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Bulletin of the London Mathematical Society Advance Access originally published online on December 10, 2008
Bulletin of the London Mathematical Society 2009 41(1):41-50; doi:10.1112/blms/bdn099
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© 2008 London Mathematical Society

Densities for Ornstein–Uhlenbeck processes with jumps

Enrico Priola

Dipartimento di Matematica
Universitàdi Torino
via Carlo Alberto 10
10123 Torino
Italy

Jerzy Zabczyk

Instytut Matematyczny
Polskiej Akademii Nauk
ul. Sniadeckich 8
00-950 Warszawa
Poland
zabczyk@impan.gov.pl

Received 8 August 2007. Accepted 2 September 2008.

We consider an Ornstein–Uhlenbeck process with values in Rn driven by a Lévy process (Zt) taking values in Rd 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 {alpha}-stable type, {alpha} isin (0, 2), we show that such density is a C{infty}-function.

2000 Mathematics Subject Classification 60H10, 60J75, 47D07.

The first author is supported by the M.I.U.R. research project Prin 2006 ‘Kolmogorov equations’ and by the Polish Ministry of Science and Education project 1PO 3A 034 29 ‘Stochastic evolution equations with Lévy noise’.

The second author is supported by the Polish Ministry of Science and Education project 1PO 3A 034 29 ‘Stochastic evolution equations with Lévy noise’.


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