On regularity properties of nonsymmetric ornstein-uhlenbeck semigroup in L^P spaces

Abstract

We study properties of the transition semigroup R_t corresponding to the Hilbert space valued nonsymmctric Ornstein-Uhlenheck process possessing an invariant measure μ Necessary and sufficient condition is given for R_tφ to be infinitely smooth in the direction of the Reproducing Kernel of μ for every bounded Borel φ. Estimates on the derivatives of R_tφ are obtained and the same estimates are obtained for the adjoint semigroup. We give also necessary and sufficient conditions for R_t to be an integral operator on L^P (H,μ) extending earlier results by Da Prato-Zabczyk and Fuhrman. It is shown also 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 compact if it is integral in L^P (H,μ) Finally, an application to some parabolic PDE's on Hilbert space is given.

Keywords:

• Ornstein-Uhlenbeck semigroup
• integral operator
• differentiability in RK direction
• Sobolev spaces
• compact semigroup
• parabolic PDE
• AMS Subject Number Classification: 60H15, 35R15