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Koopman Operator Spectrum for Random Dynamical Systems

In this paper we propose a generalized Koopman operator framework for discrete and continuous time random dynamical systems. For the particular classes of ran- dom dynamical systems, we provide the results that characterize the spectrum and the eigenfunctions of the stochastic Koopman operator. We discuss the relationship between the spectral properties of the generator of the evolution and the Koopman operator family. The numerical approximations of the spectral objects (eigenval- ues, eigenfunctions) of the stochastic Koopman operator are computed by using the state of the art DMD RRR algorithm. We explore its behavior in the stochastic case on several test examples. Moreover, the DMD RRR algorithm is applied in combination with the Hankel matrix and a convergence theorem for Hankel DMD RRR in the stochastic case is proved.

stochastic Koopman operator, random dynamical systems, stochastic differential equations, dynamic mode decomposition

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