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Hausdorff dimension of invariant sets for random dynamical systems. (English) Zbl 0927.37031
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.

MSC:
37H10 Generation, random and stochastic difference and differential equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q30 Navier-Stokes equations
35K57 Reaction-diffusion equations
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