Potential theory of special subordinators and subordinate killed stable processes

Renming Song, Zoran Vondraček

Research output: Contribution to journalArticle

Abstract

In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one-to-one correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κ-fat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D.

Original languageEnglish (US)
Pages (from-to)817-847
Number of pages31
JournalJournal of Theoretical Probability
Volume19
Issue number4
DOIs
StatePublished - Dec 2006

Keywords

  • Bernstein functions
  • Complete Bernstein functions
  • Green function
  • Harmonic functions
  • Harnack inequality
  • Killed Brownian motions
  • Killed symmetric stable processes
  • Martin boundary
  • Martin kernel
  • Subordination
  • Subordinators

ASJC Scopus subject areas

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

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