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

Let TD denote the first exit time of a Brownian motion from domain D in Rn. Given domains U,W⊆Rn containing origin, we investigate cases which are more likely to have fast exits U than W, meaning P(TUP(TWt)>P(TW>t) large. This result, applies only two dimensions, shows unit disk has lowest long stays amongst all Schlicht domains.