Error bounds for kernel-based approximations of the Koopman operator Philipp, Friedrich Schaller, Manuel Worthmann, Karl Peitz, Sebastian Nüske, Feliks 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. Springer 2024 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_6501 https://ris.uni-paderborn.de/record/38031 Philipp F, Schaller M, Worthmann K, Peitz S, Nüske F. Error bounds for kernel-based approximations of the Koopman operator. <i>Applied and Computational Harmonic Analysis </i>. 2024;71. doi:<a href="https://doi.org/10.1016/j.acha.2024.101657">10.1016/j.acha.2024.101657</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1016/j.acha.2024.101657 info:eu-repo/semantics/altIdentifier/arxiv/2301.08637 info:eu-repo/semantics/openAccess