非局部扰动下分数阶Laplacian算子的Dirichlet热核估计
Dirichlet heat kernel estimates for fractional Laplacian under non-local perturbation
摘要：
对于$d\ge 2$和$0<β<α<2$，考虑$\mathbb{R}^d$上的一族非局部算子$\mathcal{L}^{b}=Δ^{α/2}+\mathcal{S}^{b}$，其中$$ \mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-β)\int_{ \{z\in \mathbb{R}^d: |z|>\varepsilon\}} (f(x+z)-f(x))\frac{b(x,z)}{|z|^{d+β}}\,dz, $$和$b(x,z)$是$\mathbb{R}^{d}\times\mathbb{R}^{d}$上的有界可测函数，对于每个$x,z\in\mathbb{R}^{d}$都有$b(x,z)=b(x,-z)$。这里${\cal A}(d, -β)$是归一化常数，因此当$b(x, z)\equiv 1$时，$\mathcal{S}^b=-(-Δ)^{β/2}$。最近在Chen和Wang [arXiv：1312.7594 [math.PR]]中表明，当$b(x, z) \geq -\frac{\mathcal{A}(d, -α)} {\mathcal{A}(d, -β)}\, |z|^{β-α}$，则$\mathcal{L}^b$承认唯一的基本解$p^b(t, x, y)$是严格正的和连续的。核$p^b(t, x, y)$唯一地决定了保守Feller过程$X^b$，它具有强Feller性质. Feller过程$X^b$也是$(\mathcal{L}^b, \mathcal{S}(\mathbb{R}^d))$鞅问题的唯一解，其中$\mathcal{S}(\mathbb{R}^d)$表示$\mathbb{R}^d$上的调和函数空间。在本文中，我们关注的是$X^{b}$的子进程$X^{b,D}$在离开有界$C^{1,1}$开集$D\subset \mathbb{R}^d$时被杀死。我们建立了明确的尖锐的双边估计的过渡密度函数的$X^{b, D}$。
英文摘要：
For $d\ge 2$ and $0<β<α<2$, consider a family of non-local operators $\mathcal{L}^{b}=Δ^{α/2}+\mathcal{S}^{b}$ on $\mathbb{R}^d$, where $$ \mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-β)\int_{ \{z\in \mathbb{R}^d: |z|>\varepsilon\}} (f(x+z)-f(x))\frac{b(x,z)}{|z|^{d+β}}\,dz, $$ 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 ${\cal A}(d, -β)$ is a normalizing constant so that $\mathcal{S}^b=-(-Δ)^{β/2}$ when $b(x, z)\equiv 1$. It was recently shown in Chen and Wang [arXiv:1312.7594 [math.PR]] that when $b(x, z) \geq -\frac{\mathcal{A}(d, -α)} {\mathcal{A}(d, -β)}\, |z|^{β-α}$, 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 $X^{b, D}$.
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