Over the last years, several kernel-based methods for the analysis of high-dimensional data sets have been developed, many of which can be seen as nonlinear extensions of classical linear methods such as PCA, CCA, or TICA. We extend transfer operator theory to reproducing kernel Hilbert spaces and show that the resulting operators are related to Hilbert space representations of conditional distributions. One main benefit of the presented kernel-based approaches is that these methods can be applied to any domain where a similarity measure given by a kernel is available. Furthermore, we derive a data-driven method for the approximation of the Koopman generator and show how, in addition to learning the governing equations of deterministic systems, it can also be used to identify the drift and diffusion terms of stochastic differential equations. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular and fluid dynamics as well as video and text data analysis.
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