Symmetric Ornstein-Uhlenbeck semigroups and their generators
The authors consider a linear stochastic differential equation \(dZ(t)= AZ(t)dt+ \sqrt{Q}dW(t)\) on a Hilbert space \(H\) where \(A\) generates a \(C_0\)-semigroup \(\{S(t)\}_{t\geq 0}\), the operator \(Q\) is bounded selfadjoint nonnegative and \(W\) denotes a standard cylindrical Wiener process. Assuming the existence of an invariant measure \(\mu\) for the equation, necessary and sufficient conditions are provided for the transition semigroup \(\{R(t)\}_{t\geq 0}\) to be symmetric in \(L^2(H,\mu)\). This result enables the authors to characterize the domain of the generator of \(\{R(t)\}_{t\geq 0}\) in \(L^p(H,\mu)\) in terms of appropriate Sobolev spaces and to derive a formula for the size of the spectral gap of the generator. Moreover, necessary and sufficient conditions for the transition semigroup to be compact, Hilbert-Schmidt or strong Feller are given in terms of the semigroup \(\{Q^{-1/2}S(t)Q^{1/2}\}_{t\geq 0}\). The results are applied to study an Ornstein-Uhlenbeck process in a chaotic environment.
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