Exit times from cones in $\mathbf{R}^n$ of symmetric stable processes

Abstract

Let $T_\theta$ be the first exit time of an $n$-dimensional symmetric stable process with parameter $\alpha \in (0,2)$ from a cone of angle $\theta$, $0 \lt \theta \lt \pi$. Then there exists a constant $ p(\theta,\alpha,n)$ such that for $x$ in the cone, $E^x T_\theta^p \lt \infty$ if $p \lt p(\theta,\alpha,n)$, and $E^x T_\theta^p = \infty$ if $ p \gt p(\theta,\alpha,n)$. We characterize $p(\theta,\alpha,n)$ in terms of the principal eigenvalue of an operator and give upper and lower bounds for it. We also present a generalization of this result to more general cones in $\mathbf{R}^n$. These results extend the two-dimensional results of R. D. DeBlassie to $n$ dimensions and more general cones.