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

Let T_D 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(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.