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On the infinitesimal generators of Ornstein–Uhlenbeck processes with jumps in Hilbert space

Applebaum, D. (2007) On the infinitesimal generators of Ornstein–Uhlenbeck processes with jumps in Hilbert space. Potential Analysis, 26 (1). pp. 79-100. ISSN 0926-2601

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Abstract

We study Hilbert space valued Ornstein–Uhlenbeck processes (Y(t), t ≥ 0) which arise as weak solutions of stochastic differential equations of the type dY = JY + CdX(t) where J generates a C 0 semigroup in the Hilbert space H, C is a bounded operator and (X(t), t ≥ 0) is an H-valued Lévy process. The associated Markov semigroup is of generalised Mehler type. We discuss an analogue of the Feller property for this semigroup and explicitly compute the action of its generator on a suitable space of twice-differentiable functions. We also compare the properties of the semigroup and its generator with respect to the mixed topology and the topology of uniform convergence on compacta.

Item Type: Article
Copyright, Publisher and Additional Information: © 2007 Springer. This is an author produced version of a paper subsequently published in Potential Analysis. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: H-valued Lévy process; Ornstein–Uhlenbeck process; generalised Mehler semigroup; auxiliary semigroup; operator-selfdecomposability; quasi-locally equicontinuous semigroup; pseudo-Feller property; mixed topology; cylinder function; Kolmogorov–Lévy operator
Institution: The University of Sheffield
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 30 Sep 2009 13:20
Last Modified: 08 Feb 2013 16:59
Published Version: http://dx.doi.org/10.1007/s11118-006-9028-y
Status: Published
Publisher: Springer Verlag
Refereed: Yes
Identification Number: 10.1007/s11118-006-9028-y
URI: http://eprints.whiterose.ac.uk/id/eprint/9800

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