On Ornstein-Uhlenbeck Generators

Abstract

. The aim of this paper is to investigate properties of the generator L of the general nonsymmetric Ornstein-Uhlenbeck semigroup possessing an invariant measure. First we give the necessary and sufficient conditions for the Logarithmic Sobolev Inequality, which extend the result of Simon to the nonsymmetric case. Next, we present conditions for the domains of L and L to have continuous imbeddings into the Sobolev spaces W 2;2 Q1 and W 1;2 Q and into the Orlicz spaces L p LogL r , thus extending the results known for the diagonal or symmetric case. We characterize the cores of the generators L and L and obtain certain factorizations of L and L which generalize the representation of the Malliavin operator. The necessary and sufficient condition for compactness of the imbedding of the Sobolev space W 1;2 Q into L 2 is also given. Finally we give necessary and sufficient conditions for L to be symmetric. The domains are characterized more explicitly in this case. * ..