L 2-theory for non-symmetric Ornstein–Uhlenbeck semigroups on domains

Abstract

We prove that the mild solution of the stochastic evolution equation \({{d}X(t) = AX(t)\,{d}t + {d}W(t)}\) on a Banach space E has a continuous modification if the associated Ornstein–Uhlenbeck semigroup is analytic on L2 with respect to the invariant measure. This result is used to extend recent work of Da Prato and Lunardi for Ornstein–Uhlenbeck semigroups on domains \({\mathcal{O} \subseteq E}\) to the non-symmetric case. Denoting the generator of the Ornstein–Uhlenbeck semigroup by \({L_\mathcal{O}}\), we obtain sufficient conditions in order that the domain of \({\sqrt{-L_\mathcal{O}}}\) be a first-order Sobolev space.