Kolmogorov equations for measures

Abstract

We consider a semigroup of operators in the Banach space $C_b(H)$ of uniformly continuous and bounded functions on a separable Hilbert space $H$. In particular, we deal with semigroups that are related to solution of stochastic PDEs in $H$ and which are not, in general, strongly continuous. We prove an existence and uniqueness result for a measure valued equation involving this class of semigroups. Then we apply the result to a large class of second order differential operators in $C_b(H)$.