Invariance of Gaussian RKHSs Under Koopman Operators of Stochastic Differential Equations With Constant Matrix Coefficients

Abstract

We consider the Koopman operator semigroup (Kt)t≥0 associated with stochastic differential equations of the form dXt=AXtdt+BdWt with constant matrices A and B and Brownian motion Wt. We prove that the reproducing kernel Hilbert space HC generated by a Gaussian kernel with a positive definite covariance matrix C is invariant under each Koopman operator Kt if the matrices A, B, and C satisfy the following Lyapunov‐like matrix inequality: AC2+C2A⊤≤2BB⊤. In this course, we prove a characterization concerning the inclusion HC1⊂HC2 of Gaussian RKHSs for two positive definite matrices C1 and C2. The question of whether the sufficient Lyapunov‐condition is also necessary is left as an open problem.