Invariance of Gaussian RKHSs under Koopman operators of stochastic differential equations with constant matrix coefficients

We consider the Koopman operator semigroup (Kᵗ)t≥ 0 associated with stochastic differential equations of the form dXₜ = AXₜ\,dt + B\,dWₜ with constant matrices A and B and Brownian motion Wₜ. We prove that the reproducing kernel Hilbert space C 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 C₁⊂C₂ 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.