Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

ORNSTEIN–UHLENBECK PROCESSES IN BANACH SPACES AND THEIR SPECTRAL REPRESENTATIONS

James S. Grovesa1

a1 Department of Mathematics and Statistics, University of Lancaster, Lancaster LA1 4YF, UK (j.groves@lancaster.ac.uk)

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

(Received December 14 2000)

Keywords

  • Ornstein–Uhlenbeck processes;
  • Banach spaces;
  • absolutely summing operators;
  • operator groups