Symmetric Ornstein-Uhlenbeck semigroups and their generators

 We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure μ. For a reversible process the domain of its generator in L ^p (μ) is characterized in terms of appropriate Sobolev spaces thus extending the Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck operator. We provide also a formula for the size of the spectral gap of the generator. Those results are applied to study the Ornstein-Uhlenbeck process in a chaotic environment. Necessary and sufficient conditions for a transition semigroup (R _t ) to be compact, Hilbert-Schmidt and strong Feller are given in terms of the coefficients of the Ornstein-Uhlenbeck operator. We show also that the existence of spectral gap implies a smoothing property of R _t and provide an estimate for the (appropriately defined) gradient of R _t φ. Finally, in the Hilbert-Schmidt case, we show tha t for any the function R _t φ is an (almost) classical solution of a version of the Kolmogorov equation. Received: 17 September 2001 / Revised version: 3 June 2002 / Published online: 30 September 2002 This work was partially supported by the Small ARC Grant Scheme. Mathematics Subject Classification (2000): Primary: 60H15, 47F05; Secondary: 60J60, 35R15, 35K15 Key words or phrases: Ornstein-Uhlenbeck operator – Second quantization – Reversibility – Spectral gap – Sobolev spaces – Domain of generator