Boundary behavior of harmonic functions for truncated stable processes

Panki Kim, Renming Song

Research output: Contribution to journalArticlepeer-review

Abstract

For any α (0,2), a truncated symmetric α-stable process in Rd is a symmetric Lévy process in Rd 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.

Original languageEnglish (US)
Pages (from-to)287-321
Number of pages35
JournalJournal of Theoretical Probability
Volume21
Issue number2
DOIs
StatePublished - Jun 2008

Keywords

  • Boundary Harnack principle
  • Green functions
  • Harmonic functions
  • Harnack inequality
  • Martin boundary
  • Poisson kernels
  • Relative Fatou theorem
  • Relative Fatou type theorem
  • Symmetric stable processes
  • Truncated symmetric stable processes

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

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