Gibbsian dynamics and invariant measures for stochastic dissipative PDEs

We present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative PDEs. It consists of two main steps. The first step is the reduction to a finite dimensional Gibbsian dynamics of the low modes. The second step is to prove the equivalence between measures induced by different past histories using Girsanov theorem. As applications, we prove ergodicity for Ginzburg-Landau, Kuramoto-Sivashinsky and Cahn-Hilliard equations with stochastic forcing. KEY WORDS: Ergodicity; invariant measures; stationary processes; infinitedimensional random dynamical systems; stochastic partial differential equations