Determining modes for dissipative random dynamical systems

The recent theory of random attractors is becoming very useful for the study of the asymptotic behaviour of dissipative random dynamical systems. A random attractor is a random invariant compact set which attracts every trajectory as time becomes infinite. Moreover, it can be proved that, in some cases, the random attractor has finite HausdorfT dimension. It seems that the asymptotic behaviour of the random dynamical system is governed by a finite number of degrees of freedom. In this work we prove a result in this direction: we generalize a result from the (deterministic) theory of dissipative dynamical systems known in the literature as determining modes. The result is applied to Navier – Stokes equation and a problem of reaction-diffusion type, both with additive white noise. We finally prove that the random attractor associated to the stochastic Navier – Stokes equation has finite Hausdorff dimension