## Abstract

Let T_{D} denote the first exit time of a Brownian motion from a domain D in R^{n} . Given domains U, W ⊆ R^{n} containing the origin, we investigate the cases in which we are more likely to have fast exits from U than W, meaning P(T_{U} < t) > P(T_{W} < 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(T_{U} > t) > P(T_{W} > 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 language | English |
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Article number | 58 |

Number of pages | 12 |

Journal | Electronic Communications in Probability |

Volume | 27 |

DOIs | |

Publication status | Published - 2022 |

## Keywords

- Brownian motion
- capacity
- exit time distribution