Dirichlet heat kernel estimates for fractional Laplacian with gradient perturbation

Abstract

Suppose that $d\ge 2$ 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 α-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.

Published in at http://dx.doi.org/10.1214/11-AOP682 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Heat kernel estimates for the fractional Laplacian with Dirichlet conditions

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