Perturbation by non-local operators

Abstract

Suppose that $d\ge 1$ and $0<\beta <\alpha <2$. We establish the existence and uniqueness of the fundamental solution $q^b(t,x,y)$ to a class of (possibly nonsymmetric) non-local operators $L^b={\Delta }^{\alpha /2}+S^b$, where $$S^{b}f(x):=A(d,-\beta ){\int }_{R^d}(f(x+z)-f(x)-\nabla f(x)\cdot z{1}_{\{|z|\le 1\}})\frac{b(x,z)}{|z{|}^{d+\beta }}dz$$ and $b(x,z)$ is a bounded measurable function on $R^d\times R^d$ with $b(x,z)=b(x,-z)$ for $x,z\in R^d$. Here $A(d,-\beta )$ is a normalizing constant so that $S^b={\Delta }^{\beta /2}$ when $b(x,z)\equiv 1$. We show that if $b(x,z)\ge -\frac{\mathcal{A}(d,-\alpha )}{A(d,-\beta )}|z{|}^{\beta -\alpha }$, 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. The Feller process $X^b$ is the unique solution to the martingale problem of $(L^b,\mathcal{S}(R^d))$, where $\mathcal{S}(R^d)$ denotes the space of tempered functions on $R^d$. Furthermore, sharp two-sided estimates on $q^b(t,x,y)$ are derived. In stark contrast with the gradient perturbations, these estimates exhibit different behaviors for different types of $b(x,z)$. The model considered in this paper contains the following as a special case. Let $Y$ and $Z$ be (rotationally) symmetric $\alpha$-stable process and symmetric $\beta$-stable processes on $R^d$, respectively, that are independent to each other. Solution to stochastic differential equations $d{X}_t=d{Y}_t+c(X_{t-})d{Z}_t$ has infinitesimal generator $L^b$ with $b(x,z)=|c(x){|}^{\beta }$.

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