Meixner class of orthogonal polynomials of a non-commutative monotone Levy noise

TL;DR

This paper characterizes the Meixner class of orthogonal polynomials associated with non-commutative monotone Lévy noise, revealing their structure and parameterization in terms of two real parameters and their operator representation.

Contribution

It introduces the Meixner class of orthogonal polynomials in the non-commutative monotone Lévy process setting and characterizes them via two parameters, providing a new operator representation.

Findings

Orthogonal polynomials form a basis in the non-commutative $L^2$-space.

The Meixner class is characterized by two parameters, $ ext{lambda}$ and $ ext{eta}$.

Monotone Lévy noise admits a specific operator representation involving creation and annihilation operators.

Abstract

Let $(X_t)_{t\ge0}$ denote a non-commutative monotone L\'evy process. Let $\omega=(\omega(t))_{t\ge0}$ denote the corresponding monotone L\'evy noise.. A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-space $L^2(\tau)$ that has the form $\sum_{i=0}^n\langle \omega^{\otimes i},f^{(i)}\rangle$, where $f^{(i)}\in C_0(\mathbb R_+^i)$. We denote by $\mathbf{CP}$ the space of all continuous polynomials of $\omega$. For $f^{(n)}\in C_0(\mathbb R_+^n)$, the orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ is defined as the orthogonal projection of the monomial $\langle\omega^{\otimes n},f^{(n)}\rangle$ onto the subspace of $L^2(\tau)$ that is orthogonal to all continuous polynomials of $\omega$ of order $\le n-1$. We denote by $\mathbf{OCP}$ the linear span of the orthogonal polynomials. Each orthogonal polynomial $\langle P^{(n)}(\omega),f^{(n)}\rangle$ depends only on the restriction of the function $f^{(n)}$ to the set $\{(t_1,\dots,t_n)\in\mathbb R_+^n\mid t_1\ge t_2\ge\dots\ge t_n\}$. The orthogonal polynomials allow us to construct a unitary operator $J:L^2(\tau)\to\mathbb F$, where $\mathbb F$ is an extended monotone Fock space. Thus, we may think of the monotone noise $\omega$ as a distribution of linear operators acting in $\mathbb F$. We say that the orthogonal polynomials belong to the Meixner class if $\mathbf{CP}=\mathbf{OCP}$. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: $\lambda\in\mathbb R$ and $\eta\ge0$. In this case, the monotone L\'evy noise has the representation $\omega(t)=\partial_t^\dag+\lambda\partial_t^\dag\partial_t+\partial_t+\eta\partial_t^\dag\partial_t\partial_t$. Here, $\partial_t^\dag$ and $\partial_t$ are the (formal) creation and annihilation operators at $t\in\mathbb R_+$ acting in $\mathbb F$.