Exponential convergence for Markov systems



Abstract

Markov operators arising from graph directed constructions of iterated function systems are considered. Exponential convergence to an invariant measure is proved.


Keywords

Markov operator; invariant measure

1. Barnsley M.F., Demko S.G., Elton J.H., Geronimo J.S., Invariant measures for Markov processes arising from iterated function systems with place dependent probabilities, Ann. Inst. H. Poincaré Probab. Statist. 24 (1988), 367–394.
2. Hairer M., Exponential mixing properties of stochastic PDEs through asymptotic coupling, Probab. Theory Related Fields 124 (2002), 345–380.
3. Hairer M., Mattingly J., Scheutzow M., Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations, Probab. Theory Related Fields 149 (2011), no. 1, 223–259.
4. Horbacz K., Szarek T., Irreducible Markov systems on Polish spaces, Studia Math. 177 (2006), no. 3, 285–295.
5. Horbacz K., Ślęczka M., Law of large numbers for random dynamical systems, Preprint 2013, arXiv:1304.6863.
6. Kapica R., Ślęczka M., Random iteration with place dependent probabilities, Preprint 2012, arXiv:1107.0707v2.
7. Mauldin R.D., Williams S.C., Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), 811–829.
8. Mauldin R.D., Urbański M., Graph directed Markov systems: geometry and dynamics of limit sets, Cambridge University Press, Cambridge, 2003.
9. Rachev S.T., Probability metrics and the stability of stochastic models, John Wiley, New York, 1991.
10. Ślęczka M., The rate of convergence for iterated function systems, Studia Math. 205 (2011), no. 3, 201–214.
11. Werner I., Ergodic theorem for contractive Markov systems, Nonlinearity 17 (2004), 2303–2313.
12. Werner I., Contractive Markov systems, J. London Math. Soc. (2) 71 (2005), 236–258.
13. Wojewódka H., Exponential rate of convergence for some Markov operators, Statist. Probab. Lett. 83 (2013), 2337–2347.
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Published : 2015-09-30


ŚlęczkaM. (2015). Exponential convergence for Markov systems. Annales Mathematicae Silesianae, 29, 139-149. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13984

Maciej Ślęczka  sleczka@ux2.math.us.edu.pl
Instytut Matematyki, Uniwersytet Śląski w Katowicach  Poland



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