ON THE DOMAIN OF NONSYMMETRIC ORNSTEIN–UHLENBECK OPERATORS IN BANACH SPACES

Abstract

We consider the linear stochastic Cauchy problem [Formula: see text] where A generates a C0-semigroup on a Banach space E, WH is a cylindrical Brownian motion over a Hilbert space H, and B: H → E is a bounded operator. Assuming the existence of a unique minimal invariant measure μ∞, let Lp denote the realization of the Ornstein–Uhlenbeck operator associated with this problem in Lp (E, μ∞). Under suitable assumptions concerning the invariance of the range of B under the semigroup generated by A, we prove the following domain inclusions, valid for 1 < p ≤ 2: [Formula: see text] Here [Formula: see text] denotes the kth order Sobolev space of functions with Fréchet derivatives up to order k in the direction of H. No symmetry assumptions are made on Lp.