On the probability of fast exits and long stays of planar Brownian motion in simply connected domains

Let $T^D$ denote the first exit time of a planar Brownian motion from a domain $D$. Given two simply connected planar domains $U,W\mathrm{\ne }\text{SC}$ containing $0$, we investigate the cases in which we are more likely to have fast exits (meaning for instance $\mathbf{P}(T^U<t)>\mathbf{P}(T^W<t)$ for $t$ small) from $U$ than from $W$, or long stays (meaning $\mathbf{P}(T^U>t)>\mathbf{P}(T^W>t)$ for $t$ large). We prove several results on these questions. In particular, we show that the primary factor in the probability of fast exits is the proximity of the boundary to the origin, while for long stays an important factor is the moments of the exit time. The complex analytic theory that motivated our inquiry is also discussed.