MARTIN BOUNDARY AND INTEGRAL-REPRESENTATION FOR HARMONIC-FUNCTIONS OFSYMMETRICAL STABLE PROCESSES

Martin boundaries and integral representations of positive functions w hich are harmonic in a bounded domain D with respect to Brownian motio n are well understood. Unlike the Brownian case, there are two differe nt kinds of harmonicity with respect to a discontinuous symmetric stab le process. One kind are functions harmonic in D with respect to the w hole process X, and the other are functions harmonic in D with respect to the process X-D killed upon leaving D. In this paper we show that for bounded Lipschitz domains, the Martin boundary with respect to the killed stable process X-D can be identified with the Euclidean bounda ry. We further give integral representations for both kinds of positiv e harmonic functions. Also given is the conditional gauge theorem cond itioned according to Martin kernels and the limiting behaviors of the h-conditional stable process, where h is a positive harmonic function of X-D. In the case when D is a bounded C-1,C-1 domain, sharp estimate on the Martin kernel of D is obtained. (C) 1998 Academic Press.