On the eigenfunctions of the complex Ornstein-Uhlenbeck operators

TL;DR

This paper explores the eigenfunctions of complex Ornstein-Uhlenbeck operators, introducing Hermite-Laguerre-Ito polynomials and deriving the Mehler summation formula for the complex process.

Contribution

It presents two methods to derive eigenfunctions in complex Hilbert spaces and introduces Hermite-Laguerre-Ito polynomials as eigenfunctions.

Findings

Eigenfunctions are characterized as Hermite-Laguerre-Ito polynomials.

Mehler summation formula for the complex Ornstein-Uhlenbeck process is established.

Two natural methods for deriving eigenfunctions are proposed.

Abstract

Starting from the 1-dimensional complex-valued Ornstein-Uhlenbeck process, we present two natural ways to imply the associated eigenfunctions of the 2-dimensional normal Ornstein-Uhlenbeck operators in the complex Hilbert space $L_{\Cnum}^2(\mu)$. We call the eigenfunctions Hermite-Laguerre-Ito polynomials. In addition, the Mehler summation formula for the complex process are shown.