Boundary Behavior of Harmonic Functions for Truncated Stable Processes

For any α∈(0,2), a truncated symmetric α-stable process in ℝ ^d is a symmetric Lévy process in ℝ ^d with no diffusion part and with a Lévy density given by c|x|^−d−α 1_{|x|<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, 2007]) 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 this 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 in ℝ ^d called bounded roughly connected κ-fat open sets (including bounded non-convex κ-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 κ-fat open sets. The research of P. Kim is supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-331-C00037). The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167.