Harmonic functions of subordinate killed Brownian motion

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

doi:10.1016/j.jfa.2004.01.001

Harmonic functions of subordinate killed Brownian motion

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

Research output: Contribution to journal › Article › peer-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 language	English (US)
Pages (from-to)	399-426
Number of pages	28
Journal	Journal of Functional Analysis
Volume	215
Issue number	2
DOIs	https://doi.org/10.1016/j.jfa.2004.01.001
State	Published - 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

Online availability

10.1016/j.jfa.2004.01.001

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title = "Harmonic functions of subordinate killed Brownian motion",

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.",

keywords = "Boundary Harnack principle, Fractional Laplacian, Green function, Harmonic functions, Harnack inequality, Intrinsic ultracontractivity, Killed Brownian motions, Martin boundary, Martin kernel, Subordination",

author = "J. Glover and Z. Pop-Stojanovic and M. Rao and H. {\v S}iki{\'c} and R. Song and Z. Vondra{\v c}ek",

note = "Funding Information: Keywords: Killed Brownian motions; Subordination; Fractional Laplacian; Harmonic functions; Green function; Martin kernel; Martin boundary; Harnack inequality; Boundary Harnack principle; Intrinsic ultracontractivity ·Corresponding author. E-mail addresses: glover@math.ufl.edu (J. Glover), zps@math.ufl.edu (Z. Pop-Stojanovic), rao@math. ufl.edu (M. Rao), hsik ic@math.hr (H. Sˇ ik i{\'c}), rsong@math.uiuc.edu (R. Song), vondra@math.hr (Z. Vondracˇ ek ). 1The research is supported in part by MZT Grant 0037118 of the Republic of Croatia and in part by a joint US-Croatia Grant INT 0302167. 2The research is supported in part by a joint US-Croatia Grant INT 0302167. 3The research is Supported in part by MZT Grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia Grant INT 0302167.",

year = "2004",

month = oct,

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doi = "10.1016/j.jfa.2004.01.001",

language = "English (US)",

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journal = "Journal of Functional Analysis",

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T1 - Harmonic functions of subordinate killed Brownian motion

AU - Glover, J.

AU - Pop-Stojanovic, Z.

AU - Rao, M.

AU - Šikić, H.

AU - Song, R.

AU - Vondraček, Z.

N1 - Funding Information: Keywords: Killed Brownian motions; Subordination; Fractional Laplacian; Harmonic functions; Green function; Martin kernel; Martin boundary; Harnack inequality; Boundary Harnack principle; Intrinsic ultracontractivity ·Corresponding author. E-mail addresses: glover@math.ufl.edu (J. Glover), zps@math.ufl.edu (Z. Pop-Stojanovic), rao@math. ufl.edu (M. Rao), hsik ic@math.hr (H. Sˇ ik ić), rsong@math.uiuc.edu (R. Song), vondra@math.hr (Z. Vondracˇ ek ). 1The research is supported in part by MZT Grant 0037118 of the Republic of Croatia and in part by a joint US-Croatia Grant INT 0302167. 2The research is supported in part by a joint US-Croatia Grant INT 0302167. 3The research is Supported in part by MZT Grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia Grant INT 0302167.

PY - 2004/10/15

Y1 - 2004/10/15

N2 - 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.

AB - 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.

KW - Boundary Harnack principle

KW - Fractional Laplacian

KW - Green function

KW - Harmonic functions

KW - Harnack inequality

KW - Intrinsic ultracontractivity

KW - Killed Brownian motions

KW - Martin boundary

KW - Martin kernel

KW - Subordination

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JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -

Harmonic functions of subordinate killed Brownian motion

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Keywords

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