On the finite dimensionality of random attractors.

*(English)*Zbl 0888.60051Suppose \(\omega\mapsto A(\omega)\) 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)

##### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

60H99 | Stochastic analysis |

##### Keywords:

global attractor; random dynamical system; Hausdorff dimension; random attractor; stochastic reaction diffusion equation; stochastic nonlinear wave equation
PDF
BibTeX
XML
Cite

\textit{A. Debussche}, Stochastic Anal. Appl. 15, No. 4, 473--491 (1997; Zbl 0888.60051)

Full Text:
DOI

**OpenURL**

##### References:

[1] | DOI: 10.1007/BF02112325 · Zbl 0562.35067 |

[2] | DOI: 10.1007/BF01193705 · Zbl 0819.58023 |

[3] | Crauel H., Hausdorff dimension of invariant sets for random dynamical systems (1994) · Zbl 0927.37031 |

[4] | Crauel H., Random attractors (1995) |

[5] | Da Prato, The stochastic Cahn-Hilliard equation (1994) · Zbl 0838.60056 |

[6] | Foias C., J. Math. Pures Appl 58 pp 339– (1979) |

[7] | Hale J.K., Asynlptotic behaviour of dissipative dynamical systems, Mathernatical Surveys and Monographs 25 (1988) |

[8] | DOI: 10.1007/BF02112336 · Zbl 0561.76044 |

[9] | DOI: 10.1080/00036818708839678 · Zbl 0609.35009 |

[10] | Temam R., Infinite dimensional dynamical systems in mechanics and physics (1988) · Zbl 0662.35001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.