Harmonic functions of subordinate killed Brownian motion

J. Glover, Z. Pop-Stojanovic, M. Rao, H. Šikić, R. Song, Z. Vondraček

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.

Original languageEnglish (US)
Pages (from-to)399-426
Number of pages28
JournalJournal of Functional Analysis
Volume215
Issue number2
DOIs
StatePublished - Oct 15 2004

Keywords

  • Boundary Harnack principle
  • Fractional Laplacian
  • Green function
  • Harmonic functions
  • Harnack inequality
  • Intrinsic ultracontractivity
  • Killed Brownian motions
  • Martin boundary
  • Martin kernel
  • Subordination

ASJC Scopus subject areas

  • Analysis

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