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

Abstract

Suppose that $\alpha \in (0,2)$ and that $X$ is an $\alpha$-stable-like process on ${\mathbb{R}}^d$. Let $F$ be a function on ${\mathbb{R}}^d$ belonging to the class ${\mathbf{J}}_{\mathbf{d},\alpha }$ (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 $$C_1{e}^{-C_{2}t}{t}^{-\frac{d}{\alpha }}(1\wedge \frac{t^{\frac{1}{\alpha }}}{|x-y|}{)}^{d+\alpha }\le q(t,x,y)\le C_3{e}^{C_{4}t}{t}^{-\frac{d}{\alpha }}(1\wedge \frac{t^{\frac{1}{\alpha }}}{|x-y|}{)}^{d+\alpha }$$ for all $(t,x,y)\in (0,\infty )\times {\mathbb{R}}^d\times {\mathbb{R}}^d$.

27 pages