On the Ornstein-Uhlenbeck Operator in Spaces of Continuous Functions

We study the realization of the Ornstein-Uhlenbeck operator A in the space of the uniformly continuous and bounded functions in Rn. We prove that it generates a semigroup which is neither strongly continuous nor analytic, but enjoys nice smoothing properties. Such properties are used to prove a Schauder-type theorem for the elliptic equation λu − Au = f.