Random periodicity is ubiquitous in the real world. In this talk, I will provide the concepts of random periodic paths and periodic measure to mathematically describe random periodicity. These two different notions are Random periodicity is ubiquitous in the real world. In this talk, I will provide the concepts of random periodic paths and periodic measure to mathematically describe random periodicity. These two different notions are “equivalent”. An ergodic theory is established. For Markovian random dynamical systems, in the random periodic case, the infinitesimal generator of the Markovian has infinite number of equally placed simple eigenvalues including $0$ on the imaginary axis, in contrast to the mixing stationary case in which the Koopman-von Neumann Theorem says there is only one simple eigenvalue $0$ on the imaginary axis. Examples of of Markov chains, random mappings,stochastic differential equations and stochastic partial differential equations with random periodic paths or periodic measures will be provided. This theory implies law of large numbers, central limit theorems and applications to time series (touched if time permits).
