Green function estimates and Harnack inequality for subordinate Brownian motions

Murali Rao; Renming Song; Zoran Vondraček

doi:10.1007/s11118-005-9003-z

Green function estimates and Harnack inequality for subordinate Brownian motions

Murali Rao, Renming Song, Zoran Vondraček

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

Abstract

Let X be a Lévy process in ℝ^d, d ≥ 3, obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.

Original language	English (US)
Pages (from-to)	1-27
Number of pages	27
Journal	Potential Analysis
Volume	25
Issue number	1
DOIs	https://doi.org/10.1007/s11118-005-9003-z
State	Published - Aug 2006

Keywords

Capacity
Complete Bernstein function
Green function
Harnack inequality
Subordinate Brownian motion
Subordinator

ASJC Scopus subject areas

Analysis

Online availability

10.1007/s11118-005-9003-z

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title = "Green function estimates and Harnack inequality for subordinate Brownian motions",

abstract = "Let X be a L{\'e}vy process in ℝd, d ≥ 3, obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a L{\'e}vy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the L{\'e}vy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.",

keywords = "Capacity, Complete Bernstein function, Green function, Harnack inequality, Subordinate Brownian motion, Subordinator",

author = "Murali Rao and Renming Song and Zoran Vondra{\v c}ek",

note = "Funding Information: We thank Z.-Q. Chen for helpful discussions on the first version of this paper. We also thank the referee for the helpful comments. The research of Renming Song is supported in part by a joint US-Croatia grant INT 0302167. The research of Zoran Vondracˇek is supported in part by MZOS grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167.",

year = "2006",

month = aug,

doi = "10.1007/s11118-005-9003-z",

language = "English (US)",

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journal = "Potential Analysis",

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T1 - Green function estimates and Harnack inequality for subordinate Brownian motions

AU - Rao, Murali

AU - Song, Renming

AU - Vondraček, Zoran

N1 - Funding Information: We thank Z.-Q. Chen for helpful discussions on the first version of this paper. We also thank the referee for the helpful comments. The research of Renming Song is supported in part by a joint US-Croatia grant INT 0302167. The research of Zoran Vondracˇek is supported in part by MZOS grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167.

PY - 2006/8

Y1 - 2006/8

N2 - Let X be a Lévy process in ℝd, d ≥ 3, obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.

AB - Let X be a Lévy process in ℝd, d ≥ 3, obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.

KW - Capacity

KW - Complete Bernstein function

KW - Green function

KW - Harnack inequality

KW - Subordinate Brownian motion

KW - Subordinator

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DO - 10.1007/s11118-005-9003-z

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ER -

Green function estimates and Harnack inequality for subordinate Brownian motions

Abstract

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