PR ] 2 8 A pr 2 00 8 Densities for Ornstein-Uhlenbeck processes with jumps 11 april 2008

Abstract: We consider an Ornstein-Uhlenbeck process with values in Rn driven by a Lévy process (Zt) 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 rank condition and a mild assumption on the Lévy measure of (Zt), we prove that the law of the Ornstein-Uhlenbeck process at any time t > 0 has a density on Rn. Moreover, when the Lévy process is of α-stable type, α ∈ (0, 2), we show that such density is a C∞-function. 1 Supported by the Italian National Project MURST “Equazioni di Kolmogorov” and by the Polish Ministry of Science and Education project 1PO 3A 034 29 “Stochastic evolution equations with Lévy noise”. 2 Supported by the Polish Ministry of Science and Education project 1PO 3A 034 29 “Stochastic evolution equations with Lévy noise”.

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