Random Periodic Processes, Periodic Measures and Ergodicity

TL;DR

This paper establishes ergodicity results for random dynamical systems with periodic measures on Polish spaces, introduces Poincaré sections for Markov processes, and links spectral properties of generators to mixing and ergodic behavior.

Contribution

It introduces the concept of PS-ergodicity, proves its equivalence with ergodicity under certain conditions, and establishes a new class of ergodic random processes with periodic measures.

Findings

PS-ergodicity implies ergodicity for periodic measures.

Spectral conditions on the generator determine PS-ergodicity and mixing.

Established the equivalence between random periodic processes and periodic measures.

Abstract

Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincar\'e 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.