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.
|Number of pages||12|
|Journal||Electronic Communications in Probability|
|Publication status||Published - 2022|
- Brownian motion
- exit time distribution