On the duration of stays of Brownian motion in domains in Euclidean space

Dimitrios Betsakos, Maher Boudabra, Greg Markowsky

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Let TD denote the first exit time of a Brownian motion from a domain D in Rn . Given domains U, W ⊆ Rn 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.

Original languageEnglish
Article number58
Number of pages12
JournalElectronic Communications in Probability
Publication statusPublished - 2022


  • Brownian motion
  • capacity
  • exit time distribution

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