The First Exit Time of a Two-Dimensional Symmetric Stable Process from a Wedge

Abstract

Let $T_\theta$ be the first exit time of a symmetric stable process [with parameter $\alpha \in (0, 2)$] from a wedge of angle $2\theta, 0 < \theta < \pi$. Then there are constants $p_{\theta, \alpha} > 0$ such that for starting points $x$ in the wedge, $E_xT^p_\theta < \infty$ if $0 < p < p_{\theta, \alpha}$ and $E_xT^p_\theta = \infty$ if $p > p_{\theta, \alpha}$. We characterize $p_{\alpha, \theta}$ and obtain upper and lower bounds.