GRADIENT ESTIMATES AND DOMAIN IDENTIFICATION FOR ANALYTIC ORNSTEIN-UHLENBECK OPERATORS

Let P be the Ornstein-Uhlenbeck semigroup associated with the stochastic Cauchy problem dU(t) = AU(t) dt + dWH(t), where A is the generator of a C0-semigroup S on a Banach space E, H is a Hilbert subspace of E, and WH is an H-cylindrical Brownian motion. Assum-ing that S restricts to a C0-semigroup on H, we obtain Lp-bounds for DHP (t). We show that if P is analytic, then the invariance assumption is fulfilled. As an application we determine the Lp-domain of the generator of P explicitly in the case where S restricts to a C0-semigroup on H which is similar to an analytic contraction semigroup