We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.
@article{arxiv.2301.08637,
title = {Error bounds for kernel-based approximations of the Koopman operator},
author = {Friedrich Philipp and Manuel Schaller and Karl Worthmann and Sebastian Peitz and Feliks Nüske},
journal= {arXiv preprint arXiv:2301.08637},
year = {2023}
}