Harmonic measures for symmetric stable processes

Let D be an open set in R (n ≥ 2) and ω(·,D) be the harmonic measure on Dc with respect to the symmetric α-stable process (0 < α < 2) killed upon leaving D. We study inequalities on volumes or capacities which imply that a set S on ∂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 Dc \S that determine whether ω(S,D) is zero or positive. We study null sets of harmonic measures for symmetric α-stable processes. A symmetric α-stable process X on R is a Lévy process whose transition density p(t, x−y) relative to Lebesgue measure is uniquely determined by its Fourier transform Rn eix·ξp(t, x) dx = e−t|ξ| α . Here α must be in the interval (0, 2]. When α = 2, it is a Brownian motion running with a time clock twice as fast as the standard one. From now on, we assume 0 < α < 2, when referring to symmetric stable processes. Unlike the Brownian motion, the generator of a symmetric α-stable process is nonlocal, as it is the fractional Laplacian −(−∆)α/2, an integro-differential operator. A symmetric stable process has discontinuous sample paths and heavy tails, while Brownian motion has continuous sample paths and exponentially decaying tails. Basic properties of symmetric stable processes and their potential-theoretic formulations in terms of Riesz kernels can be found in [BG] and [L]. From now on, D is an open set in R (n ≥ 2), 0 < α < 2, X is the symmetric α-stable processX killed upon leavingD, and τD is the first exit time. An α-harmonic function u in D is a locally integrable function in R continuous in D satisfying |x|>1 |u(x)| · |x|−n−α dx <∞ and (0.1) u(x) = Eu(XτB(x,r)) 2000 Mathematics Subject Classification: 60J45, 31C99. The research is supported in part by NSF Grant DMS-0070312.