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Synchronisation of almost all trajectories of a random dynamical system

The author wishes gratefully to acknowledge funding from an EPSRC Doctoral Training Account and an EPSRC Doctoral Prize, both at Imperial College London

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  • It has been shown by Le Jan that, given a memoryless-noise random dynamical system together with an ergodic distribution for the associated Markov transition probabilities, if the support of the ergodic distribution admits locally asymptotically stable trajectories, then there is a random attracting set consisting of finitely many points, whose basin of forward-time attraction includes a random full measure open set. In this paper, we present necessary and sufficient conditions for this attracting set to be a singleton. Our result does not require the state space to be compact, but holds on general Lusin metric spaces (in both discrete and continuous time).

    Mathematics Subject Classification: Primary: 37H05; Secondary: 60F15.

    Citation:

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  • [1] V. A. Antonov, Modeling of processes of cyclic evolution type. Synchronization by a random signal, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 1984, 67–76.
    [2] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
    [3] P. H. Baxendale, Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms, in Spatial Stochastic Processes, Progr. Probab., 19, Birkhäuser Boston, Boston, MA, 1991,189–218. doi: 10.1007/978-1-4612-0451-0_9.
    [4] P. BertiL. Pratelli and P. Rigo, Almost sure weak convergence of random probability measures, Stochastics, 78 (2006), 91-97.  doi: 10.1080/17442500600745359.
    [5] M. CranstonB. Gess and M. Scheutzow, Weak synchronization for isotropic flows, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3003-3014.  doi: 10.3934/dcdsb.2016084.
    [6] H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11, Taylor & Francis, London, 2002.
    [7] H. Crauel and F. Flandoli, Additive noise destroys a pitchfork bifurcation, J. Dynam. Differential Equations, 10 (1998), 259-274.  doi: 10.1023/A:1022665916629.
    [8] F. FlandoliB. Gess and M. Scheutzow, Synchronization by noise, Probab. Theory Related Fields, 168 (2017), 511-556.  doi: 10.1007/s00440-016-0716-2.
    [9] F. FlandoliB. Gess and M. Scheutzow, Synchronization by noise for order-preserving random dynamical systems, Ann. Probab., 45 (2017), 1325-1350.  doi: 10.1214/16-AOP1088.
    [10] D. H. Fremlin, Measure Theory. 4: Topological Measure Spaces, Torres Fremlin, Colchester, 2006.
    [11] A. J. Homburg, Synchronization in iterated function systems, preprint, arXiv: 1303.6054v1.
    [12] T. Kaijser, On stochastic perturbations of iterations of circle maps, Phys. D, 68 (1993), 201-231.  doi: 10.1016/0167-2789(93)90081-B.
    [13] Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10, Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.
    [14] S. Kuksin and  A. ShirikyanMathematics of Two-Dimensional Turbulence, Cambridge Tracts in Mathematics, 194, Cambridge University Press, Cambridge, 2012.  doi: 10.1017/CBO9781139137119.
    [15] Y. Le Jan, \'{E}quilibre statistique pour les produits de difféomorphismes aléatoires indépendants, Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 111-120. 
    [16] D. Malicet, Random walks on Homeo(S1), Commun. Math. Phys., 356 (2017), 1083-1116.  doi: 10.1007/s00220-017-2996-5.
    [17] J. Newman, Ergodic theory for semigroups of Markov kernels, 2015. Available from: http://wwwf.imperial.ac.uk/ jmn07/Ergodic_Theory_for_Semigroups_of_Markov_Kernels.pdf.
    [18] J. Newman, Synchronisation in invertible random dynamical systems on the circle, preprint, arXiv: 1502.07618v2.
    [19] J. Newman, Necessary and sufficient conditions for stable synchronization in random dynamical systems, Ergodic Theory Dynam. Systems, 38 (2018), 1857-1875.  doi: 10.1017/etds.2016.109.
    [20] A. S. Pikovskii, Synchronization and stochastization of array of self-excited oscillators by external noise, Radiophys. Quantum Electron., 27 (1984), 390-395.  doi: 10.1007/BF01044784.
    [21] S. M. Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics, 180, Springer-Verlag, New York, 1998. doi: 10.1007/978-3-642-85473-6.
    [22] R. ToralC. R. MirassoE. Hernández-García and O. Piro, Analytical and numerical studies of noise-induced synchronization of chaotic systems, Chaos, 11 (2001), 665-673.  doi: 10.1063/1.1386397.
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