Potential theory of subordinate Brownian motions with Gaussian components

Panki Kim, Renming Song, Zoran Vondraček

Research output: Contribution to journalArticle

Abstract

In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in C1,1 open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded C1,1 open set D and identify the Martin boundary of D with respect to the subordinate Brownian motion with the Euclidean boundary.

Original languageEnglish (US)
Pages (from-to)764-795
Number of pages32
JournalStochastic Processes and their Applications
Volume123
Issue number3
DOIs
StatePublished - 2013

Keywords

  • Boundary Harnack principle
  • Exit distribution
  • Green function
  • Harmonic function
  • Lévy system
  • Martin boundary
  • Subordinate Brownian motion

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics

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