Open Access
March 2020 Hitting probabilities of a Brownian flow with radial drift
Jong Jun Lee, Carl Mueller, Eyal Neuman
Ann. Probab. 48(2): 646-671 (March 2020). DOI: 10.1214/19-AOP1368


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.


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Jong Jun Lee. Carl Mueller. Eyal Neuman. "Hitting probabilities of a Brownian flow with radial drift." Ann. Probab. 48 (2) 646 - 671, March 2020.


Received: 1 February 2018; Revised: 1 March 2019; Published: March 2020
First available in Project Euclid: 22 April 2020

zbMATH: 07199857
MathSciNet: MR4089490
Digital Object Identifier: 10.1214/19-AOP1368

Primary: 60H10
Secondary: 37C10 , 60J45 , 60J60

Keywords: Bessel process , hitting , Stochastic differential equations , stochastic flow

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 2 • March 2020
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