Symmetric stable processes stay in thick sets

Abstract

Let $X(t)$ be the symmetric $\alpha$-stable process in $\Bbb R^d (0<\alpha<2, d\ge 2)$. Then let $W(f)$ be the thorn $\{x \in \Bbb R^d\dvtx 0<x_1<1, (x_2^2 +\cdots + x_d^2)^{1/2} < f(x_1) \}$ where $f\dvtx (0,1)\rightarrow(0,1)$ is continuous, increasing with $f(0^+) = 0$. Recently Burdzy and Kulczycki gave an exact integral condition on f for the existence of a random time s such that $X(t)$ remains in the thorn $X(s)+ \overline{W(f)}$ for all $t \in [s,s+1)$. We extend their theorem to general open sets W with $0 \in \partial W$. In general, $\alpha$-processes may stay in sets which are quite lacunary and are not locally connected at 0.