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

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@article{42aa8caccb154699a23f5d26ef42fcfc,

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,

day = "15",

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

language = "English (US)",

volume = "215",

pages = "399--426",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

publisher = "Academic Press Inc.",

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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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DO - 10.1016/j.jfa.2004.01.001

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VL - 215

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EP - 426

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

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