Dirichlet heat kernel estimates for fractional laplacian with gradient perturbation

Zhen Qing Chen; Panki Kim; Renming Song

doi:10.1214/11-AOP682

Dirichlet heat kernel estimates for fractional laplacian with gradient perturbation

Zhen Qing Chen^*, Panki Kim, Renming Song

^*Corresponding author for this work

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

66 Citations (Scopus)

Abstract

Suppose that d ≥ 2 and α ∈ (1, 2). Let D be a bounded C^1,1 open set in R^d and b an R^d -valued function on R^d whose components are in a certain Kato class of the rotationally symmetric α-stable process. In this paper, we derive sharp two-sided heat kernel estimates for L^b = δ^α/2 + b. ∇ in D with zero exterior condition. We also obtain the boundary Harnack principle for L^b in D with explicit decay rate.

Original language	English
Pages (from-to)	2483-2538
Number of pages	56
Journal	Annals of Probability
Volume	40
Issue number	6
DOIs	https://doi.org/10.1214/11-AOP682
Publication status	Published - 2012
Externally published	Yes

Keywords

Boundary harnack inequality
Exit time
Gradient operator
Green function
Heat kernel
Kato class
Lévy system
Symmetric α-stable process
Transition density

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10.1214/11-AOP682

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title = "Dirichlet heat kernel estimates for fractional laplacian with gradient perturbation",

abstract = "Suppose that d ≥ 2 and α ∈ (1, 2). Let D be a bounded C1,1 open set in Rd and b an Rd -valued function on Rd whose components are in a certain Kato class of the rotationally symmetric α-stable process. In this paper, we derive sharp two-sided heat kernel estimates for Lb = δα/2 + b. ∇ in D with zero exterior condition. We also obtain the boundary Harnack principle for Lb in D with explicit decay rate.",

keywords = "Boundary harnack inequality, Exit time, Gradient operator, Green function, Heat kernel, Kato class, L{\'e}vy system, Symmetric α-stable process, Transition density",

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T1 - Dirichlet heat kernel estimates for fractional laplacian with gradient perturbation

AU - Chen, Zhen Qing

AU - Kim, Panki

AU - Song, Renming

PY - 2012

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N2 - Suppose that d ≥ 2 and α ∈ (1, 2). Let D be a bounded C1,1 open set in Rd and b an Rd -valued function on Rd whose components are in a certain Kato class of the rotationally symmetric α-stable process. In this paper, we derive sharp two-sided heat kernel estimates for Lb = δα/2 + b. ∇ in D with zero exterior condition. We also obtain the boundary Harnack principle for Lb in D with explicit decay rate.

AB - Suppose that d ≥ 2 and α ∈ (1, 2). Let D be a bounded C1,1 open set in Rd and b an Rd -valued function on Rd whose components are in a certain Kato class of the rotationally symmetric α-stable process. In this paper, we derive sharp two-sided heat kernel estimates for Lb = δα/2 + b. ∇ in D with zero exterior condition. We also obtain the boundary Harnack principle for Lb in D with explicit decay rate.

KW - Boundary harnack inequality

KW - Exit time

KW - Gradient operator

KW - Green function

KW - Heat kernel

KW - Kato class

KW - Lévy system

KW - Symmetric α-stable process

KW - Transition density

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DO - 10.1214/11-AOP682

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SN - 0091-1798

VL - 40

SP - 2483

EP - 2538

JO - Annals of Probability

JF - Annals of Probability

IS - 6

ER -

Dirichlet heat kernel estimates for fractional laplacian with gradient perturbation

Abstract

Keywords

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