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

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