On the Estimates of the Density of Feynman-Kac Semigroups of $alpha$-Stable-like Processes

Suppose that $alpha in (0,2)$ and that $X$ is an $alpha$-stable-like process on $R^d$. Let $F$ be a function on $R^d$ belonging to the class $bf{J_{d,alpha}}$ (see Introduction) and $A_{t}^{F}$ be $sum_{s le t}F(X_{s-},X_{s}), t> 0$, a discontinuous additive functional of $X$. With neither $F$ nor $X$ being symmetric, under certain conditions, we show that the Feynman-Kac semigroup ${S_{t}^{F}:t ge 0}$ defined by $$ S_{t}^{F}f(x)=mathbb{E}_{x}(e^{-A_{t}^{F}}f(X_{t}))$$ has a density $q$ and that there exist positive constants $C_1,C_2,C_3$ and $C_4$ such that $$C_{1}e^{-C_{2}t}t^{-frac{d}{alpha}}(1 wedge frac{t^{frac{1}{alpha}}}{|x-y|})^{d+alpha} leq q(t,x,y) leq C_{3}e^{C_{4}t}t^{-frac{d}{alpha}}(1 wedge frac{t^{frac{1}{alpha}}}{|x-y|})^{d+alpha}$$ for all $(t,x,y)in (0,infty) times R^d times R^d$.