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This paper investigates additive processes with respect to several different independences in non-commutative probability in terms of the convolution hemigroups of the distributions of the increments of the processes. In particular, we focus on the relation of monotone convolution hemigroups and chordal Loewner chains, a special kind of family of conformal mappings. Generalizing the celebrated Loewner differential equation, we introduce the concept of ``generator'' for a class of Loewner chains. The locally uniform convergence of Loewner chains is then equivalent to a suitable convergence of generators. Using generators, we define homeomorphisms between the aforementioned class of chordal Loewner chains, the set of monotone convolution hemigroups with finite second moment, and the set of classical convolution hemigroups on the real line with finite second moment. Moreover, we define similar homeomorphisms between classical, free, and boolean convolution hemigroups on the real line, but without any assumptions on the moments., Comment: 67 pages

Motivated by recent progress of structural optimization problems, the paper presents a new method for constructing the distance function to the boundary of given sets of interest, which simplifies the optimization procedure. We extend the celebrated Varadhan's elliptic equation theory in 1967 by adding the source term to the equation, in which we encode the information about the set. We will also establish the rate of convergence in this new framework, which is sharp at least in the one-dimensional case.

Free probability of type B was invented by Biane--Goodman--Nica, and then it was generalized by Belinschi--Shlyakhtenko and F\'evrier--Nica to infinitesimal free probability. The latter found its applications to eigenvalues of perturbed random matrices in the work of Shlyakhtenko and C\'ebron--Dahlqvist--Gabriel. This paper offers a new framework, called ``free probability of type B${}^\prime$'', which appears in the large size limit of independent unitarily invariant random matrices with perturbations. Our framework is related to boolean, free, (anti)monotone, cyclic-(anti)monotone and conditionally free independences. We then apply the new framework to the principal minor of unitarily invariant random matrices, which leads to the definition of a multivariate inverse Markov--Krein transform., Comment: 24 pages; Revision from reviewer comments

This paper calculates the fluctuations of eigenvalues of polynomials on large Haar unitaries cut by finite rank deterministic matrices. When the eigenvalues are all simple, we can give a complete algorithm for computing the fluctuations. When multiple eigenvalues are involved, we present several examples suggesting that a general algorithm would be much more complex., Comment: 18 pages, 6 figures

Belinschi et al. [Adv. Math., 226 (2011), 3677--3698] proved that the normal distribution is freely infinitely divisible. This paper establishes a certain monotonicity, real analyticity and asymptotic behavior of the density of the free L\'{e}vy measure. The monotonicity property strengthens the result in Hasebe et al. [Int. Math. Res. Not. (2019), 1758--1787] that the normal distribution is freely selfdecomposable., Comment: 18 pages, 6 figures

In this paper, uniform convergence on compacta is studied on the space of functions taking values in the set of finite Borel measures. Related limit theorems, including L\'evy's continuity theorem and functional limit theorems for (classical and non-commutative) additive processes, are also described., Comment: (v1) 20 pages. (v2) 11 pages. The title has been changed

The main purpose of this paper is to study time-inhomogeneous one-dimensional branching processes (mainly on a continuous but also on a discrete state space) with the help of recent achievements in Loewner Theory dealing with evolution families of holomorphic self-maps in simply connected domains of the complex plane. Under a suitable stochastic continuity condition, we show that the families of the Laplace exponents of branching processes on $[0,\infty]$ can be characterized as topological (i.e. depending continuously on the time parameters) reverse evolution families whose elements are Bernstein functions. For the case of a stronger regularity w.r.t. time, we establish a Loewner - Kufarev type ODE for the Laplace exponents and characterize branching processes with finite mean in terms of the vector field driving this ODE. Similar results are obtained for families of probability generating functions for branching processes on the discrete state space $\{0,1,2,\ldots\}\cup\{\infty\}$. In addition, we find necessary and sufficient conditions for "spatial" embeddability of such branching processes into branching processes on $[0,\infty]$. Finally, we give some probabilistic interpretations of the Denjoy - Wolff point at $0$ and at $\infty$., Comment: Reference [16] has been updated

