Perturbation by non-local operators

Suppose that $d\ge 1$ and $0<β<α<2$. We establish the existence and uniqueness of the fundamental solution $q^b(t, x, y)$ to a class of (possibly nonsymmetric) non-local operators $L^b=Δ^{α/2}+S^b$, where $S^bf(x):=A(d, -β) \int_{R^d} ( f(x+z)-f(x)- \nabla f(x) \cdot z 1_{\{|z|\leq 1\}} ) \frac{b(x, z)}{|z|^{d+β}}dz$ 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 $A(d, -β)$ is a normalizing constant so that $S^b=Δ^{β/2}$ when $b(x, z)\equiv 1$. We show that if $b(x, z) \geq -\frac{{\cal A}(d, -α)}{A(d, -β)}\, |z|^{β-α}$, 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 $(L^b, {\cal S} (R^d))$, where ${\cal S}(R^d)$ 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 $α$-stable process and symmetric $β$-stable processes on $R^d$, respectively, that are independent to each other. Solution to stochastic differential equations $dX_t=dY_t + c(X_{t-})dZ_t$ has infinitesimal generator $L^b$ with $b(x, z)=| c(x)|^β$.

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