Hausdorff dimension of a random invariant set

The notion of random attractor for a dissipative stochastic dynamical syste m has recently been introduced. It generalizes the concept of global attrac tor in the deterministic theory. It has been shown that many stochastic dyn amical systems associated to a dissipative partial differential equation pe rturbed by noise do possess a random attractor. In this paper, we prove tha t, as in the case of the deterministic attractor, the Hausdorff dimension o f the random attractor can be estimated by using global Lyapunov exponents. The result is obtained under very natural assumptions. As an application, we consider a stochastic reaction-diffusion equation and show that its rand om attractor has finite Hausdorff dimension. (C) Elsevier, Paris.