In this paper, we continue our investigation of dissipative PDE's forced by random bounded kick-forces and of the corresponding random dynamical system (RDS) in function spaces. It has been proved that the domain \mathcal{A} of attainability from zero (which is a compact subset of a function space) is invariant for the RDS associated with the original equation and carries a stationary measure \mu, which is unique among all measures supported by \mathcal{A}. Here we show that \mu is the unique stationary measure for the RDS in the whole space and study its ergodic properties.