On the mean exit time from a ball for a symmetric stable process

TL;DR

The paper addresses the mean exit time from a ball for symmetric $\alpha$-stable processes and generalizes Getoor's result from the standard isotropic case to arbitrary symmetric stable processes. It employs a reduction to the one-dimensional Getoor identity, a generator-domain argument, and a local Ito formula for jump processes to derive a universal exit-time formula. The main contribution is the explicit expression $E^x\tau^X_{B_r} = \frac{\kappa_\alpha}{\nu(B_1^c)} (r^2 - |x|^2)_+^{\alpha/2}$ with $\kappa_\alpha = \frac{1}{\Gamma(1-\alpha/2)\Gamma(1+\alpha/2)}$. This result highlights a robust, dimensionally consistent exit-time behavior for a broad class of symmetric stable processes and connects to potential theory through the Lévy measure structure.

Abstract

Getoor in [3] calculated the mean exit time from a ball for the standard isotropic $α$-stable process in $\mathbb{R}^d$ starting from the interior of the ball. The purpose of this note is to show that, up to multplicative constant, the same formula is valid for any symmetric $α$-stable process.

Theorems & Definitions (6)

• Theorem 1.1
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• proof : proof of Theorem \ref{['main']}