Exit Times for Symmetric Stable Processes in $\mathbb{R}^n$

Abstract

Let $X_t$ be a symmetric stable process of index $\alpha$ in $\mathbb{R}^n$ and $\tau = \inf\{t: X_t \not\in D\}$ where $D$ is a connected open region in $\mathbb{R}^n$. If $0 < p < \alpha$ two sided $L^p$ inequalities are obtained between $\tau^{1/\alpha}$ and the maximal function $X^\ast_\tau = \sup_{t < \tau} |X_t|$. Analytic conditions for $\tau^{1/\alpha} \in L^p$ are given in terms of domination of $|x|^p, x \in D^c$ by a function $u(x) \alpha$-harmonic in $D$. Also, the boundary behavior of $\alpha$-harmonic functions is studied by obtaining two-sided $L^p$ inequalities, $0 < p < \infty$, between a random and deterministic maximal function of non-negative $\alpha$-harmonic functions.