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 P-alpha = {(x, y) is an element of R x R: x > 0, \y\ < Axalpha}, 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 Banuelos et al. (Ann. Probab. 29 (2001) 882) and for general alpha by Li (Ann. Probab. 31 (2003) 1078). A sharp version of this result is also proved in higher dimensions. (C) 2004 Elsevier Inc. All rights reserved.