## The Annals of Probability

- Ann. Probab.
- Volume 11, Number 3 (1983), 578-588.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 11, Number 3 (1983), 578-588.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993502

**Digital Object Identifier**

doi:10.1214/aop/1176993502

**Mathematical Reviews number (MathSciNet)**

MR704544

**Zentralblatt MATH identifier**

0516.60085

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Secondary: 60G46: Martingales and classical analysis

**Keywords**

Symmetric stable process maximal function exit times

#### Citation

Bass, R. F.; Cranston, M. Exit Times for Symmetric Stable Processes in $\mathbb{R}^n$. Ann. Probab. 11 (1983), no. 3, 578--588. doi:10.1214/aop/1176993502. https://projecteuclid.org/euclid.aop/1176993502