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

We consider the Koopman operator semigroup $(K^t{)}_{t\ge 0}$ associated with stochastic differential equations of the form $d{X}_t=A{X}_{t}dt+Bd{W}_t$ with constant matrices $A$ and $B$ and Brownian motion $W_t$. We prove that the reproducing kernel Hilbert space ${\text{bH}}_C$ generated by a Gaussian kernel with a positive definite covariance matrix $C$ is invariant under each Koopman operator $K^t$ if the matrices $A$, $B$, and $C$ satisfy the following Lyapunov-like matrix inequality: $A{C}^2+C^2{A}^{\mathrm{\top }}\le 2B{B}^{\mathrm{\top }}$. In this course, we prove a characterization concerning the inclusion ${\text{bH}}_{C_1}\subset {\text{bH}}_{C_2}$ of Gaussian RKHSs for two positive definite matrices $C_1$ and $C_2$. The question of whether the sufficient Lyapunov-condition is also necessary is left as an open problem.