Gradient Estimate for Ornstein-Uhlenbeck Jump Processes

By using absolutely continuous lower bounds of the L\'evy measure, explicit gradient estimates are derived for the semigroup of the corresponding L\'evy process with a linear drift. A derivative formula is presented for the conditional distribution of the process at time $t$ under the condition that the process jumps before $t$. Finally, by using bounded perturbations of the L\'evy measure, the resulting gradient estimates are extended to linear SDEs driven by L\'evy-type processes.