**Ornstein-Uhlenbeck processes in Banach spaces a...**280 KB, PDF-document

3/11/15

Research output: Contribution to journal › Journal article

Published

<mark>Journal publication date</mark> | 06/2002 |
---|---|

<mark>Journal</mark> | Proceedings of the Edinburgh Mathematical Society |

Issue number | 2 |

Volume | 45 |

Number of pages | 25 |

Pages (from-to) | 301-325 |

<mark>State</mark> | Published |

<mark>Original language</mark> | English |

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

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.