Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation

We extend the unified kernel framework for transport equations and Koopman eigenfunctions, developed in previous work by the authors for deterministic systems, to stochastic differential equations (SDEs). In the deterministic setting, three analytically grounded constructions-Lions-type variational principles, Green's function convolution, and resolvent operators along characteristic flows--were shown to yield identical reproducing kernels. For stochastic systems, the Koopman generator includes a second-order diffusion term, transforming the first-order hyperbolic transport equation into a second-order elliptic-parabolic PDE. This fundamental change necessitates replacing the method of characteristics with probabilistic representations based on the Feynman--Kac formula. Our main contributions include: (i) extension of all three kernel constructions to stochastic systems via Feynman--Kac path-integral representations; (ii) proof of kernel equivalence under uniform ellipticity assumptions; (iii) a collocation-based computational framework incorporating second-order differential operators; (iv) error bounds separating RKHS approximation error from Monte Carlo sampling error; (v) analysis of how diffusion affects numerical conditioning; and (vi) connections to generator EDMD, diffusion maps, and kernel analog forecasting. Numerical experiments on Ornstein--Uhlenbeck processes, nonlinear SDEs with varying diffusion strength, and multi-dimensional systems validate the theoretical developments and demonstrate that moderate diffusion can improve numerical stability through elliptic regularization.