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Koopman Operator Spectrum for Random Dynamical Systems

  • Nelida Črnjarić-ŽicEmail author
  • Senka Maćešić
  • Igor Mezić
Article
  • 11 Downloads

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 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

Stochastic Koopman operator Random dynamical systems Stochastic differential equations Dynamic mode decomposition 

Mathematics Subject Classification

37H10 47B33 37M99 65P99 

Notes

Acknowledgements

This research has been supported by the DARPA Contract HR0011-16-C-0116 and HR0011-18-9-0033, AFOSR Grants FA9550-08-1-0217 and FA9550-17-C-0012, and support of scientific research of the University of Rijeka, Project No. 18-118-1257. We are thankful to Milan Korda, Allan Avila and anonymous referees for useful comments that helped to substantially improve the manuscript.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Nelida Črnjarić-Žic
    • 1
    Email author
  • Senka Maćešić
    • 1
  • Igor Mezić
    • 2
  1. 1.Faculty of EngineeringUniversity of RijekaRijekaCroatia
  2. 2.Faculty of Mechanical Engineering and MathematicsUniversity of California, Santa BarbaraSanta BarbaraUSA

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