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Ornstein-Uhlenbeck processes in Banach spaces and their spectral representations.

Groves, James S. (2002) Ornstein-Uhlenbeck processes in Banach spaces and their spectral representations. Proceedings of the Edinburgh Mathematical Society, 45 (2). pp. 301-325. ISSN 0013-0915

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    Abstract

    For Q the variance of some centred Gaussian random vector in a separable Banach space it is shown that, necessarily, Q factors through $\ell^2$ as a product of 2-summing operators. This factorization condition is sufficient when the Banach space is of Gaussian type 2. The stochastic integral of a deterministic family of operators with respect to a Q-Wiener process is shown to exist under a continuity condition involving the 2-summing norm. A Langevin equation $$ \rd\bm{Z}_t+\sLa\bm{Z}_t\,\rd t=\rd\bm{B}_t, $$ with values in a separable Banach space, is studied. The operator $\sLa$ is closed and densely defined. A weak solution $(\bm{Z}_t,\bm{B}_t)$, where $\bm{Z}_t$ is centred, Gaussian and stationary, while $\bm{B}_t$ is a Q-Wiener process, is given when $\ri\sLa$ and $\ri\sLa^*$ generate $C_0$ groups and the resolvent of $\sLa$ is uniformly bounded on the imaginary axis. Both $\bm{Z}_t$ and $\bm{B}_t$ are stochastic integrals with respect to a spectral Q-Wiener process. AMS 2000 Mathematics subject classification: Primary 60G15. Secondary 46E40; 47B10; 47D03; 60H10

    Item Type: Article
    Journal or Publication Title: Proceedings of the Edinburgh Mathematical Society
    Additional Information: http://journals.cambridge.org/action/displayJournal?jid=PEM The final, definitive version of this article has been published in the Journal, Proceedings of the Edinburgh Mathematical Society, 45 (2), pp 301-325 2002, © 2002 Cambridge University Press.
    Subjects: Q Science > QA Mathematics
    Departments: Faculty of Science and Technology > Mathematics and Statistics
    ID Code: 19268
    Deposited By: ep_ss_importer
    Deposited On: 20 Nov 2008 09:18
    Refereed?: Yes
    Published?: Published
    Last Modified: 26 Jul 2012 15:27
    Identification Number:
    URI: http://eprints.lancs.ac.uk/id/eprint/19268

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