Let denote the first exit time of a Brownian motion from a domain D in . Given domains containing the origin, we investigate the cases in which we are more likely to have fast exits from U than W, meaning 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 for t large. This result, which applies only in two dimensions, shows that the unit disk has the lowest probability of long stays amongst all Schlicht domains.
The authors are grateful to two anonymous referees for helpful comments.
"On the duration of stays of Brownian motion in domains in Euclidean space." Electron. Commun. Probab. 27 1 - 12, 2022. https://doi.org/10.1214/22-ECP498