  We consider an infinite-dimensional dynamical system with polynomial
nonlinearity and additive noise given by a finite number of Wiener processes.
By studying how randomness is spread by the system we develop a counterpart of
Hormander's classical theory in this setting. We study the distributions of
finite-dimensional projections of the solutions and give conditions that
provide existence and smoothness of densities of these distributions with
respect to the Lebesgue measure. We also apply our results to concrete SPDEs
such as Stochastic Reaction Diffusion Equation and Stochastic 2D Navier--Stokes
System.
