Gradient estimates and domain identification for analytic Ornstein-Uhlenbeck operators

Abstract

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 C 0-semigroup S on a Banach space E, H is a Hilbert subspace of E, and WH is an H-cylindrical Brownian motion. Assuming that S restricts to a C 0-semigroup on H, we obtain L p-bounds for DH P(t). We show that if P is analytic, then the invariance assumption is fulfilled. As an application we determine the L p-domain of the generator of P explicitly in the case where S restricts to a C 0-semigroup on H which is similar to an analytic contraction semigroup. The results are applied to the 1D stochastic heat equation driven by additive space-time white noise.