Random Periodic Processes, Periodic Measures and Ergodicity

Abstract

Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincaré sections is introduced. It is proved that if the periodic measure is PS-ergodic, then it is ergodic. Moreover, if the infinitesimal generator of the Markov semigroup only has equally placed simple eigenvalues including $0$ on the imaginary axis, then the periodic measure is PS-ergodic and has positive minimum period. Conversely if the periodic measure with the positive minimum period is PS-mixing, then the infinitesimal generator only has equally placed simple eigenvalues (infinitely many) including $0$ on the imaginary axis. Moreover, under the spectral gap condition, PS-mixing of the periodic measure is proved. The ``equivalence" of random periodic processes and periodic measures is established. This is a new class of ergodic random processes. Random periodic paths of stochastic perturbation of the periodic motion of an ODE is obtained.

38 pages

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Cited by in corpus (14)

• Numerical Approximation of Random Periodic Solutions of Stochastic Differential Equations
• Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations
• Ergodicity of Invariant Capacity
• Expected Exit Time for Time-Periodic Stochastic Differential Equations and Applications to Stochastic Resonance
• Ergodic Numerical Approximation to Periodic Measures of Stochastic Differential Equations
• Ergodicity of Sublinear Markovian Semigroups
• Random Almost Periodic Solutions of Random Dynamical Systems
• Anticipating Random Periodic Solutions--II. SPDEs with Multiplicative Linear Noise
• A Sufficient and Necessary Condition of PS-ergodicity of Periodic Measures and Generated Ergodic Upper Expectations
• Random periodic solutions of non-autonomous stochastic differential equations
• A random dynamical systems perspective on stochastic resonance
• Weak Random Periodic Solutions of Random Dynamical Systems
• The Galerkin analysis for the random periodic solution of semilinear stochastic evolution equations
• Averaging principle for stochastic differential equations in the random periodic regime