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Random dynamical systems with jumps

Part of: General theory of linear operators

Published online by Cambridge University Press:  14 July 2016

Katarzyna Horbacz
Katarzyna Horbacz*
Affiliation:
University of Silesia
*
∗ Postal address: Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland. Email address: horbacz@ux2.math.us.edu.pl
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Abstract

We consider random dynamical systems with randomly chosen jumps on infinite-dimensional spaces. The choice of deterministic dynamical systems and jumps depends on a position. The system generalizes dynamical systems corresponding to learning systems, Poisson driven stochastic differential equations, iterated function system with infinite family of transformations and random evolutions. We will show that distributions which describe the dynamics of this system converge to an invariant distribution. We use recent results concerning asymptotic stability of Markov operators on infinite-dimensional spaces obtained by T. Szarek.

Keywords

Dynamical systemMarkov semigroupinvariant measure

MSC classification

Primary: 47A35: Ergodic theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 890 - 910
Copyright
Copyright © Applied Probability Trust 2004 

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References

Arnold, L. (1998). Random Dynamical Systems. Springer, Berlin.10.1007/978-3-662-12878-7Google Scholar
Barnsley, M. F., Demko, S. G., Elton, J. H., and Geronimo, J. S. (1988). Invariant measures arising from iterated function systems with place dependent probabilities. Ann. Inst. H. Poincaré Prob. Statist. 24, 367–394.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization. Chapman and Hall, London.10.1007/978-1-4899-4483-2Google Scholar
Diekmann, O., Heijmans, H. J. A. M., and Thieme, H. R. (1984). On the stability of the cell size distribution. J. Math. Biol. 19, 227–248.10.1007/BF00277748Google Scholar
Fortet, R., and Mourier, B. (1953). Convergence de la répartition empirique vers la répartition théorétique. Ann. Sci. École Norm. Sup. 70, 267–285.10.24033/asens.1013Google Scholar
Frisch, U. (1968). Wave propagation in random media. In Probabilistic Methods in Applied Mathematics, ed. Bharucha-Reid, A. T., Academic Press, New York, pp. 75–198.Google Scholar
Griego, R. J., and Hersh, R. (1969). Random evolutions, Markov chains and systems of partial differential equations. Proc. Nat. Acad. Sci. USA 62, 305–308.10.1073/pnas.62.2.305Google Scholar
Griego, R. J., and Hersh, R. (1971). Theory of random evolutions with applications to partial differential equation. Trans. Amer. Math. Soc. 156, 405–418.10.1090/S0002-9947-1971-0275507-7Google Scholar
Horbacz, K. (1998). Randomly connected dynamical systems—asymptotic stability. Ann. Polon. Math. 68, 31–50.10.4064/ap-68-1-31-50Google Scholar
Horbacz, K. (2002). Invariant measure related with randomly connected Poisson driven differential equations. Ann. Polon. Math. 79, 31–43.10.4064/ap79-1-3Google Scholar
Horbacz, K. (2002). Randomly connected differential equations with Poisson type perturbations. Nonlinear Stud. 9, 81–98.Google Scholar
Horbacz, K., and Szarek, T. (2001). Randomly connected dynamical systems on Banach space. Stoch. Anal. Appl. 19, 519–543.10.1081/SAP-100002100Google Scholar
Horbacz, K., Myjak, J., and Szarek, T. (2004). On stability of some general random dynamical system. In preparation.Google Scholar
Keller, J. B. (1964). Stochastic equations and wave propagation in random media. Proc. Symp. Appl. Math. 16, 1456–1470.Google Scholar
Kertz, R. (1978). Limit theorems for semigroups with perturbed generators with applications to multi-scaled random evolutions. J. Funct. Anal. 27, 215–233.10.1016/0022-1236(78)90028-9Google Scholar
Lasota, A. (1995). From fractals to stochastic differential equations. In Chaos—The Interplay Between Stochastic and Deterministic Behaviour (Lecture Notes Phys. 457), eds Garbaczewski, P., Wolf, M. and Weron, A., Springer, Berlin, pp. 235–255.10.1007/3-540-60188-0_58Google Scholar
Lasota, A., and Mackey, M. C. (1999). Cell division and the stability of cellular populations. J. Math. Biol. 38, 241–261.10.1007/s002850050148Google Scholar
Lasota, A., and Traple, J. (2003). Invariant measures related with Poisson driven stochastic differential equation. Stoch. Process. Appl. 106, 81–93.10.1016/S0304-4149(03)00017-6Google Scholar
Lasota, A., and Yorke, J. A. (1994). Lower bound technique for Markov operators and iterated function systems. Random Comput. Dynam. 2, 41–77.Google Scholar
Malczak, J. (1993). Statistical stability of Poisson driven differential equation. Bull. Pol. Acad. Math. 41, 159–176.Google Scholar
Meyn, S., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, Berlin.10.1007/978-1-4471-3267-7Google Scholar
Myjak, J., and Szarek, T. (2004). On the existence of an invariant measure for Markov-Feller operators. J. Math. Anal. Appl. 294, 215–222.10.1016/j.jmaa.2004.02.011Google Scholar
Pinsky, M. A. (1991). Lectures on Random Evolution. World Scientific, River Edge, NJ.10.1142/1328Google Scholar
Szarek, T. (2003). Invariant measures for Markov operators with applications to function systems. Studia Math. 154, 207–222.10.4064/sm154-3-2Google Scholar
Szarek, T. and Wedrychowicz, S. (2002). Markov semigroups generated by a Poisson driven differential equation. Nonlinear Anal. A 50, 41–54.10.1016/S0362-546X(01)00724-6Google Scholar
Traple, J. (1996). Markov semigroups generated by Poisson driven differential equations. Bull. Pol. Acad. Math. 44, 161–182.Google Scholar