Hausdorff dimension of invariant sets for random dynamical systems (Q1271035)

The authors introduce the notion of invariance of a random set with respect to random dynamical systems (shortly RDS) and describe the concept of random attractor. Moreover they show that the Hausdorff dimension of a compact invariant random set of an RDS with an ergodic base flow is almost surely constant. As in the deterministic case under the conditions that the RDS contract volumes of a sufficiently high dimension, the authors estimate the Hausdorff dimension of a compact random invariant domain. The results are applied to a reaction-diffusion equation with additive noise and to two-dimensional Navier-Stokes equations with bounded noise.