Koopman Operator Spectrum for Random Dynamical Systems

TL;DR

This paper analyzes the spectral properties of the Koopman operator in random dynamical systems, introduces a numerical method for spectral approximation, and demonstrates its effectiveness through various examples.

Contribution

It characterizes the spectrum and eigenfunctions of the stochastic Koopman operator and develops a convergent numerical algorithm for spectral analysis.

Findings

Spectral objects can be effectively approximated in stochastic systems.

The stochastic Hankel DMD algorithm converges reliably.

Application to examples reveals spectral structures and aids model reduction.

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.