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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 (Rt) 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 Rt and provide an estimate for the (appropriately defined) gradient of Rtφ. Finally, in the Hilbert-Schmidt case, we show that for any φ ∈ L p (µ) the function Rtφ is an (almost) classical solution of a version of the Kolmogorov equation. Contents 1. INTRODUCTION 2. CHARACTERIZATION OF SYMMETRIC OU SEMIGROUPS 2.1. General OU Process 2.2. Characterizations of the Symmetry 2.3. The First Consequences of Symmetry 3. IDENTIFICATION OF DOMAINS 4. SPECTRAL GAP AND REGULARITY

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