SDEs and Transfer Operators

PhD project lies at the intersection of stochastic differential equations and transfer operators (e.g. Koopman and Perron-Frobenius operators). Since this interdisciplinary topic has not been previously thoroughly explored, there are many open research questions and directions we are keen on investigating. The research question we are currently working towards is applying the standard EDMD algorithm to mean field stochastic differential equations, with the goal of proving the accuracy of these algorithms both theoretically and computationally. Moreover, after these results are achieved, we will illustrate a practical application of this algorithm on a real-life (science) model. Some other directions we are interested in exploring (after sufficiently exhausting the above research question), is how the eigenvalues, eigenvectors of transfer operators change when an SDE becomes non-reversible. Moreover, we'd like to investigate other algorithms which are used to find eigenvalues and eigenfunctions of Koopman operators such as kernel-based methods (via the use of reproducing kernel Hilbert spaces).