The Annals of Probability

Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation

Zhen-Qing Chen, Panki Kim, and Renming Song

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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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Ann. Probab., Volume 40, Number 6 (2012), 2483-2538.

First available in Project Euclid: 26 October 2012

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Zentralblatt MATH identifier

Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 60J75: Jump processes
Secondary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}

Symmetric $\alpha$-stable process gradient operator heat kernel transition density Green function exit time Lévy system boundary Harnack inequality Kato class


Chen, Zhen-Qing; Kim, Panki; Song, Renming. Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation. Ann. Probab. 40 (2012), no. 6, 2483--2538. doi:10.1214/11-AOP682.

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