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 Probability Surveys > Vol. 12 (2015) open journal systems 


Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes

David Applebaum, University of Sheffield, UK


Abstract
We review the probabilistic properties of Ornstein-Uhlenbeck processes in Hilbert spaces driven by Lévy processes. The emphasis is on the different contexts in which these processes arise, such as stochastic partial differential equations, continuous-state branching processes, generalised Mehler semigroups and operator self-decomposable distributions. We also examine generalisations to the case where the driving noise is cylindrical.

AMS 2000 subject classifications: Primary 60G51; secondary 60H15, 60H10, 60E07, 60J80.

Keywords: Lévy process, Ornstein-Uhlenbeck process, Mehler semigroup, skew–convolution semigroup, branching property, invariant measure, operator self–decomposability, Urbanik semigroup, cylindrical process.

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Applebaum, David, Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes, Probability Surveys, 12, (2015), 33-54 (electronic). DOI: 10.1214/14-PS249.

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