Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

Let $D$ be an open set in ${\mathbb R}^n\ (n \ge 2)$ and $\omega (\cdot ,D)$ be the harmonic measure on $D^{\rm c}$ with respect to the symmetric $\alpha $-stable process $(0 < \alpha < 2)$ killed upon leaving $D$. We study inequalities on volumes or capacities which imply that a set $S$ on $\partial D$ has zero harmonic measure and others which imply that $S$ has positive harmonic measure. In general, it is the relative sizes of the sets $S$ and $D^{\rm c}\setminus S$ that determine whether $\omega (S,D)$ is zero or positive.