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

Abstract

In the present paper we consider the transition semigroup Pt related to some stochastic reaction-diffusion equations with the non-linear 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(script O sign̄), where script O sign ⊂ double-struck R signd is a bounded open set. In L2(script O sign) the only result proved is the strong Feller property, for d = 1. Here we are able to prove that if f ∈ C∞(double-sruck R sign) and d ≤ 3, then Ptφ ∈ C∞b (C(script O sign̄)) for any φ ∈ Bb(C(script O sign̄)) 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.