Random Delta-Hausdorff-attractors

Michael Scheutzow; Maite Wilke-Berenguer

doi:10.3934/dcdsb.2018148

Article Contents

2018, Volume 23, Issue 3: 1199-1217. Doi: 10.3934/dcdsb.2018148

This issue Previous Article Generalized KdV equation subject to a stochastic perturbation Next Article On the Oseledets-splitting for infinite-dimensional random dynamical systems

Random Delta-Hausdorff-attractors

Michael Scheutzow^1, and
Maite Wilke-Berenguer^1,

1.
Technische Universität Berlin, Fak. Ⅱ, Institut für Mathematik, Sekr. MA 7-5, Straße des 17. Juni 136,10623 Berlin, Germany

Received: February 28, 2017

Revised: July 31, 2017

Early access: February 2018

Published: June 2018

The second author is supported by the DFG-SPP 1590

Abstract Full Text(HTML) Related Papers Cited by

Abstract

Global random attractors and random point attractors for random dynamical systems have been studied for several decades. Here we introduce two intermediate concepts: Δ-Hausdorff-attractors are characterized by attracting all deterministic compact sets of Hausdorff dimension at most Δ, where Δ is a non-negative number, while cc-attractors attract all countable compact sets. We provide two examples showing that a given random dynamical system may have various different Δ-Hausdorff-attractors for different values of Δ. It seems that both concepts are new even in the context of deterministic dynamical systems.

Keywords:

Mathematics Subject Classification: Primary: 37H99, 37B25, 37C70; Secondary: 37H10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.
	S. Bernstein , Solution of a mathematical problem connected with the theory of heredity, Ann. Math. Statist., 13 (1942) , 53-61. doi: 10.1214/aoms/1177731642.
	I. Chueshov and M. Scheutzow , On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Syst., 19 (2004) , 127-144. doi: 10.1080/1468936042000207792.
	H. Crauel and F. Flandoli , Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994) , 365-393. doi: 10.1007/BF01193705.
	H. Crauel , Random point attractors versus random set attractors, Journal of the London Mathematical Society, 63 (2001) , 413-427. doi: 10.1017/S0024610700001915.
	H. Crauel, Random Probability Measures on Polish Spaces, Volume 11 of Stochastics Monographs, Taylor & Francis, London, 2002.
	H. Crauel and M. Scheutzow, Minimal random attractors, arXiv: 1712.08692.
	R. R. Davronov , U. U. Jamilov and M. Ladra , Conditional cubic stochastic operator, Journal of Difference Equations and Applications, 21 (2015) , 1163-1170. doi: 10.1080/10236198.2015.1062481.
	S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986.
	F. Flandoli , B. Gess and M. Scheutzow , Synchronization by noise, Probab. Theory Related Fields, 168 (2017) , 511-556. doi: 10.1007/s00440-016-0716-2.
	R. N. Ganikhodzhaev , Quadratic Stochastic Operators, Lyapunov functions, and tournaments, Russian Academy of Sciences. Sbornik Mathematics, 76 (1993) , 489-506.
	N. N. Ganikhodzaev , The random models of heredity in the random environments, Dokl. Akad. Nauk Ruz, 12 (2000) , 6-8.
	U. U. Jamilov , M. Scheutzow and M. Wilke-Berenguer , On the random dynamics of Volterra quadratic operators, Ergodic Theory and Dynamical Systems, 37 (2017) , 228-243. doi: 10.1017/etds.2015.30.
	B. J. Mamurov and U. A. Rozikov, On cubic stochastic operators and processes Journal of Physics: Conference Series, 697 (2016), 012017. doi: 10.1088/1742-6596/697/1/012017.
	P. Mandl, Analytical Treatment of one-dimensional Markov Processes, Die Grundlehren der mathematischen Wissenschaften, Band 151. Academia Publishing House of the Czechoslovak Academy of Sciences, Prague; Springer-Verlag New York Inc., New York, 1968.
	P. Mörters and Y. Peres, Brownian Motion, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. With an appendix by Oded Schramm and Wendelin Werner.
	G. Ochs, Weak random attractors, Report 499, Institut für Dynamische Systeme, Universität Bremen, 1999.
	M. Scheutzow , Comparison of various concepts of a random attractor: A case study, Archiv der Mathematik, 78 (2002) , 233-240.
	M. Wilke Berenguer, A Selection of Stochastic Processes Emanating from the Natural Sciences, Ph. D thesis, Technische Universität Berlin, 2016.