Laplacian perturbed by non-local operators

Jie-Ming Wang

Published 2014-02-26Version 1

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 $\L^b=\Delta+S^b$, where $$ S^bf(x):=\int_{\R^d} \left( f(x+z)-f(x)- \nabla f(x) \cdot z\1_{\{|z|\leq 1\}} \right) \frac{b(x, z)}{|z|^{d+\beta}}dz$$ and $b(x, z)$ is a bounded measurable function on $\R^d\times \R^d$ with $b(x, z)=b(x, -z)$ for $x, z\in \R^d$. We show that if for each $x\in\R^d, b(x, z) \geq 0$ for a.e. $z\in\R^d$, 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.