On the eigenfunctions of the complex Ornstein–Uhlenbeck operators

Abstract

Starting from the $1$-dimensional complex-valued Ornstein–Uhlenbeck process, we present two natural ways to obtain the associated eigenfunctions of the $2$-dimensional normal Ornstein–Uhlenbeck operator in the complex Hilbert space $L_{\mathbb{C}}^2\left(\mu \right)$. We call the eigenfunctions Hermite–Laguerre–Itô polynomials. In addition, the Mehler summation formula for the complex process is shown.