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

ABSTRACT We consider the Koopman operator semigroup associated with stochastic differential equations of the form with constant matrices and and Brownian motion . We prove that the reproducing kernel Hilbert space generated by a Gaussian kernel with a positive definite covariance matrix is invariant under each Koopman operator if the matrices , , and satisfy the following Lyapunov‐like matrix inequality: . In this course, we prove a characterization concerning the inclusion of Gaussian RKHSs for two positive definite matrices and . The question of whether the sufficient Lyapunov‐condition is also necessary is left as an open problem.