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.


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
Full Text: DOI


[1] Babin, A. V.; Vishik, M. I.: Attractors of Navier-Stokes systems and of parabolic equations and estimates for their dimensions. J. soviet math. 28, No. no 5, 619-627 (1985) · Zbl 0562.35067
[2] Babin, A. V.; Vishik, M. I.: Attractors for evolution equations. (1992) · Zbl 0778.58002
[3] Castaing, C.; Valadier, M.: Convex analysis and mesurable multifunctions. Lecture notes in mathematics 580 (1977) · Zbl 0346.46038
[4] Constantin, P.; Foias, C.; Temam, R.: Attractors representing turbulent flows. Memoirs of the amer. Math. soc. 53, No. no 314 (1985) · Zbl 0567.35070
[5] Crauel, H.; Flandoli, F.: Attractors for random dynamical systems. Prob. theory and related fields 100, 365-393 (1994) · Zbl 0819.58023
[6] H. Crauel, F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, To appear in Journal of Dyn. and Diff. Eq... · Zbl 0927.37031
[7] Crauel, H.; Debussche, A.; Flandoli, F.: Random attractors. Journal of dyn. And diff. Eq. 9, No. no 2, 307-341 (1997) · Zbl 0884.58064
[8] Da Prato, G.; Zabczyck, J.: Ergodicity for infinite dimensional systems. London mathematical society, lecture note series 229 (1996)
[9] Debussche, A.: On the finite dimensionality of random attractors. Stochastic analysis and applications 15, No. no 4, 473-492 (1997) · Zbl 0888.60051
[10] Douady, A.; Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. sci. Paris 290, 1135-1138 (1980) · Zbl 0443.58016
[11] Dunford, N.; Schwartz, J. T.: Linear operators. (1958)
[12] Foias, C.; Temam, R.: Some analytic and geometrical properties of the solutions of Navier-Stokes equations. J. math. Pures appl. 58, 339-368 (1979) · Zbl 0454.35073
[13] Hale, J. K.: Asymptotic behaviour of dissipative dynamical systems. Mathematical surveys and monographs 25 (1988) · Zbl 0642.58013
[14] Ladyzhenskaya, O. A.: On the finitness of the dimension of bounded invariant sets for the Navier-Stokes equations and other related dissipative systems. J. soviet math. 28, No. no 5, 714-725 (1985) · Zbl 0561.76044
[15] Lions, J. -L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. (1969) · Zbl 0189.40603
[16] Mallet-Paret, J.: Negatively invariant sets of compact maps and an extension of a theorem of cartwright. Journal of diff. Eq. 22, 331 (1976) · Zbl 0354.34072
[17] Marion, M.: Attractors for reaction-diffusion equations ; existence and estimate of their dimension. Appl. anal. 25, 101-147 (1987) · Zbl 0609.35009
[18] Reed, M.; Simon, B.: Methods of modern mathematical physics. (1978) · Zbl 0401.47001
[19] Schmalfuß, B.: Measure attractors and stochastic attractors. Technical report 332 (1995) · Zbl 0944.60071
[20] B. Schmalfuß, The stochastic attractor of the stochastic Lorenz system, To appear. · Zbl 0887.34057
[21] Temam, R.: Infinite dimensional dynamical systems in mechanics and physics. (1997) · Zbl 0871.35001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.