A generalization of the Koopman operator framework, originally developed for deterministic dynamical systems, to discrete and continuous time random dynamical systems (RDS) results with the stochastic Koopman operators. We provide the results that characterize the spectrum and the eigenfunctions of the stochastic Koopman operator associated with different types of linear RDS. Then we consider the RDS for which the associated Koopman operator family is a semigroup, especially those for which the generator can be determined. We propose different approaches for using the data-driven DMD algorithms in the stochastic framework to approximate numerically the spectral objects (eigenvalues, eigenfunctions) of the stochastic Koopman operator. We prove that, under certain assumptions, the outputs of the stochastic Hankel DMD (sHankel-DMD) algorithm converge to the true stochastic Koopman eigenvalues and eigenfunctions. We apply the methodology to a variety of examples, revealing objects in spectral expansions of the stochastic Koopman operator.
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