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.

💡 Research Summary

The paper extends a previously developed unified kernel framework for deterministic dynamical systems to stochastic differential equations (SDEs). In deterministic settings, three analytically grounded constructions—Lions‑type variational principles, Green’s function convolution, and resolvent operators along characteristic flows—produce identical reproducing kernels that can be used to approximate Koopman eigenfunctions. When stochastic forcing is introduced, the Koopman generator acquires a second‑order diffusion term, turning the first‑order hyperbolic transport equation into a second‑order elliptic‑parabolic PDE. This change eliminates the applicability of the method of characteristics and requires probabilistic representations via the Feynman‑Kac formula.

The authors’ contributions are as follows: (i) they generalize all three kernel constructions to stochastic systems by replacing characteristic‑based formulas with Feynman‑Kac path‑integral representations; (ii) under uniform ellipticity and mild regularity assumptions they prove a Kernel Equivalence Theorem showing that the variational kernel, the Green‑function‑based kernel, and the stochastic resolvent kernel are mathematically identical; (iii) they develop a collocation method for the second‑order Koopman PDE, deriving explicit formulas for kernel matrices that involve second derivatives of the kernel (e.g., diffusion matrix entries D_{ij}=½ tr