Densities for Ornstein-Uhlenbeck processes with jumps

We consider an Ornstein-Uhlenbeck process with values in R^n driven by a Lévy process (Z_t) taking values in R^d with d possibly smaller than n. The Lévy noise can have a degenerate or even vanishing Gaussian component. Under a controllability condition and an assumption on the Lévy measure of (Z_t), we prove that the law of the Ornstein-Uhlenbeck process at any time t>0 has a density on R^n. Moreover, when the Lévy process is of $α$-stable type, $α\in (0,2)$, we show that such density is a $C^{\infty}$-function.

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