Orateur
Prof.
Huaizhong Zhao
(Loughborough University)
Description
An ergodic theorem and a mean ergodic theorem in the random periodic regime on a Polish space is proved.
The idea of Poincaré sections is introduced and under the strong Feller and irreducible assumptions on Poincaré
sections, the weak convergence of the transition probabilities at the discrete time of integral multiples of the period is
obtained. Thus the Khas'minskii-Doob type theorem is established and the ergodicity of the invariant measure, which
is the mean of the periodic measure over a period interval, is obtained. The Krylov-Bogoliubov type theorem for the
existence of periodic measures by considering the Markovian semigroup on a Poincaré section at discrete times of
integral multiples of the period is also proved. It is proved that three equivalent criteria give necessary and sufficient
conditions to classify between random periodic and stationary regimes. The three equivalent criteria are given in terms
of three different notion respectively, namely Poincaré sections, angle variable and infinitesimal generator of the induced
linear transformation of the canonical dynamical system associated with the invariant measure. It is proved that infinitesimal
generator has only two simple eigenvalues, which are $0$ and the quotient of $2\pi$ by the minimal period, while the
classical Koopman-von Neumann theorem says that the generator has only one simple eigenvalue $0$ in the stationary
and mixing case. The ``equivalence" of random periodic processes and periodic measures is established.
The strong law of large numbers (SLLN) is also proved for random periodic processes.
This is a joint work with Chunrong Feng.
Auteur principal
Prof.
Huaizhong Zhao
(Loughborough University)
