  We study the asymptotic properties of the trajectories of a discrete-time
random dynamical system in an infinite-dimensional Hilbert space. Under some
natural assumptions on the model, we establish a multiplica-tive ergodic
theorem with an exponential rate of convergence. The assumptions are satisfied
for a large class of parabolic PDEs, including the 2D Navier--Stokes and
complex Ginzburg--Landau equations perturbed by a non-degenerate bounded random
kick force. As a consequence of this er-godic theorem, we derive some new
results on the statistical properties of the trajectories of the underlying
random dynamical system. In particular , we obtain large deviations principle
for the occupation measures and the analyticity of the pressure function in a
setting where the system is not irreducible. The proof relies on a refined
version of the uniform Feller property combined with some contraction and
bootstrap arguments.

Analysis of PDEs;Dynamical Systems;Probability