}}^n$ with initial point ${\varphi }_0(x,\omega )=x$, driven by a single $n$-dimensional Brownian motion, and with an outward radial drift of magnitude $\frac{F(\Vert {\varphi }_t(x)\Vert )}{\Vert {\varphi }_t(x)\Vert }$, 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 $C^{\ast },c^{\ast }>0$ not depending on $n$, such that if $F>C^{\ast }n$ then the image of the initial set under the flow has probability 0 of hitting the origin. If $0\le F\le c^{\ast }{n}^{3/4}$, and if the initial set has nonempty interior, then the image of the set has positive probability of hitting the origin.

34 pages, 3 figures