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Random Delta-Hausdorff-attractors
| 1. | Technische Universität Berlin, Fak. Ⅱ, Institut für Mathematik, Sekr. MA 7-5, Straße des 17. Juni 136,10623 Berlin, Germany |
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.
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L. Arnold,
Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. |
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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. |
| [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. |
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H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
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H. Crauel,
Random point attractors versus random set attractors, Journal of the London Mathematical Society, 63 (2001), 413-427.
doi: 10.1017/S0024610700001915. |
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H. Crauel,
Random Probability Measures on Polish Spaces, Volume 11 of Stochastics Monographs, Taylor & Francis, London, 2002. |
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H. Crauel and M. Scheutzow, Minimal random attractors, arXiv: 1712.08692. |
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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. |
| [9] |
S. N. Ethier and T. G. Kurtz,
Markov Processes: Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986. |
| [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. |
| [11] |
R. N. Ganikhodzhaev,
Quadratic Stochastic Operators, Lyapunov functions, and tournaments, Russian Academy of Sciences. Sbornik Mathematics, 76 (1993), 489-506.
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N. N. Ganikhodzaev,
The random models of heredity in the random environments, Dokl. Akad. Nauk Ruz, 12 (2000), 6-8.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
G. Ochs, Weak random attractors, Report 499, Institut für Dynamische Systeme, Universität Bremen, 1999. |
| [18] |
M. Scheutzow,
Comparison of various concepts of a random attractor: A case study, Archiv der Mathematik, 78 (2002), 233-240.
|
| [19] |
M. Wilke Berenguer,
A Selection of Stochastic Processes Emanating from the Natural Sciences, Ph. D thesis, Technische Universität Berlin, 2016. |
show all references
References:
| [1] |
L. Arnold,
Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. |
| [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. |
| [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. |
| [4] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
| [5] |
H. Crauel,
Random point attractors versus random set attractors, Journal of the London Mathematical Society, 63 (2001), 413-427.
doi: 10.1017/S0024610700001915. |
| [6] |
H. Crauel,
Random Probability Measures on Polish Spaces, Volume 11 of Stochastics Monographs, Taylor & Francis, London, 2002. |
| [7] |
H. Crauel and M. Scheutzow, Minimal random attractors, arXiv: 1712.08692. |
| [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. |
| [9] |
S. N. Ethier and T. G. Kurtz,
Markov Processes: Characterization and Convergence, John Wiley & Sons, Inc., New York, 1986. |
| [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. |
| [11] |
R. N. Ganikhodzhaev,
Quadratic Stochastic Operators, Lyapunov functions, and tournaments, Russian Academy of Sciences. Sbornik Mathematics, 76 (1993), 489-506.
|
| [12] |
N. N. Ganikhodzaev,
The random models of heredity in the random environments, Dokl. Akad. Nauk Ruz, 12 (2000), 6-8.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
G. Ochs, Weak random attractors, Report 499, Institut für Dynamische Systeme, Universität Bremen, 1999. |
| [18] |
M. Scheutzow,
Comparison of various concepts of a random attractor: A case study, Archiv der Mathematik, 78 (2002), 233-240.
|
| [19] |
M. Wilke Berenguer,
A Selection of Stochastic Processes Emanating from the Natural Sciences, Ph. D thesis, Technische Universität Berlin, 2016. |
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