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Weak random periodic solutions of random dynamical systems

Part of: Random dynamical systems Measure-theoretic ergodic theory Stochastic processes

Published online by Cambridge University Press:  12 July 2023

Wei Sun  and
Zuo-Huan Zheng
Wei Sun*
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montreal, QC H3G 1M8, Canada
Zuo-Huan Zheng
Affiliation:
Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China e-mail: zhzheng@amt.ac.cn
*
e-mail: wei.sun@concordia.ca
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Abstract

We first introduce the concept of weak random periodic solutions of random dynamical systems. Then, we discuss the existence of such periodic solutions. Further, we introduce the definition of weak random periodic measures and study their relationship with weak random periodic solutions. In particular, we establish the existence of invariant measures of random dynamical systems by virtue of their weak random periodic solutions. We use concrete examples to illustrate the weak random periodic phenomena of dynamical systems induced by random and stochastic differential equations.

Keywords

Random dynamical systemstochastic differential equationweak random periodic solutionweak random periodic measureinvariant measure

MSC classification

Primary: 37H05: Foundations, general theory of cocycles, algebraic ergodic theory
Secondary: 28D10: One-parameter continuous families of measure-preserving transformations 60G17: Sample path properties
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 1 , March 2024 , pp. 137 - 151
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

W. Sun acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada (Grant No. 4394-2018). Z.-H. Zheng acknowledges financial supports of the NSF of China (Grant Nos. 12090014, 12031020, and 11671382), CAS Key Project of Frontier Sciences (Grant No. QYZDJ-SSW-JSC003), the Key Lab of Random Complex Structures and Data Sciences, CAS, and the National Center for Mathematics and Interdisciplinary Sciences, CAS.

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