  Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to
approximate the Koopman operator for deterministic and stochastic (control)
systems. This operator is linear and encompasses full information on the
(expected stochastic) dynamics. In this paper, we analyze a kernel-based EDMD
algorithm, known as kEDMD, where the dictionary consists of the canonical
kernel features at the data points. The latter are acquired by i.i.d. samples
from a user-defined and application-driven distribution on a compact set. We
prove bounds on the prediction error of the kEDMD estimator when sampling from
this (not necessarily ergodic) distribution. The error analysis is further
extended to control-affine systems, where the considered invariance of the
Reproducing Kernel Hilbert Space is significantly less restrictive in
comparison to invariance assumptions on an a-priori chosen dictionary.
