Pseudogenerators of Spatial Transfer Operators

Metastable behavior in dynamical systems may be a significant challenge for a simulation-based analysis. In recent years, transfer operator--based approaches to problems exhibiting metastability have matured. In order to make these approaches computationally feasible for larger systems, various reduction techniques have been proposed: For example, Schütte introduced a spatial transfer operator which acts on densities on configuration space, while Weber proposed to avoid trajectory simulation (like Froyland, Junge, and Koltai) by considering a discrete generator. In this paper, we show that even though the family of spatial transfer operators is not a semigroup, it possesses a well-defined generating structure. What is more, the pseudogenerators up to order 4 in the Taylor expansion of this family have particularly simple explicit expressions involving no momentum averaging. This makes collocation methods particularly easy to implement and computationally efficient, which in turn may open the door for further efficiency improvements in, e.g., the computational treatment of conformation dynamics. We experimentally verify the predicted properties of these pseudogenerators by means of two academic examples.