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Marek Bozejko

Publications and source records attributed to Marek Bozejko.

15 recordsLinked to original sources
arXiv2024

Deformations and q-convolutions. Old and new results

This paper is the survey of some of our results related to $q$-deformations of the Fock spaces and related to $q$-convolutions for probability measures on the real line $\mathbb{R}$. The main idea is done by the combinatorics of moments of the measures and related $q$-cumulants of different types. The main and interesting $q$-convolutions are related to classical continuous (discrete) $q$-Hermite polynomial. Among them are classical ($q=1$) convolutions, the case $q=0$, gives the free and Boolean relations, and the new class of $q$-analogue of classical convolutions done by Carnovole, Koornwinder, Biane, Anshelovich, and Kula. The paper contains many questions and problems related to the positivity of that class of $q$-convolutions. The main result is the construction of Brownian motion related to $q$-Discrete Hermite polynomial of type I.

math-ph↗
arXiv2015

An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

Let $ν$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighborhood of zero. An anyon Lévy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langleω,φ\rangle$ in the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R,dx\otimesν)$. Here $φ=φ(x)$ runs over a space of test functions on $\mathbb R^d$, while $ω=ω(x)$ is interpreted as an operator-valued distribution on $\mathbb R^d$. Let $L^2(τ)$ be the noncommutative $L^2$-space generated by the algebra of polynomials in variables $\langle ω,φ\rangle$, where $τ$ is the vacuum expectation state. We construct noncommutative orthogonal polynomials in $L^2(τ)$ of the form $\langle P_n(ω),f^{(n)}\rangle$, where $f^{(n)}$ is a test function on $(\mathbb R^d)^n$. Using these orthogonal polynomials, we derive a unitary isomorphism $U$ between $L^2(τ)$ and an extended anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathbf F(L^2(\mathbb R^d,dx))$. The usual anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathcal F(L^2(\mathbb R^d,dx))$, is a subspace of $\mathbf F(L^2(\mathbb R^d,dx))$. Furthermore, we have the equality $\mathbf F(L^2(\mathbb R^d,dx))=\mathcal F(L^2(\mathbb R^d,dx))$ if and only if the measure $ν$ is concentrated at one point, i.e., in the Gaussian/Poisson case. Using the unitary isomorphism $U$, we realize the operators $\langle ω,φ\rangle$ as a Jacobi (i.e., tridiagonal) field in $\mathbf F(L^2(\mathbb R^d,dx))$. We derive a Meixner-type class of anyon Lévy white noise for which the respective Jacobi field in $\mathbf F(L^2(\mathbb R^d,dx))$ has a relatively simple structure.

math.PR↗
arXiv2014

On free infinite divisibility for classical Meixner distributions

We prove that symmetric Meixner distributions, whose probability densities are proportional to $|Γ(t+ix)|^2$, are freely infinitely divisible for $0<t\leq\frac{1}{2}$. The case $t=\frac{1}{2}$ corresponds to the law of Lévy's stochastic area whose probability density is $\frac{1}{\cosh(πx)}$. A logistic distribution, whose probability density is proportional to $\frac{1}{\cosh^2(πx)}$, is freely infinitely divisible too.

math.PR↗
arXiv2013

Generalized Gaussian processes and relations with random matrices and positive definite functions on permutation groups

The main purpose of this paper of the paper is an explicite construction of generalized Gaussian process with function $t_b(V)=b^{H(V)}$, where $H(V)=n-h(V)$, $h(V)$ is the number of singletons in a pair-partition $V \in \st{P}_2(2n)$. This gives another proof of Theorem of A. Buchholtz \cite{Buch} that $t_b$ is positive definite function on the set of all pair-partitions. Some new combinatorial formulas are also presented. Connections with free additive convolutions probability measure on $\mathbb{R}$ are also done. Also new positive definite functions on permutations are presented and also it is proved that the function $H$ is norm (on the group $S(\infty)=\bigcup S(n)$.

math.PR↗
arXiv2011

Non-commutative Lévy processes for generalized (particularly anyon) statistics

Let $T=\mathbb R^d$. Let a function $Q:T^2\to\mathbb C$ satisfy $Q(s,t)=\bar{Q(t,s)}$ and $|Q(s,t)|=1$. A generalized statistics is described by creation operators $\partial_t^†$ and annihilation operators $\partial_t$, $t\in T$, which satisfy the $Q$-commutation relations. From the point of view of physics, the most important case of a generalized statistics is the anyon statistics, for which $Q(s,t)$ is equal to $q$ if $s t$. Here $q\in\mathbb C$, $|q|=1$. We start the paper with a detailed discussion of a $Q$-Fock space and operators $(\partial_t^†,\partial_t)_{t\in T}$ in it, which satisfy the $Q$-commutation relations. Next, we consider a noncommutative stochastic process (white noise) $ω(t)=\partial_t^†+\partial_t+λ\partial_t^†\partial_t$, $t\in T$. Here $λ\in\mathbb R$ is a fixed parameter. The case $λ=0$ corresponds to a $Q$-analog of Brownian motion, while $λ\ne0$ corresponds to a (centered) $Q$-Poisson process. We study $Q$-Hermite ($Q$-Charlier respectively) polynomials of infinitely many noncommutatative variables $(ω(t))_{t\in T}$. The main aim of the paper is to explain the notion of independence for a generalized statistics, and to derive corresponding Lévy processes. To this end, we recursively define $Q$-cumulants of a field $(ξ(t))_{t\in T}$. This allows us to define a $Q$-Lévy process as a field $(ξ(t))_{t\in T}$ whose values at different points of $T$ are $Q$-independent and which possesses a stationarity of increments (in a certain sense). We present an explicit construction of a $Q$-Lévy process, and derive a Nualart-Schoutens-type chaotic decomposition for such a process.

