A Sufficient and Necessary Condition of PS-ergodicity of Periodic Measures and Generated Ergodic Upper Expectations

TL;DR

This paper establishes a comprehensive criterion for the ergodicity of periodic measures in random dynamical systems and explores the implications for sublinear expectations, with applications to Brownian motion.

Contribution

It provides a necessary and sufficient condition for PS-ergodicity of periodic measures and links ergodicity of these measures to ergodic properties of associated sublinear expectations.

Findings

Periodic measure ergodicity is characterized by the underlying noise system's ergodicity.

Trace of the periodic path determines PS-ergodicity in Markov systems.

Constructed sublinear expectations inherit ergodicity from periodic measures.

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 Poincar\'{e} section or completely outside a Poincar\'{e} 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.