Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

Panki Kim; Renming Song

doi:10.1016/j.jmaa.2006.10.014

Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

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Abstract

In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains. Informally the Schrödinger-type operators we consider are of the form L + μ ṡ ∇ + ν where L is a uniformly elliptic second order differential operator, μ is a vector-valued signed measure belonging to K_{d, 1} and ν is a signed measure belonging to K_{d, 2}. In this paper, we establish two-sided estimates for the heat kernels of Schrödinger-type operators in bounded C^{1, 1}-domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrödinger-type operators in bounded Lipschitz domains.

Original language	English (US)
Pages (from-to)	57-80
Number of pages	24
Journal	Journal of Mathematical Analysis and Applications
Volume	332
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2006.10.014
State	Published - Aug 1 2007

Keywords

3G theorem
Boundary Harnack principle
Brownian motion
Diffusion
Diffusion process
Green function
Harmonic function
Heat kernel
Kato class
Lipschitz domain
Measure-valued drift
Non-symmetric diffusion
Schrödinger operator
Transition density

ASJC Scopus subject areas

Analysis
Applied Mathematics

Online availability

10.1016/j.jmaa.2006.10.014

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title = "Estimates on Green functions and Schr{\"o}dinger-type equations for non-symmetric diffusions with measure-valued drifts",

abstract = "In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains. Informally the Schr{\"o}dinger-type operators we consider are of the form L + μ ṡ ∇ + ν where L is a uniformly elliptic second order differential operator, μ is a vector-valued signed measure belonging to Kd, 1 and ν is a signed measure belonging to Kd, 2. In this paper, we establish two-sided estimates for the heat kernels of Schr{\"o}dinger-type operators in bounded C1, 1-domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schr{\"o}dinger-type operators in bounded Lipschitz domains.",

keywords = "3G theorem, Boundary Harnack principle, Brownian motion, Diffusion, Diffusion process, Green function, Harmonic function, Heat kernel, Kato class, Lipschitz domain, Measure-valued drift, Non-symmetric diffusion, Schr{\"o}dinger operator, Transition density",

author = "Panki Kim and Renming Song",

note = "Funding Information: * Corresponding author. Fax: 82 2 887 4694. E-mail addresses: [email protected] (P. Kim), [email protected] (R. Song). URLs: http://www.math.snu.ac.kr/~pkim (P. Kim), http://www.math.uiuc.edu/~rsong (R. Song). 1 The research of this author is supported by Research Settlement Fund for the new faculty of Seoul National University. 2 The research of this author is supported in part by a joint US–Croatia grant INT 0302167. 3 Fax: 1 217 333 9576.",

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doi = "10.1016/j.jmaa.2006.10.014",

language = "English (US)",

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PY - 2007/8/1

Y1 - 2007/8/1

N2 - In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains. Informally the Schrödinger-type operators we consider are of the form L + μ ṡ ∇ + ν where L is a uniformly elliptic second order differential operator, μ is a vector-valued signed measure belonging to Kd, 1 and ν is a signed measure belonging to Kd, 2. In this paper, we establish two-sided estimates for the heat kernels of Schrödinger-type operators in bounded C1, 1-domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrödinger-type operators in bounded Lipschitz domains.

AB - In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains. Informally the Schrödinger-type operators we consider are of the form L + μ ṡ ∇ + ν where L is a uniformly elliptic second order differential operator, μ is a vector-valued signed measure belonging to Kd, 1 and ν is a signed measure belonging to Kd, 2. In this paper, we establish two-sided estimates for the heat kernels of Schrödinger-type operators in bounded C1, 1-domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrödinger-type operators in bounded Lipschitz domains.

KW - 3G theorem

KW - Boundary Harnack principle

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KW - Diffusion

KW - Diffusion process

KW - Green function

KW - Harmonic function

KW - Heat kernel

KW - Kato class

KW - Lipschitz domain

KW - Measure-valued drift

KW - Non-symmetric diffusion

KW - Schrödinger operator

KW - Transition density

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Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

Abstract

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