### Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise

Eugene Lytvynov , Irina Rodionova

Infinite Dimensional Analysis, Quantum Probability and Related Topics, Volume: 21, Issue: 02, Start page: 1850011

Swansea University Authors:

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DOI (Published version): 10.1142/S021902571850011X

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, i.e., formally $\omega(t)=\frac d{dt}X_t$. A continuous polynomial of $\omega$ is an element of the corresponding non-commutative $L^2$-spa...

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Published in: Infinite Dimensional Analysis, Quantum Probability and Related Topics 0219-0257 1793-6306 2018 https://cronfa.swan.ac.uk/Record/cronfa36409 No Tags, Be the first to tag this record!
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, i.e., formally $\omega(t)=\frac d{dt}X_t$. 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 $\omega(t)=\partial_t^\dag+\lambda\partial_t^\dag\partial_t+\partial_t+\eta\partial_t^\dag\partial_t\partial_t$. Here, $\partial_t^\dag$ and $\partial_t$ are the (formal) creation and annihilation operators at $t\in\mathbb R_+$ acting in $\mathbb F$. Monotone independence, monotone Levy noise, monotone Levy process, Meixner class of orthogonal polynomials. College of Science 02 1850011