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2022 On the duration of stays of Brownian motion in domains in Euclidean space
Dimitrios Betsakos, Maher Boudabra, Greg Markowsky
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Electron. Commun. Probab. 27: 1-12 (2022). DOI: 10.1214/22-ECP498


Let TD denote the first exit time of a Brownian motion from a domain D in Rn. Given domains U,WRn containing the origin, we investigate the cases in which we are more likely to have fast exits from U than W, meaning P(TU<t)>P(TW<t) for t small. We show that the primary factor in the probability of fast exits from domains is the proximity of the closest regular part of the boundary to the origin. We also prove a result on the complementary question of longs stays, meaning P(TU>t)>P(TW>t) for t large. This result, which applies only in two dimensions, shows that the unit disk D has the lowest probability of long stays amongst all Schlicht domains.


The authors are grateful to two anonymous referees for helpful comments.


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Dimitrios Betsakos. Maher Boudabra. Greg Markowsky. "On the duration of stays of Brownian motion in domains in Euclidean space." Electron. Commun. Probab. 27 1 - 12, 2022.


Received: 6 June 2022; Accepted: 6 November 2022; Published: 2022
First available in Project Euclid: 22 November 2022

MathSciNet: MR4368695
zbMATH: 1504.31014
Digital Object Identifier: 10.1214/22-ECP498

Primary: 31A15 , 31B15 , 60J45 , 60J65

Keywords: Brownian motion , capacity , exit time distribution

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