Preliminaries 1.1 Function spaces 1.2 Basic facts from measure theory 1.3 Markov processes and random dynamical systems Notes and comments 2 Two-dimensional Navier-Stokes equations 2.1 Cauchy problem for the deterministic system 2.2 Stochastic Navier-Stokes equations 2.3 Navier-Stokes equations perturbed by a random kick force 2.4 Navier-Stokes equations perturbed by spatially regular white noise 2.5 Existence of a stationary distribution 2.6 Appendix: some technical proofs Notes and comments 3 Uniqueness of stationary measure and mixing 3.1 Three results on uniqueness and mixing 3.2 Dissipative RDS with bounded kicks 3.3 Navier-Stokes system perturbed by white noise 3.4 Navier-Stokes system with unbounded kicks 3.5 Further results and generalisations 3.6 Appendix: some technical proofs 3.7 Relevance of the results for physics Notes and comments vii www.cambridge.org

We consider the d -dimensional nonlinear Schrödinger equation under periodic boundary conditions:

Together they move on a rotational torus of finite or infinite dimension, depending on how many modes are excited. In particular, for every choice J = { j 1 < j 2 < · · · < j n } ⊂ N of n ≥ 1 basic modes there is an invariant linear space E J of complex dimension n which is completely foliated into rotational tori:In addition, each such torus is linearly stable, and all solutions have vanishing Lyapunov exponents. -This is the linear situation.Upon restoring the nonlinearity f the invariant manifolds E J will not persist in their entirety due to resonances among the modes and the strong perturbing effect of f for large amplitudes. We show, however, that in a sufficiently small neighbourhood of the origin a large Cantor subfamily of rotational n -tori persists and is only slightly deformed.

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