This paper introduces the class of selfdecomposable distributions concerning Boolean convolution. A general regularity property of Boolean selfdecomposable distributions is established; in particular the number of atoms is at most two and the singular continuous part is zero. We then analyze how shifting probability measures changes Boolean selfdecomposability. Several examples are presented to supplement the above results. Finally, we prove that the standard normal distribution $N(0,1)$ is Boolean selfdecomposable but the shifted one $N(m,1)$ is not for sufficiently large $|m|$., Comment: 20 pages, 3 figures

One-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous Markov processes. A suitable analogue of one-parameter semigroups in the inhomogeneous setting is the notion of a (reverse) evolution family. In this paper we study evolution families formed by Bernstein functions, which play the role of Laplace exponents for inhomogeneous continuous-state branching processes. In particular, we characterize all Herglotz vector fields that generate such evolution families and give a complex-analytic proof of a qualitative description equivalent to Silverstein's representation formula for the infinitesimal generators of one-parameter semigroups of Bernstein functions. We also establish several sufficient conditions for families of holomorphic self-maps, satisfying the algebraic part in the definition of an evolution family, to be absolutely continuous and hence to be described as solutions to the generalized Loewner - Kufarev differential equation. Most of these results are then applied in the sequel paper [https://doi.org/10.48550/arXiv.2211.12442] to study continuous-state branching processes., Comment: A reference to the sequel preprint "Loewner Theory for Bernstein functions II" is added in the abstract. Ver.2: References [19] and [32] have been updated and a small error on page 22 (the very end of the proof of Theorem 3) has been corrected Ver.3: the grant info has been updated

The present paper introduces a modified version of cyclic-monotone independence which originally arose in the context of random matrices, and also introduces its natural analogy called cyclic-Boolean independence. We investigate formulas for convolutions, limit theorems for sums of independent random variables, and also classify infinitely divisible distributions with respect to cyclic-Boolean convolution. Finally, we provide applications to the eigenvalues of the adjacency matrices of iterated star products of graphs and also iterated comb products of graphs., Comment: 34 pages; EU-logo added, no other changes

We attack the classification problem of multi-faced independences, the first non-trivial example being Voiculescu's bi-freeness. While the present paper does not achieve a complete classification, it formalizes the idea of lifting an operator on a pre-Hilbert space in a "universal" way to a larger product space, which is key for the construction of (old and new) examples. It will be shown how universal lifts can be used to construct very well-behaved (multi-faced) independences in general. Furthermore, we entirely classify universal lifts to the tensor product and to the free product of pre-Hilbert spaces. Our work brings to light surprising new examples of 2-faced independences. Most noteworthy, for many known 2-faced independences, we find that they admit continuous deformations within the class of 2-faced independences, showing in particular that, in contrast with the single faced case, this class is infinite (and even uncountable)., Comment: This update corrects a technical error in the latex source file, which caused several passages not to display in the pdf

We prove a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy; for example, this class includes GUE (Gaussian Unitary Ensemble) and Wishart matrices. More precisely, if the EED of the whole matrix converges to some deterministic probability measure $\mathfrak{m}$, then its fluctuation from the EED of the principal submatrix, after a rescaling, concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with $\mathfrak{m}$ by the Markov--Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of $\mathfrak{m}$. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions., Comment: 15 pages

Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems of free probability, we not only show the multiplicativity violation of maximum output norms in the asymptotic regimes but also prove the additivity violation via Haagerup inequality for a new class of random quantum channels constructed by rectifying the above completely positive maps based on strong convergence., Comment: 24 pages. Appendix D was added and minor corrections and changes were made

We introduce new homomorphisms relative to additive convolutions and max-convolutions in free, boolean and classical cases. Crucial roles are played by the limit distributions for free multiplicative law of large numbers., Comment: 15 pages

This paper defines the notion of generators for a class of decreasing radial Loewner chains which are only continuous with respect to time. For this purpose, "Loewner's integral equation" which generalizes Loewner's differential equation is defined and analyzed. The definition of generators is motivated by the L\'evy-Khintchine representation for additive processes on the unit circle. Actually, we can and do introduce a homeomorphism between the above class of Loewner chains and the set of the distributions of increments of additive processes equipped with suitable topologies. On the other hand, from the viewpoint of non-commutative probability theory, the above generators also induce bijections with some other objects: in particular, monotone convolution hemigroups and free convolution hemigroups. Finally, the generators of Loewner chains constructed from free convolution hemigroups via subordination are computed., Comment: 34 pages

