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

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Journal publication date 06/2002 Proceedings of the Edinburgh Mathematical Society 2 45 25 301-325 Published English

### 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

### Bibliographic note

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.