Spectral Analysis of Fokker--Planck and Related Operators Arising From Linear Stochastic Differential Equations

We study spectral properties of certain families of linear second-order differential operators arising from linear stochastic differential equations. We construct a basis in the Hilbert space of square-integrable functions using modified Hermite polynomials, and obtain a representation for these operators from which their eigenvalues and eigenfunctions can be computed. In particular, we completely describe the spectrum of the Fokker–Planck operator on an appropriate invariant subspace of rapidly decaying functions. The eigenvalues of the Fokker–Planck operator provide information about the speed of convergence of the corresponding probability distribution to steady state, which is important for stochastic estimation and control applications. We show that the operator families under consideration can be realized as solutions of differential equations in the double bracket form on an operator Lie algebra, which leads to a simple expression for the flow of their eigenfunctions.