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 domainDwith respect to Brownian motion are well understood. Unlike the Brownian case, there are two different kinds of harmonicity with respect to a discontinuous symmetric stable process. One kind are functions harmonic inDwith respect to the whole processX, and the other are functions harmonic inDwith respect to the processX^Dkilled upon leavingD. In this paper we show that for bounded Lipschitz domains, the Martin boundary with respect to the killed stable processX^Dcan 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 theh-conditional stable process, wherehis a positive harmonic function ofX^D. In the case whenDis a boundedC^1,1domain, sharp estimate on the Martin kernel ofDis obtained.