Koopman Operator Spectrum for Random Dynamical Systems
Abstract
In this paper we consider the Koopman operator associated with the discrete and the continuous time random dynamical system (RDS). We provide 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 define a stochastic Hankel DMD (sHankel-DMD) algorithm for numerical approximations of the spectral objects (eigenvalues, eigenfunctions) of the stochastic Koopman operator and prove its convergence. We apply the methodology to a variety of examples, revealing objects in spectral expansions of the stochastic Koopman operator and enabling model reduction.
Keywords
Cite
@article{arxiv.1711.03146,
title = {Koopman Operator Spectrum for Random Dynamical Systems},
author = {Nelida Črnjarić-Žic and Senka Maćešić and Igor Mezić},
journal= {arXiv preprint arXiv:1711.03146},
year = {2019}
}