Boundary Behavior of Harmonic Functions for Truncated Stable Processes

Panki Kim, Renming Song

Published 2006-06-28, updated 2007-05-13Version 4

For any \alpha in (0, 2), a truncated symmetric \alpha-stable process is a symmetric Levy process with no diffusion part and with a Levy density given by c|x|^{-d-\alpha} 1_{|x|< 1} for some constant c. In previous paper we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of a truncated symmetric stable process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in non-convex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets called bounded roughly connected \kappa-fat open sets (including bounded non-convex \kappa-fat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected \kappa-fat open sets.