## 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 language | English (US) |
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Pages (from-to) | 3917-3969 |

Number of pages | 53 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 6 |

DOIs | |

State | Published - Mar 15 2019 |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics