Dirichlet heat kernel estimates for fractional Laplacian under non-local perturbation

Abstract

For $d\ge 2$ and $0<\beta <\alpha <2$, consider a family of non-local operators ${\mathcal{L}}^b={\Delta }^{\alpha /2}+{\mathcal{S}}^b$ on ${\mathbb{R}}^d$, where $${\mathcal{S}}^{b}f(x):=\underset{\epsilon \to 0}{\lim }\mathcal{A}(d,-\beta ){\int }_{\{z\in {\mathbb{R}}^d:|z|>\epsilon \}}(f(x+z)-f(x))\frac{b(x,z)}{|z{|}^{d+\beta }}dz,$$ and $b(x,z)$ is a bounded measurable function on ${\mathbb{R}}^d\times {\mathbb{R}}^d$ with $b(x,z)=b(x,-z)$ for every $x,z\in {\mathbb{R}}^d$. Here $\mathcal{A}(d,-\beta )$ is a normalizing constant so that ${\mathcal{S}}^b=-(-\Delta {)}^{\beta /2}$ when $b(x,z)\equiv 1$. It was recently shown in Chen and Wang [arXiv:1312.7594 [math.PR]] that when $b(x,z)\ge -\frac{\mathcal{A}(d,-\alpha )}{\mathcal{A}(d,-\beta )}|z{|}^{\beta -\alpha }$, then ${\mathcal{L}}^b$ admits a unique fundamental solution $p^b(t,x,y)$ which is strictly positive and continuous. The kernel $p^b(t,x,y)$ uniquely determines a conservative Feller process $X^b$, which has strong Feller property. The Feller process $X^b$ is also the unique solution to the martingale problem of $({\mathcal{L}}^b,\mathcal{S}({\mathbb{R}}^d))$, where $\mathcal{S}({\mathbb{R}}^d)$ denotes the space of tempered functions on ${\mathbb{R}}^d$. In this paper, we are concerned with the subprocess $X^{b,D}$ of $X^b$ killed upon leaving a bounded $C^{1,1}$ open set $D\subset {\mathbb{R}}^d$. We establish explicit sharp two-sided estimates for the transition density function of $X^{b,D}$.