  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 nonempty interior, then the image of the set has positive
probability of hitting the origin.
