  Let $T^D$ denote the first exit time of a planar Brownian motion from a
domain $D$. Given two simply connected planar domains $U,W \neq \SC$ 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 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.
