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May  2018, 23(3): 1199-1217. doi: 10.3934/dcdsb.2018148

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  March 2017 Revised  August 2017 Published  May 2018 Early access  February 2018

Fund Project: The second author is supported by the DFG-SPP 1590.

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: Random attractor, Delta-Hausdorff-attractor, pullback attractor, weak attractor, forward attractor, cc-attractor, random dynamical system, Hausdorff dimension.
Mathematics Subject Classification: Primary: 37H99, 37B25, 37C70; Secondary: 37H10.
Citation: Michael Scheutzow, Maite Wilke-Berenguer. Random Delta-Hausdorff-attractors. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1199-1217. doi: 10.3934/dcdsb.2018148
References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[5]

H. Crauel, Random point attractors versus random set attractors, Journal of the London Mathematical Society, 63 (2001), 413-427.  doi: 10.1017/S0024610700001915.  Google Scholar

[6]

H. Crauel, Random Probability Measures on Polish Spaces, Volume 11 of Stochastics Monographs, Taylor & Francis, London, 2002.  Google Scholar

[7]

H. Crauel and M. Scheutzow, Minimal random attractors, arXiv: 1712.08692. Google Scholar

[8]

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.  Google Scholar

[9]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986.  Google Scholar

[10]

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.  Google Scholar

[11]

R. N. Ganikhodzhaev, Quadratic Stochastic Operators, Lyapunov functions, and tournaments, Russian Academy of Sciences. Sbornik Mathematics, 76 (1993), 489-506.   Google Scholar

[12]

N. N. Ganikhodzaev, The random models of heredity in the random environments, Dokl. Akad. Nauk Ruz, 12 (2000), 6-8.   Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

G. Ochs, Weak random attractors, Report 499, Institut für Dynamische Systeme, Universität Bremen, 1999. Google Scholar

[18]

M. Scheutzow, Comparison of various concepts of a random attractor: A case study, Archiv der Mathematik, 78 (2002), 233-240.   Google Scholar

[19]

M. Wilke Berenguer, A Selection of Stochastic Processes Emanating from the Natural Sciences, Ph. D thesis, Technische Universität Berlin, 2016. Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[5]

H. Crauel, Random point attractors versus random set attractors, Journal of the London Mathematical Society, 63 (2001), 413-427.  doi: 10.1017/S0024610700001915.  Google Scholar

[6]

H. Crauel, Random Probability Measures on Polish Spaces, Volume 11 of Stochastics Monographs, Taylor & Francis, London, 2002.  Google Scholar

[7]

H. Crauel and M. Scheutzow, Minimal random attractors, arXiv: 1712.08692. Google Scholar

[8]

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.  Google Scholar

[9]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986.  Google Scholar

[10]

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.  Google Scholar

[11]

R. N. Ganikhodzhaev, Quadratic Stochastic Operators, Lyapunov functions, and tournaments, Russian Academy of Sciences. Sbornik Mathematics, 76 (1993), 489-506.   Google Scholar

[12]

N. N. Ganikhodzaev, The random models of heredity in the random environments, Dokl. Akad. Nauk Ruz, 12 (2000), 6-8.   Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

G. Ochs, Weak random attractors, Report 499, Institut für Dynamische Systeme, Universität Bremen, 1999. Google Scholar

[18]

M. Scheutzow, Comparison of various concepts of a random attractor: A case study, Archiv der Mathematik, 78 (2002), 233-240.   Google Scholar

[19]

M. Wilke Berenguer, A Selection of Stochastic Processes Emanating from the Natural Sciences, Ph. D thesis, Technische Universität Berlin, 2016. Google Scholar

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