Smoothing properties of transition semigroups relative to SDEs with values in Banach spaces

In the present paper we consider the transition semigroup Pt related to some stochastic reaction-diffusion equations with the nonlinear term f having polynomial growth and satisfying some dissipativity conditions. We are proving that it has a regularizing effect in the Banach space of continuous functions C(<(theta)over bar>), where theta subset of R-d is a bounded open set. In L-2(theta) the only result proved is the strong Feller property, for d = 1. Here we are able to prove that if f is an element of C-infinity(R) and d less than or equal to 3, then P(t)phi is an element of C-b(infinity)(C(<(theta)over bar>)) for any phi is an element of B-b(C(<(theta)over bar>)) and t > 0. An important application is to the study of the ergodic properties of the system. These results are also of interest for some problem in stochastic control.