Hitting Probabilities of a Brownian flow with Radial Drift

TL;DR

This paper analyzes the probability of a stochastic flow with radial drift hitting the origin, showing thresholds for the drift magnitude that determine whether the flow can reach the origin.

Contribution

It establishes bounds on the radial drift strength that guarantee or prevent the flow from hitting the origin, extending understanding of Brownian flows with radial drift.

Findings

Flow with drift > C* n almost surely avoids the origin.

Flow with drift ≤ c* n^{3/4} has positive probability to hit the origin.

Results are independent of the dimension n.

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