Stability of random dynamical systems on Banach spaces. (English) Zbl 1121.37037
The authors consider a stochastic process generated by a randomly chosen dynamical system on a separable Banach space. A weak ergodic theorem is proved for this process under some suitable assumptions.
Reviewer: Anatoliy Swishchuk (Calgary)
MSC:
| 37H99 | Random dynamical systems |
| 37C75 | Stability theory for smooth dynamical systems |
| 47A35 | Ergodic theory of linear operators |
| 47D07 | Markov semigroups and applications to diffusion processes |
| 60G57 | Random measures |
| 60J35 | Transition functions, generators and resolvents |
References:
| [1] | L. Arnold, Random Dynamical Systems, Springer - Verlag, Berlin (1998). · Zbl 0906.34001 |
| [2] | G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge (1992). · Zbl 0761.60052 |
| [3] | M.H.A. Davis Markov, Models and Optimization, Chapman and Hall, London (1993). |
| [4] | R.M. Dudley, Probabilities and Matrics, Aarhus Universitet (1976). |
| [5] | S. Ethier, T. Kurtz, Markov Processes, John Wiley and Sons, New York (1986). |
| [6] | R. Fortet, B. Mourier, Convergence de la répartition empirique vers la répartition théorétique, Ann. Sci. École Norm. Sup., 70 (1953), 267–285. · Zbl 0053.09601 |
| [7] | U. Frisch, Wave propagation in random media, In: A.T. Bharucha-Reid (ed) Probabilistic Methods in Applied Mathematics, Academic Press (1968). · Zbl 0181.22303 |
| [8] | K. Horbacz, Dynamical systems with multiplicative perturbations: The strong convergence of measures, Ann. Polon. Math., 58 (1993), 85–93. · Zbl 0782.47007 |
| [9] | K. Horbacz, Randomly connected dynamical systems – asymptotic stability, Ann. Pol. Math. (1998), 31–50. · Zbl 0910.47003 |
| [10] | K. Horbacz, T. Szarek, Randomly connected dynamical systems on Banach spaces, Stochastic Anal. Appl., 19 (2001), 519–543. · Zbl 1062.47506 |
| [11] | J.B. Keller, Stochastic equations and wave propagation in random media, Proc. Symp. Appl. Math., 16 (1964), 1456–170. · Zbl 0202.46703 |
| [12] | M. Kuczma, B. Choczewski, R. Ger, Iterative Functional Equations, Cambridge Univ. Press, New York - Cambridge (1985). · Zbl 0703.39005 |
| [13] | A. Lasota, J. Traple, Invariant measures related with Poisson driven stochastic differential equation, Stoch. Proc. and Their Appl., 106 (2003), 81–93. · Zbl 1075.60535 |
| [14] | A. Lasota, J.A. Yorke, Lower bound technique for Markov operators and iterated function systems, Random Comput. Dynam., 2 (1994) 41–77. · Zbl 0804.47033 |
| [15] | S. Meyn, R. Tweedie, Markov Chains and Stochastic Stability, Springer - Verlag, Berlin (1993). · Zbl 0925.60001 |
| [16] | D. Snyder, Random Point Processes, John Wiley, New York (1975). · Zbl 0385.60052 |
| [17] | T. Szarek, S. Wedrychowicz, Markov semigroups generated by a Poisson driven differential equation, Nonlin. Anal. TMA, 50(1) (2002) 41–54. · Zbl 1004.60062 |
| [18] | T. Szarek, Invariant measures for Markov operators with applications to function systems, Studia Math., 154(3) (2003), 207–222. · Zbl 1036.47003 |
| [19] | J. Traple, Markov semigroups generated by Poisson driven differential equations Bull. Pol. Ac.:Math., 44 (1996), 161–182. · Zbl 0861.45008 |
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.
