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.
Categories: math.PR