Strong Unique Ergodicity of Random Dynamical Systems on Polish Spaces

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Abstract

In this paper we want to show the existence of a form of asymptotic stability of random dynamical systems in the sense of L. Arnold using arguments analogous to those presented by T. Szarek in [6], that is showing it using conditions generalizing the notion of tightness of measures. In order to do that we use tightness theory for random measures as developed by H. Crauel in [2].

References

  • [1] Arnold L., Random dynamical systems, Springer Monographs in Mathematics, Springer, Berlin, 1998.

  • [2] Crauel H., Random probability measures on Polish spaces, Series Stochastics Monographs, Vol. 11, Taylor & Francis, London, 2002.

  • [3] Crauel H., Flandoli F., Attractors for random dynamical systems, Probab. Theory Relat. Fields 100 (1994), 365–393.

  • [4] Lasota A., Yorke J.A., Lower bound technique for Markov operators and iterated function systems, Random Comput. Dynam. 2 (1994), 41–77.

  • [5] Szarek T., The stability of Markov operators on Polish spaces, Studia Math. 143 (2000), 145–152.

  • [6] Szarek T., Invariant measures for non-expansive Markov operators on Polish spaces, Dissertationes Math. 415 (2003), 62 pp.

  • [7] Valadier M., Young measures, in: Methods of Nonconvex Analysis (Varrenna 1989), Lecture Notes in Math. 1446, Springer, Berlin, 1990, pp. 152–188.

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researchers in all branches of pure and applied mathematics

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