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Random periodic processes, periodic measures and ergodicity

Feng, Chunrong and Zhao, Huaizhong (2020) 'Random periodic processes, periodic measures and ergodicity.', Journal of Differential Equations, 269 (9). pp. 7382-7428.


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.

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Publisher statement:© 2020 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (
Date accepted:31 May 2020
Date deposited:06 October 2021
Date of first online publication:05 June 2020
Date first made open access:06 October 2021

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