Random periodic processes, periodic measures and ergodicity

Feng, Chunrong; Zhao, Huaizhong

Abstract

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.

Citation

Feng, C., & Zhao, H. (2020). Random periodic processes, periodic measures and ergodicity. Journal of Differential Equations, 269(9), 7382-7428. https://doi.org/10.1016/j.jde.2020.05.034

• Journal Article Type: Article
• Acceptance Date: May 31, 2020
• Online Publication Date: Jun 5, 2020
• Publication Date: 2020-10
• Deposit Date: Jan 5, 2021
• Publicly Available Date: Oct 6, 2021
• Journal: Journal of Differential Equations
• Print ISSN: 0022-0396
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 269
• Issue: 9
• Pages: 7382-7428
• DOI: https://doi.org/10.1016/j.jde.2020.05.034