A sufficient and necessary condition of PS-ergodicity of periodic measures and generated ergodic upper expectations

Feng, Chunrong; Qu, Baoyou; Zhao, Huaizhong

doi:10.1088/1361-6544/ab9584

A sufficient and necessary condition of PS-ergodicity of periodic measures and generated ergodic upper expectations

Feng, Chunrong; Qu, Baoyou; Zhao, Huaizhong

Authors

Dr Chunrong Feng chunrong.feng@durham.ac.uk
Associate Professor

Baoyou Qu

Professor Huaizhong Zhao huaizhong.zhao@durham.ac.uk
Professor

Abstract

This paper contains two parts. In the first part, we study the ergodicity of periodic measures of random dynamical systems on a separable Banach space. We obtain that the periodic measure of the continuous time skew-product dynamical system generated by a random periodic path is ergodic if and only if the underlying noise metric dynamical system at discrete time of integral multiples of the period is ergodic. For the Markov random dynamical system case, we prove that the periodic measure of a Markov semigroup is PS-ergodic if and only if the trace of the random periodic path at integral multiples of period either entirely lies on a Poincaré section or completely outside a Poincaré section almost surely. In the second part of this paper, we construct sublinear expectations from periodic measures and obtain the ergodicity of the sublinear expectations from the ergodicity of periodic measures. We give some examples including the ergodicity of the discrete time Wiener shift of Brownian motions. The latter result would have some independent interests.

Citation

Feng, C., Qu, B., & Zhao, H. (2020). A sufficient and necessary condition of PS-ergodicity of periodic measures and generated ergodic upper expectations. Nonlinearity, 33(10), https://doi.org/10.1088/1361-6544/ab9584

Journal Article Type	Article
Acceptance Date	May 21, 2020
Online Publication Date	Sep 1, 2020
Publication Date	2020-10
Deposit Date	Oct 7, 2020
Publicly Available Date	Oct 7, 2020
Journal	Nonlinearity
Print ISSN	0951-7715
Electronic ISSN	1361-6544
Publisher	IOP Publishing
Peer Reviewed	Peer Reviewed
Volume	33
Issue	10
DOI	https://doi.org/10.1088/1361-6544/ab9584

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