Abstract
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.
Citation
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. https://doi.org/10.1214/11-AOP682
Information