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On the finite dimensionality of random attractors. (English) Zbl 0888.60051
Suppose \(\omega\mapsto A(\omega)\) is the global attractor of a random dynamical system on a separable Hilbert space. Generalizing a method used by O. A. Ladyzhenskaya [J. Sov. Math. 28, 714-726 (1985); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 115, 137-155 (1982; Zbl 0535.76033)] and by A. V. Babin and M. I. Vishik [ibid. 28, 619-627 (1985); resp. ibid. 115, 3-15 (1982; Zbl 0507.35076)], conditions for finiteness of the Hausdorff dimension of the attractor are given. The basic condition is the existence of a deterministic finite-dimensional projection \(P\), such that the distance of \(P\)-images of any two solutions on the attractor has an integrable exponential growth rate, and that further distances of \(I-P\)-images are of suitably bounded exponential growth. This type of conditions is already needed for the approach to work in the deterministic case. The additional assumption needed for the stochastic case essentially is integrability of the diameter of the random attractor. Upper bounds for the dimension in terms of the rank of \(P\) are obtained. The result is applied to a stochastic reaction diffusion equation and to a stochastic nonlinear wave equation, both with (finite-dimensional) additive white noise.
Reviewer: H.Crauel (Berlin)

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
60H99 Stochastic analysis
Full Text: DOI
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