We prove that the marginal law $\sigma_{t}\boxtimes\nu$ of free positive multiplicative Brownian motion is log-unimodal for all $t>0$ if $\nu$ is a multiplicatively symmetric log-unimodal distribution, and that $\sigma_{t}\boxtimes\nu$ is log-unimodal for sufficiently large $t$ if $\nu$ is supported on a suitably chosen finite interval. Counterexamples are given when $\nu$ is not assumed to be symmetric or having a bounded support., Comment: 14 pages

In this paper, we characterize idempotent distributions with respect to the bi-free multiplicative convolution on the bi-torus. Also, the bi-free analogous Levy triplet of an infinitely divisible distribution on the bi-torus without non-trivial idempotent factors is obtained. This triplet is unique and generates a homomorphism from the bi-free multiplicative semigroup of infinitely divisible distributions to the classical one. The relevances of the limit theorems associated with four convolutions, classical and bi-free additive convolutions and classical and bi-free multiplicative convolutions, are analyzed. The analysis relies on the convergence criteria for limit theorems and the use of push-forward measures induced by the wrapping map from the plane to the bi-torus. Different from the bi-free circumstance, the classical multiplicative L\'{e}vy triplet is not always unique. Due to this, some conditions are furnished to ensure uniqueness.

We study unimodality for free multiplicative convolution with free normal distributions $\{\lambda_t\}_{t>0}$ on the unit circle. We give four results on unimodality for $\mu\boxtimes\lambda_t$: (1) if $\mu$ is a symmetric unimodal distribution on the unit circle then so is $\mu\boxtimes \lambda_t$ at any time $t>0$; (2) if $\mu$ is a symmetric distribution on $\mathbb{T}$ supported on $\{e^{i\theta}: \theta \in [-\varphi,\varphi]\}$ for some $\varphi \in (0,\pi/2)$, then $\mu \boxtimes \lambda_t$ is unimodal for sufficiently large $t>0$; (3) ${\bf b} \boxtimes \lambda_t$ is not unimodal at any time $t>0$, where ${\bf b}$ is the equally weighted Bernoulli distribution on $\{1,-1\}$; (4) $\lambda_t$ is not freely strongly unimodal for sufficiently small $t>0$. Moreover, we study unimodality for classical multiplicative convolution (with Poisson kernels), which is useful in proving the above four results., Comment: 19 pages, 4 figures

We prove one-to-one correspondences between certain decreasing Loewner chains in the upper half-plane, a special class of real-valued Markov processes, and quantum stochastic processes with monotonically independent additive increments. This leads us to a detailed investigation of probability measures on $\mathbb{R}$ with univalent Cauchy transform. We discuss several subclasses of such measures and obtain characterizations in terms of analytic and geometric properties of the corresponding Cauchy transforms. Furthermore, we obtain analogous results for the setting of decreasing Loewner chains in the unit disk, which correspond to quantum stochastic processes of unitary operators with monotonically independent multiplicative increments.

We investigate certain analytical properties of the free $\alpha-$stable densities on the line. We prove that they are all classically infinitely divisible when $\alpha\le 1$, and that they belong to the extended Thorin class when $\alpha \leq 3/4.$ The L\'evy measure is explicitly computed for $\alpha =1,$ showing that the free 1-stable random variables are not Thorin except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when $\alpha > 1.$ In the one-sided case we prove, refining unimodality, that the densities are whale-shaped that is their successive derivatives vanish exactly once. Finally, we derive a collection of results connected to the fine structure of the one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, a new identity for the Beta-Gamma algebra, and several intrinsic properties of whale-shaped densities., Comment: 44 pages

We consider different limit theorems for additive and multiplicative free L\'evy processes. The main results are concerned with positive and unitary multiplicative free L\'evy processes at small time, showing convergence to log free stable laws for many examples. The additive case is much easier, and we establish the convergence at small or large time to free stable laws. During the investigation we found out that a log free stable law with index $1$ coincides with the Dykema-Haagerup distribution. We also consider limit theorems for positive multiplicative Boolean L\'evy processes at small time, obtaining log Boolean stable laws in the limit., Comment: 41 pages

