Spectral properties of the stochastic Koopman operator and its numerical approximations

A generalization of the Koopman operator framework, originally developed for de- terministic dynamical systems, to discrete and continuous time random dynamical systems results with the stochastic Koopman operators. The study of the spectral objects of these operators (Koopman eigenvalues, eigenfunctions and modes) could be useful in analysing the behaviour of the considered random dynamical system. We provide the results that characterize the spectrum and the eigenfunctions of the stochastic Koopman operators for the particular classes of random and stochastic dynamical systems. The numerical approximations of the eigenvalues and eigen- functions of the stochastic Koopman operator are computed by using DMD RRR algorithm. Its behaviour in the stochastic framework is explored on several test examples. Furthermore, we introduce the isostables and isochrones associated to the random dynamical systems and compute their numerical approximations on the chosen example.