math.PR↗
arXiv2010

The normal distribution is $\boxplus$-infinitely divisible

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a subfamily Askey-Wimp-Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution.

math.OA↗
arXiv2009

Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization

Let $T$ be an underlying space with a non-atomic measure $σ$ on it (e.g. $T=\mathbb R^d$ and $σ$ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of $T$, with freely independent values. Such a process (field), $ω=ω(t)$, $t\in T$, is given a rigorous meaning through smearing out with test functions on $T$, with $\int_T σ(dt)f(t)ω(t)$ being a (bounded) linear operator in a full Fock space. We define a set $\mathbf{CP}$ of all continuous polynomials of $ω$, and then define a con-commutative $L^2$-space $L^2(τ)$ by taking the closure of $\mathbf{CP}$ in the norm $\|P\|_{L^2(τ)}:=\|PΩ\|$, where $Ω$ is the vacuum in the Fock space. Through procedure of orthogonalization of polynomials, we construct a unitary isomorphism between $L^2(τ)$ and a (Fock-space-type) Hilbert space $\mathbb F=\mathbb R\oplus\bigoplus_{n=1}^\infty L^2(T^n,γ_n)$, with explicitly given measures $γ_n$. We identify the Meixner class as those processes for which the procedure of orthogonalization leaves the set $\mathbf {CP}$ invariant. (Note that, in the general case, the projection of a continuous monomial of oder $n$ onto the $n$-th chaos need not remain a continuous polynomial.) Each element of the Meixner class is characterized by two continuous functions $λ$ and $η\ge0$ on $T$, such that, in the $\mathbb F$ space, $ω$ has representation $ω(t)=\di_t^†+λ(t)\di_t^†\di_t+\di_t+η(t)\di_t^†\di^2_t$, where $\di_t^†$ and $\di_t$ are the usual creation and annihilation operators at point $t$.

math.PR↗
arXiv2006

On a class of free Levy laws related to a regression problem

The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum. We apply this result to describe free Levy processes with quadratic conditional variances, and to prove a converse implication related to asymptotic freeness of random Wishart matrices.

math.OA↗
arXiv1996

$q$-Gaussian processes: non-commutative and classical aspects

We examine, for $-1<q<1$, $q$-Gaussian processes, i.e. families of operators (non-commutative random variables) $X_t=a_t+a_t^*$ -- where the $a_t$ fulfill the $q$-commutation relations $a_sa_t^*-qa_t^*a_s=c(s,t)\cdot \id$ for some covariance function $c(\cdot,\cdot)$ -- equipped with the vacuum expectation state. We show that there is a $q$-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on $q$-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of $q$-Gaussian processes possess a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret \cite{FB}.

funct-an↗
arXiv1994

Convolution and Limit Theorems for Conditionally Free Random Variables

We introduce the notion of a conditionally free product and conditionally free convolution. We describe this convolution both from a combinatorial point of view, by showing its connection with the lattice of non-crossing partitions, and from an analytic point of view, by presenting the basic formula for its $R$-transform. We calculate explicitly the distributions of the conditionally free Gaussian and conditionally free Poisson distribution.

funct-an↗
arXiv1994

Interpolations between Bosonic and Fermionic Relations given by Generalized Brownian Motions (revised version)

We present an interpolation between the bosonic and fermionic relations. This interpolation is given by an object which we call `generalized Brownian motion' and which is characterized by a generalization of the pairing rule for the calculation of the moments of bosonic and fermionic fields. We develop some basic theory for such generalized Brownian motions and consider more closely one example, which turns out to be intimately connected with Voiculescu's concept of `free product'.

funct-an↗
arXiv1994

Completely Positive Maps on Coxeter Groups, Deformed Commutation Relations, and Operator Spaces

In this article we prove that quasi-multiplicative (with respect to the usual length function) mappings on the permutation group $\SSn$ (or, more generally, on arbitrary amenable Coxeter groups), determined by self-adjoint contractions fulfilling the braid or Yang-Baxter relations, are completely positive. We point out the connection of this result with the construction of a Fock representation of the deformed commutation relations $d_id_j^*-\sum_{r,s} t_{js}^{ir} d_r^*d_s=δ_{ij}\id$, where the matrix $t_{js}^{ir}$ is given by a self-adjoint contraction fulfilling the braid relation. Such deformed commutation relations give examples for operator spaces as considered by Effros, Ruan and Pisier. The corresponding von Neumann algebras, generated by $G_i=d_i+d_i^*$, are typically not injective.

funct-an↗