Moments of the lifetime of conditioned Brownian motion in cones

Abstract:

Let $\tau$ be the time it takes standard d-dimensional Brownian motion, started at a point inside a cone $\Gamma$ in ${\mathbb {R}^d}$ which has aperture angle $\theta$, to leave the cone. Burkholder has determined the smallest p, denoted $p(\theta ,d)$, such that $E{\tau ^p} = \infty$. We show that if $y \in \partial \Gamma$ then the smallest p, such that $E({\tau ^p}|{B_\tau } = y) = \infty$, is $p = 2p(\theta ,d) + (d - 2)/2$.