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 ${\mathcal{L}}^b=\Delta +S^b$, where $$S^{b}f(x):={\int }_{{\mathbb{R}}^d}\left(f(x+z)-f(x)-\nabla f(x)\cdot z{1}_{\{|z|\le 1\}}\right)\frac{b(x,z)}{|z{|}^{d+\beta }}dz$$ 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 $x\in {\mathbb{R}}^d,b(x,z)\ge 0$ 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.

arXiv admin note: substantial text overlap with arXiv:1312.7594

References in corpus (1)

• Perturbation by non-local operators