Sharp Integrability for Brownian Motion in Parabola-shaped Regions

We study the sharp order of integrability of the exit position of Brownian motion from the planar domains ${\cal P}_\alpha = \{(x,y)\in \bR\times \bR\colon x> 0, |y| < Ax^{\alpha}\}$, $0<\alpha<1$. Together with some simple good-$\lambda$ type arguments, this implies the order of integrability for the exit time of these domains; a result first proved for $\alpha =1/2$ by Ba\~nuelos, DeBlassie and Smits \cite{ba} and for general $\alpha$ by Li \cite{li}. A sharp version of this result is also proved in higher dimensions.