The Cauchy problem for a class of Markov-type semigroups

We study the Cauchy problem for a class of Markov-type semigroups (not strongly continuous in general) in the space of all real, uniformly continuous and bounded functions on a separable metric space. In this class there are many transition Markov semigroups corresponding to stochastic differential equations in infinite dimensions as the heat semigroup and the one of Ornstein-Uhlenbeck. We define appropriate notions of solution and give existence and uniqueness theorems. Additional regularity results about the Cauchy problem associated with the Ornstein-Uhlenbeck semigroup are also proved.