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

Abstract

Let $(X_t{)}_{t\ge 0}$ denote a non-commutative monotone Lévy process. Let $\omega =(\omega (t){)}_{t\ge 0}$ denote the corresponding monotone Lévy noise.. A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-space $L^2(\tau )$ that has the form ${\sum }_{i=0}^n⟨{\omega }^{\otimes i},f^{(i)}⟩$, 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 $⟨P^{(n)}(\omega ),f^{(n)}⟩$ is defined as the orthogonal projection of the monomial $⟨{\omega }^{\otimes n},f^{(n)}⟩$ 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 $⟨P^{(n)}(\omega ),f^{(n)}⟩$ depends only on the restriction of the function $f^{(n)}$ to the set $\{(t_1,\dots ,t_n)\in {\mathbb{R}}_{+}^n\mid t_1\ge t_2\ge \cdots \ge t_n\}$. 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 $\eta \ge 0$. In this case, the monotone Lévy noise has the representation $\omega (t)={\partial }_t^{†}+\lambda {\partial }_t^{†}{\partial }_t+{\partial }_t+\eta {\partial }_t^{†}{\partial }_t{\partial }_t$. Here, ${\partial }_t^{†}$ and ${\partial }_t$ are the (formal) creation and annihilation operators at $t\in {\mathbb{R}}_{+}$ acting in $\mathbb{F}$.