Free Infinite Divisibility for Q-Gaussians

Free Brownian motion and evolution towards boxplus-infinite divisibility for k-tuples

Let D be the space of non-commutative distributions of k-tuples of selfadjoints in a C*-probability space (for a fixed k). We introduce a semigroup of transformations B_t of D, such that every distribution in D evolves under the B_t towards infinite divisibility with respect to free additive convolution. The very good properties of B_t come from some special connections that we put into evidence between free additive convolution and the operation of Boolean convolution. On the other hand we put into evidence a relation between the transformations B_t and free Brownian motion. More precisely, w…

An Analogue of Hinucin's Characterization of Infinite Divisibility for Operator-Valued Free Probability

Let $B$ be a finite, separable von Neumann algebra. We prove that a $B$-valued distribution $μ$ that is the weak limit of an infinitesimal array is infinitely divisible. The proof of this theorem utilizes the Steinitz lemma and may be adapted to provide a nonstandard proof of this type of theorem for various other probabilistic categories. We also develop weak topologies for this theory and prove the corresponding compactness and convergence results.

Free infinite divisibility for powers of random variables

We prove that $X^r$ follows an FID distribution if: (1) $X$ follows a free Poisson distribution without an atom at 0 and $r\in(-\infty,0]\cup[1,\infty)$; (2) $X$ follows a free Poisson distribution with an atom at 0 and $r\geq1$; (3) $X$ follows a mixture of some HCM distributions and $|r|\geq1$; (4) $X$ follows some beta distributions and $r$ is taken from some interval. In particular, if $S$ is a standard semicircular element then $|S|^r$ is freely infinitely divisible for $r\in(-\infty,0]\cup[2,\infty)$. Also we consider the symmetrization of the above probability measures, and in particula…

Eta-diagonal distributions and infinite divisibility for R-diagonals

The class of R-diagonal *-distributions is fairly well understood in free probability. In this class, we consider the concept of infinite divisibility with respect to the operation $\boxplus$ of free additive convolution. We exploit the relation between free probability and the parallel (and simpler) world of Boolean probability. It is natural to introduce the concept of an eta-diagonal distribution that is the Boolean counterpart of an R-diagonal distribution. We establish a number of properties of eta-diagonal distributions, then we examine the canonical bijection relating eta-diagonal distr…

保持自由无限可分性的二次多项式

Quadratic polynomials preserving free infinite divisibility

该研究将保持自由无限可分性的二次非交换多项式刻画为二次自伴李多项式，扩展了齐二次型的相关结果，通过彩色非交叉划分对合与福克空间算子模型完成证明，且发现高阶李多项式不具备该性质。

自由无限可分性、分数卷积幂与Appell多项式

Free infinite divisibility, fractional convolution powers, and Appell polynomials

研究通过有限自由R变换，将多项式多次求导的极限与自由无限可分分布联系起来，揭示了Appell序列在有限自由概率中的作用。

Semigroups related to additive and multiplicative, free and Boolean convolutions

Belinschi and Nica introduced a composition semigroup on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know whether a probability measure is freely infinitely divisible or not. In this paper we further investigate this indicator, introduce a multiplicative version of it and are able to show many properties. Specifically, on the first half of the paper, we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to …

On monotone convolution and monotone infinite divisivility

This article focuses on properties of monotone convolutions. A criterion for infinite divisibility and time evolution of convolution semigroups are mainly studied. In particular, we clarify that many analogues of the classical results of Lévy processes hold such as characterizations of subordinators and strictly stable distributions.

Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation

In a previous paper (called "Rectangular random matrices. Related covolution"), we defined, for $λ\in [0,1]$, the rectangular free convolution with ratio $λ$. Here, we investigate the related notion of infinite divisiblity, which happens to be closely related to the classical infinite divisibility: there exists a bijection between the set of classical symmetric infinitely divisible distributions and the set of distributions infinitely divisible with respect to this convolution, which preserves limit theorems. We give an interpretation of this correspondance in term of random matrices: we const…

On the Law of Free Subordinators

We study the freely infinitely divisible distributions that appear as the laws of free subordinators. This is the free analog of classically infinitely divisible distributions supported on [0,\infty), called the free regular measures. We prove that the class of free regular measures is closed under the free multiplicative convolution, t-th boolean power for $0\leq t\leq 1$, t-th free multiplicative power for $t\geq 1$ and weak convergence. In addition, we show that a symmetric distribution is freely infinitely divisible if and only if its square can be represented as the free multiplicative co…

Non-Commutative Functions and Non-Commutative Free Levy-Hincin Formula

The paper is discussing infinite divisibility in the setting of operator-valued boolean, free and, more general, c-free independences. Particularly, using Hilbert bimodules and non-commutative functions techniques, we obtain analogues of the Levy-Hincin integral representation for infinitely divisible real measures.

Two-faced Families of Non-commutative Random Variables Having Bi-free Infinitely Divisible Distributions

We study two-faced families of random variables having bi-free infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced pairs of random variables within a triangular array. Then, by using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free …

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 fr…

Infinite Mode Quantum Gaussian States

Quantum Gaussian states on Bosonic Fock spaces are quantum versions of Gaussian distributions. In this paper, we explore infinite mode quantum Gaussian states. We extend many of the results of Parthasarathy in \cite{Par10} and \cite{Par13} to the infinite mode case, which includes various characterizations, convexity and symmetry properties.