Cylindrical Levy processes in Banach spaces

Riedle, Markus and Applebaum, David (2009) Cylindrical Levy processes in Banach spaces. [MIMS Preprint]

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Cylindrical probability measures are finitely additive measures on Banach spaces that have sigma-additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and hence weak (cylindrical) stochastic processes. In this paper we focus on cylindrical Levy processes. These have (weak) Levy-Ito decompositions and an associated Levy-Khintchine formula. If the process is weakly square integrable, its covariance operator can be used to construct a reproducing kernel Hilbert space in which the process has a decomposition as an infinite series built from a sequence of uncorrelated bona fide one-dimensional Levy processes. This series is used to define cylindrical stochastic integrals from which cylindrical Ornstein-Uhlenbeck processes may be constructed as unique solutions of the associated Cauchy problem. We demonstrate that such processes are cylindrical Markov processes and study their (cylindrical) invariant measures.

Item Type: MIMS Preprint
Uncontrolled Keywords: cylindrical probability measure, cylindrical Levy process, reproducing kernel Hilbert space, Cauchy problem, cylindrical Ornstein-Uhlenbeck process, cylindrical invariant measure.
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
Depositing User: Dr Markus Riedle
Date Deposited: 19 May 2009
Last Modified: 08 Nov 2017 18:18

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