Laplacian perturbed by non-local operators

Abstract

Suppose that $d\ge 1$ and $0<\beta <2$. We establish the existence and uniqueness of the fundamental solution $q^b(t, x, y)$ to the operator ${\mathcal {L}}^b=\Delta +{\mathcal {S}}^b$, where

$$\begin{aligned} {\mathcal {S}}^bf(x):= \int _{{\mathbb {R}}^d} \left( f(x+z)-f(x)- \nabla f(x) \cdot z{1\!\!1}_{\{|z|\le 1\}} \right) \frac{b(x, z)}{|z|^{d+\beta }}dz \end{aligned}$$

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 $x, z\in {\mathbb {R}}^d$. We show that if for each $x\in {\mathbb {R}}^d, b(x, z) \ge 0$ for a.e. $z\in {\mathbb {R}}^d$, then $q^b(t, x, y)$ is a strictly positive continuous function and it uniquely determines a conservative Feller process $X^b$, which has strong Feller property. Furthermore, sharp two-sided estimates on $q^b(t, x, y)$ are derived.