journal of functional analysis 159, 267294 (1998) Martin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes

Martin boundaries and integral representations of positive functions which are harmonic in a bounded domain D with respect to Brownian motion are well under-stood. Unlike the Brownian case, there are two different kinds of harmonicity with respect to a discontinuous symmetric stable process. One kind are functions harmonic in D with respect to the whole process X, and the other are functions harmonic in D with respect to the process XD killed upon leaving D. In this paper we show that for bounded Lipschitz domains, the Martin boundary with respect to the killed stable process XD can be identified with the Euclidean boundary. We further give integral representations for both kinds of positive harmonic functions. Also given is the conditional gauge theorem conditioned according to Martin kernels and the limiting behaviors of the h-conditional stable process, where h is a positive harmonic function of XD. In the case when D is a bounded C1, 1 domain, sharp estimate on the Martin kernel of D is obtained. 1998 Academic Pres