Random periodic processes, periodic measures and ergodicity
By Chunrong Feng, Huaizhong Zhao. 2014
65 citations recorded by the source corpus.
Ergodicity of random dynamical systems with a periodic measure is obtained on a Polish space. In the Markovian case, the idea of Poincar\'e 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.
Subfields: Control and Systems Engineering, Finance, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/bb0dda0b7f523401eadbe188e89d0af0a7aa1777
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Citations in the corpus
- Anticipating Random Periodic Solutions--I. SDEs with Multiplicative Linear Noise (2015, background): “t+ s;!) + z(t+ s) = Y(t+ s;!); and the random periodicity of Y is obvious. The last claim follows from the existence of a random periodic solution and the result on periodic and invariant measures in [15]. In the above theorem, the main assumption is that the deterministic system (when noise is swit”