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

摘要

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 $$ dm{Z}_t+sLam{Z}_t,d t=dm{B}_t, $$ with values in a separable Banach space, is studied. The operator $sLa$ is closed and densely defined. A weak solution $(m{Z}_t,m{B}_t)$, where $m{Z}_t$ is centred, Gaussian and stationary, while $m{B}_t$ is a Q-Wiener process, is given when $isLa$ and $isLa^*$ generate $C_0$ groups and the resolvent of $sLa$ is uniformly bounded on the imaginary axis. Both $m{Z}_t$ and $m{B}_t$ are stochastic integrals with respect to a spectral Q-Wiener process. AMS 2000 Mathematics subject classification: Primary 60G15. Secondary 46E40; 47B10; 47D03; 60H10