Boundary Harnack Principle and Martin Boundary at Infinity for Subordinate Brownian Motions

Panki Kim; Renming Song; Zoran Vondraček

doi:10.1007/s11118-013-9375-4

Boundary Harnack Principle and Martin Boundary at Infinity for Subordinate Brownian Motions

Panki Kim, Renming Song, Zoran Vondraček

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of κ-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.

Original language	English (US)
Pages (from-to)	407-441
Number of pages	35
Journal	Potential Analysis
Volume	41
Issue number	2
DOIs	https://doi.org/10.1007/s11118-013-9375-4
State	Published - Jun 2014

Keywords

Boundary Harnack principle
Harmonic functions
Lévy processes
Martin boundary
Martin kernel
Poisson kernel
Subordinate Brownian motion

ASJC Scopus subject areas

Analysis

Online availability

10.1007/s11118-013-9375-4

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title = "Boundary Harnack Principle and Martin Boundary at Infinity for Subordinate Brownian Motions",

abstract = "In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of κ-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.",

keywords = "Boundary Harnack principle, Harmonic functions, L{\'e}vy processes, Martin boundary, Martin kernel, Poisson kernel, Subordinate Brownian motion",

author = "Panki Kim and Renming Song and Zoran Vondra{\v c}ek",

note = "Funding Information: P. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MEST) (2013004822). Funding Information: Z. Vondracˇek was supported in part by the MZOS grant 037-0372790-2801. Funding Information: R. Song was supported in part by a grant from the Simons Foundation (208236).",

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doi = "10.1007/s11118-013-9375-4",

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AU - Vondraček, Zoran

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PY - 2014/6

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N2 - In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of κ-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.

AB - In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of κ-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.

KW - Boundary Harnack principle

KW - Harmonic functions

KW - Lévy processes

KW - Martin boundary

KW - Martin kernel

KW - Poisson kernel

KW - Subordinate Brownian motion

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Boundary Harnack Principle and Martin Boundary at Infinity for Subordinate Brownian Motions

Abstract

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