Symmetric Ornstein-Uhlenbeck Semigroups and their Generators

Abstract

We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure $\mu$. For a reversible process the domain of its generator in $L^p(\mu )$ 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 $R_t\varphi$. Finally, in the Hilbert-Schmidt case, we show that for any $\varphi \in L^p(\mu )$ the function $R_t\varphi$ is an (almost) classical solution of a version of the Kolmogorov equation.