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Random periodic processes, periodic measures and ergodicity
journal contributionposted on 2020-07-20, 12:59 authored by Chunrong Feng, Huaizhong Zhao
© 2020 The Author(s) Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincaré sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including 0 on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simple eigenvalues (infinitely many) including 0 on the imaginary axis. Moreover, under the spectral gap condition, PS-mixing of the periodic measure is proved. The “equivalence” of random periodic processes and periodic measures is established. This is a new class of ergodic random processes. Random periodic paths of stochastic perturbation of the periodic motion of an ODE is obtained.
Random Periodicity in Dynamics with Uncertainty
Engineering and Physical Sciences Research CouncilFind out more...
Royal Society Newton fund grant (ref. NA150344)
- Mathematical Sciences
Published inJournal of Differential Equations
Pages7382 - 7428
- VoR (Version of Record)
Rights holder© The authors
Publisher statementThis is an Open Access Article. It is published by Elsevier under the Creative Commons Attribution 4.0 Unported Licence (CC BY). Full details of this licence are available at: http://creativecommons.org/licenses/by/4.0/