Harmonic measures for symmetric stable processes

Jang-Mei Wu¹

¹ Department of Mathematics University of Illinois 1409 West Green Street Urbana, IL 61801, U.S.A.

Studia Mathematica, Tome 149 (2002) no. 3, pp. 279-291

Résumé

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.

Détail

DOI : 10.4064/sm149-3-5

Keywords: set mathbb omega cdot harmonic measure respect symmetric alpha stable process alpha killed leaving study inequalities volumes capacities which imply set partial has zero harmonic measure others which imply has positive harmonic measure general relative sizes sets setminus determine whether omega zero positive