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Hausdorff dimension of a random invariant set. (English) Zbl 0919.58044
The author proves that, as in the case of the deterministic attractor, the Hausdorff dimension of the random attractor for a dissipative stochastic dynamical system can be estimated by using global Lyapunov exponents. The result is obtained under assumptions that are satisfied by many stochastic dynamical systems originating in dissipative evolution equations. As an application, the author considers a stochastic reaction-diffusion equation and shows that its random attractor has finite Hausdorff dimension.

MSC:
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
37A99 Ergodic theory
34D35 Stability of manifolds of solutions to ordinary differential equations
35K57 Reaction-diffusion equations
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