Laplacian perturbed by non-local operators

Abstract

Suppose that $d\geq 1$ and $0<\beta<2$. We establish the existence and uniqueness of the fundamental solution $q^b(t, x, y)$ to the operator $\mathcal{L}^b=\Delta+S^b$, where $$S^bf(x) := \int_{\mathbb{R}^d} \left( f(x+z) - f(x) - \nabla f(x) \cdot z\mathbb{1}_{\{|z| \leq 1\}} \right) \frac{b(x, z)}{|z|^{d+\beta}} dz$$ 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 $x, z\in \mathbb{R}^d$. We show that if for each $x\in\mathbb{R}^d, b(x, z) \geq 0$ 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.