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

Feng, Chunrong and Qu, Baoyou and Zhao, Huaizhong (2020) 'A sufficient and necessary condition of PS-ergodicity of periodic measures and generated ergodic upper expectations.', Nonlinearity., 33 (10). p. 5324.


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.

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Publisher statement:Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Date accepted:21 May 2020
Date deposited:07 October 2020
Date of first online publication:01 September 2020
Date first made open access:07 October 2020

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