Some Properties for Ornstein-Uhlenbeck Jump Processes

Abstract

Consider the linear SDE on $\R^n$: $$\d X_t= A X_t\d t+ B\d L_t,$$ where $A$ is a real $n\times n$ matrix, $B$ is a real $n\times d$ real matrix and $L_t$ is a L\'evy process with L\'evy measure $\nu$ on $\R^d$. Assume that $\nu(\d z)\ge \rr_0(z)\d z$ for some $\rr_0\ge 0$. If $A \le 0, \Rank (B)=n$ and $\int_{\{|z-z_0|\le\vv\}} \rr_0(z)^{-1}\d z<\infty$ holds for some $z_0\in \R^d$ and some $\vv>0$, then the associated Markov transition probability $P_t(x,\d y)$ satisfies $$\|P_t (x, \cdot)- P_t (y, \cdot)\|_{var} \le \ff{C(1+|x-y|)}{\ss t}, x,y\in \R^d, t>0$$ for some constant $C>0$, which is sharp for large $t$ and implies that the process has successful couplings. Harnack inequality, ultracontractivity and strong Feller property are also investigated for the transition semigroup.