Potential theory of subordinate killed Brownian motion

Panki Kim, Renming Song, Zoran Vondraček

Research output: Contribution to journalArticlepeer-review

Abstract

Let W D be a killed Brownian motion in a domain D ⊂ R d and S an independent subordinator with Laplace exponent φ. The process Y D defined by Y t D = W S D t is called a subordinate killed Brownian motion. It is a Hunt process with infinitesimal generator −φ(−Δ| D ), where Δ| D is the Dirichlet Laplacian. In this paper we study the potential theory of Y D under a weak scaling condition on the derivative of φ. We first show that non-negative harmonic functions of Y D satisfy the scale invariant Harnack inequality. Subsequently we prove two types of scale invariant boundary Harnack principles with explicit decay rates for non-negative harmonic functions of Y D . The first boundary Harnack principle deals with a C 1,1 domain D and non-negative functions which are harmonic near the boundary of D, whilethesecondoneis for a more general domain D and non-negative functions which are harmonic near the boundary of an interior open subset of D. The obtained decay rates are not the same, reflecting different boundary and interior behaviors of Y D .

Original languageEnglish (US)
Pages (from-to)3917-3969
Number of pages53
JournalTransactions of the American Mathematical Society
Volume371
Issue number6
DOIs
StatePublished - Mar 15 2019

Keywords

  • Boundary Harnack principle
  • Harmonic functions
  • Harnack inequality
  • Subordinate Brownian motion
  • Subordinate killed Brownian motion

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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