On the probability of fast exits and long stays of a planar Brownian motion in simply connected domains

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:493

Article

Betsakos, Dimitrios¹  Boudabra, Maher²  Markowsky, Greg² 

[1] Aristotle Univ Thessaloniki, Dept Math, Thessaloniki, Greece

[2] Monash Univ, Dept Math, Clayton, Vic, Australia

关键词: Planar Brownian motion;    Simply connected domains;    Exit times;   

DOI  :  10.1016/j.jmaa.2020.124454

来源: Elsevier

【 摘 要 】

Let T-D denote the first exit time of a planar Brownian motion from a domain D. Given two simply connected planar domains U, W not equal C containing 0, we investigate the cases in which we are more likely to have fast exits (meaning P(T-U < t) > P(T-W < t) for t small) from U than from W, or long stays (meaning P(T-U > t) > P(T-W > t) for t large). We prove several results on these questions. In particular, we show that the primary factor in the probability of fast exits is the proximity of the boundary to the origin, while for long stays an important factor is the moments of the exit time. The complex analytic theory that motivated our inquiry is also discussed. (c) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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