## On regularity properties of nonsymmetric Ornstein-Uhlenbeck semigroup in $$L^ p$$-spaces.(English)Zbl 0876.60039

The authors study properties of the transition semigroup $$R_t$$ corresponding to the Hilbert space valued nonsymmetric Ornstein-Uhlenbeck process possessing an invariant measure $$\mu$$. A necessary and sufficient condition is given for $$R_t\varphi$$ to be infinitely smooth in the direction of the reproducing kernel of $$\mu$$ for every bounded Borel $$\varphi$$. Also necessary and sufficient conditions for $$R_t$$ to be an integral operator on $$L^P(H,\mu)$$ are given. It is also shown that the integral kernel possesses strong integrability properties. The transition semigroup is also investigated in the scale of Sobolev spaces generalizing those of Malliavin calculus. The transition semigroup turns out to be strongly continuous in those spaces and to be compact if it is integral in $$L^p(H,\mu)$$.

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
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