Summary
The asymptotic behaviour of random dynamical systems in Polish spaces is considered. Under the assumption of existence of a random compact absorbing set, assumption supposed to hold path by path, a candidate pathwise attractorA(ω) is defined. The goal of the paper is to show that, in the case of stationary dynamical systems,A(ω) attracts bounded sets, is measurable with respect to the σ-algebra of invariant sets, and is independent of ω when the system is ergodic. An application to a general class of Navier-Stokes type equations perturbed by a multiplicative ergodic real noise is discussed in detail.
References
Arnold, L., Wihstutz, V. (eds.): Lyapunov Exponents, (Lect. Notes Math., vol. 1186) Berlin Heidelberg New York: Springer 1986
Breiman, L.: Probability. Reading, Mass.: Addison-Wesley 1968
Brzeźniak, Z., Capiński, M., Flandoli, F.: Stochastic partial differential equations and turbulence. Math. Models Methods Appl. Sci.1(1), 41–59 (1991)
Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. (Lect. Notes Math., vol. 550) Berlin Heidelberg New York: Springer 1977
Da Prato, G.: Some results on linear stochastic evolution equations in Hilbert spaces by the semigroup method. Stochastic Anal. Appl.1, 57–88 (1983)
Flandoli, F.: Stochastic flow and Lyapunov exponents for abstract stochastic PDEs of parabolic type. In: Proceedings Conf. on Lyapunov Exponents. Oberwolfach 1990 Arnold, L., Crauel, H., Eckmann, J.P. (eds.) (Lect. Notes Math., vol. 1486) Berlin Heidelberg New York: Springer 1991
Ichikawa, A., Stability of semilinear stochastic evolution equations. J. Math. Anal.90, 12–44 (1982)
Kurtz, Ethier, T.: Markov processes, characterization and convergence. (Wiley Ser. Probab. Math. Stat.) New York: Wiley 1986
Kuratowski, K.: Topology, vol. I. New York London: Academic Press 1966
Lions, J. L., Magenes, E.: Non-homogeneous boundary value problems and applications. Berlin Heidelberg New York: Springer 1972
Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Berlin Heidelberg New York: Springer 1983
Temam, R.: Infinite dimensional dynamical systems in mechanics and physics. Berlin Heidelberg New York: Springer 1988
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brzeźniak, Z., Capiński, M. & Flandoli, F. Pathwise global attractors for stationary random dynamical systems. Probab. Th. Rel. Fields 95, 87–102 (1993). https://doi.org/10.1007/BF01197339
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01197339
Keywords
- Dynamical System
- Stochastic Process
- Asymptotic Behaviour
- Probability Theory
- General Class