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November 2012 Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation
Zhen-Qing Chen, Panki Kim, Renming Song
Ann. Probab. 40(6): 2483-2538 (November 2012). DOI: 10.1214/11-AOP682


Suppose that $d\geq2$ and $\alpha\in(1,2)$. Let $D$ be a bounded $C^{1,1}$ open set in ${\mathbb{R}}^{d}$ and $b$ an ${\mathbb{R}}^{d}$-valued function on ${\mathbb{R}}^{d}$ whose components are in a certain Kato class of the rotationally symmetric $\alpha$-stable process. In this paper, we derive sharp two-sided heat kernel estimates for $\mathcal{L} ^{b}=\Delta^{\alpha/2}+b\cdot\nabla$ in $D$ with zero exterior condition. We also obtain the boundary Harnack principle for $\mathcal{L} ^{b}$ in $D$ with explicit decay rate.


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Zhen-Qing Chen. Panki Kim. Renming Song. "Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation." Ann. Probab. 40 (6) 2483 - 2538, November 2012.


Published: November 2012
First available in Project Euclid: 26 October 2012

zbMATH: 1264.60060
MathSciNet: MR3050510
Digital Object Identifier: 10.1214/11-AOP682

Primary: 47G20 , 60J35 , 60J75
Secondary: 47D07

Keywords: boundary Harnack inequality , Exit time , gradient operator , Green function , heat kernel , Kato class , Lévy system , Symmetric $\alpha$-stable process , Transition density

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 6 • November 2012
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