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Random periodic processes, periodic measures and ergodicity

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journal contribution
posted on 20.07.2020 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.

Funding

Random Periodicity in Dynamics with Uncertainty

Engineering and Physical Sciences Research Council

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Royal Society Newton fund grant (ref. NA150344)

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Journal of Differential Equations

Volume

269

Issue

9

Pages

7382 - 7428

Publisher

Elsevier

Version

VoR (Version of Record)

Rights holder

© The authors

Publisher statement

This 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/

Acceptance date

31/05/2020

Publication date

2020-06-05

Copyright date

2020

ISSN

0022-0396

eISSN

1090-2732

Language

en

Depositor

Huaizhong Zhao Deposit date: 20 July 2020

Licence

Exports