Random periodic processes, periodic measures and ergodicity

Chunrong Feng,Huaizhong Zhao

doi:10.1016/j.jde.2020.05.034

Chunrong Feng, Huaizhong Zhao

Open Access

https://doi.org/10.1016/j.jde.2020.05.034

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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.

Highlights

Ergodicity is significant for the theory of random dynamical systems in describing their large time behaviour and irreducibility
In this paper we will break this restriction to provide an ergodic theory when “periodicity” exists. This scenario, regarded as random periodic, is defined in a very general situation of a separable Banach space, applicable for both discrete random mappings and continuous time stochastic flows. It is well known but still worth mentioning in this context that the notion of periodic paths has been a major concept in the theory of dynamical systems since Poincare’s pioneering work ([27])
For Markovian random dynamical systems, we introduce the idea of Poincare sections {Ls}s≥0 with Ls+τ = Ls such that for any x ∈ Lt, P (s, x, Ls+t) = 1, s, t ≥ 0

Summary

Introduction

Ergodicity is significant for the theory of random dynamical systems in describing their large time behaviour and irreducibility. It is noted that the spectral structure of the Markov semigroup is more fruitful than that of the transformation operator on the path space In this context, it is worthy mentioning that in the case of the stationary regime, many results on spectral gaps have been obtained, which give how far the rest of spectra of the generator are away from the 0 eigenvalue (c.f. see [6],[32] etc). The spectral gap gives mixing property and convergence rate of transitional probability to the invariant measure This fundamentally important result has brought many powerful analysis tools to the study of ergodicity and mixing of the invariant measure of stochastic systems. We will publish these results in a different publication ([15])

Random periodic paths and examples

Periodic measures

Poincare sections and ergodicity with periodicity

Random periodic verses stationary: sufficient-necessary conditions

Spectral gap and PS-mixing

Construction of random periodic paths from a periodic measure