We define spreadability systems as a generalization of exchangeability systems in order to unify various notions of independence and cumulants known in noncommutative probability. In particular, our theory covers monotone independence and monotone cumulants which do not satisfy exchangeability. To this end we study generalized zeta and M\"obius functions in the context of the incidence algebra of the semilattice of ordered set partitions and prove an appropriate variant of Faa di Bruno's theorem. With the aid of this machinery we show that our cumulants cover most of the previously known cumulants. Due to noncommutativity of independence the behaviour of these cumulants with respect to independent random variables is more complicated than in the exchangeable case and the appearance of Goldberg coefficients exhibits the role of the Campbell-Baker-Hausdorff series in this context. In a final section we exhibit an interpretation of the Campbell-Baker-Hausdorff series as a sum of cumulants in a particular spreadability system, thus providing a new derivation of the Goldberg coefficients., Comment: many corrections and rearrangements suggested by the referee, 64 pages; to appear in Doc Math

We prove that classical and free Brownian motions with initial distributions are unimodal for sufficiently large time, under some assumption on the initial distributions. The assumption is almost optimal in some sense. Similar results are shown for a symmetric stable process with index 1 and a positive stable process with index $1/2$. We also prove that free Brownian motion with initial symmetric unimodal distribution is unimodal, and discuss strong unimodality for free convolution., Comment: 19 pages, 18 figures; accepted by ALEA

We study here properties of free Generalized Inverse Gaussian distributions (fGIG) in free probability. We show that in many cases the fGIG shares similar properties with the classical GIG distribution. In particular we prove that fGIG is freely infinitely divisible, free regular and unimodal, and moreover we determine which distributions in this class are freely selfdecomposable. In the second part of the paper we prove that for free random variables $X,Y$ where $Y$ has a free Poisson distribution one has $X\stackrel{d}{=}\frac{1}{X+Y}$ if and only if $X$ has fGIG distribution for special choice of parameters. We also point out that the free GIG distribution maximizes the same free entropy functional as the classical GIG does for the classical entropy.

In this article, the notion of bi-monotonic independence is introduced as an extension of monotonic independence to the two-faced framework for a family of pairs of algebras in a non-commutative space. The associated cumulants are defined and a moment-cumulant formula is derived in the bi-monotonic setting. In general the bi-monotonic product of states is not a state and the bi-monotonic convolution of probability measures on the plane is not a probability measure. This provides an additional example of how positivity need not be preserved under conditional bi-free convolutions., Comment: accepted by Journal of Theoretical Probability

In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of selfadjoint commuting random variables in an infinitesimal triangular array are determined. These distributions are characterized by their bi-freely infinite divisibility, and moreover, a transfer principle is established for limit theorems in classical probability theory and Voiculescu's bi-free probability theory. Complete descriptions of bi-free stability and fullness of planar probability distributions are also set down. All these results reveal one important feature about the theory of bi-free probability that it parallels the classical theory perfectly well. The emphasis in the whole work is not on the tool of bi-free combinatorics but only on the analytic machinery., Comment: 31 pages

The class of selfdecomposable distributions in free probability theory was introduced by Barndorff-Nielsen and the third named author. It constitutes a fairly large subclass of the freely infinitely divisible distributions, but so far specific examples have been limited to Wigner's semicircle distributions, the free stable distributions, two kinds of free gamma distributions and a few other examples. In this paper, we prove that the (classical) normal distributions are freely selfdecomposable. More generally it is established that the Askey-Wimp-Kerov distribution $\mu_c$ is freely selfdecomposable for any $c$ in $[-1,0]$. The main ingredient in the proof is a general characterization of the freely selfdecomposable distributions in terms of the derivative of their free cumulant transform., Comment: 22 pages

Richard P. Stanley conjectured that finite trees can be distinguished by their chromatic symmetric functions. In this paper, we prove an analogous statement for posets: Finite rooted trees can be distinguished by their order quasisymmetric functions., Comment: 16 pages, 5 figures, referees' suggestions incorporated

We construct a deformed Fock space and a Brownian motion coming from Coxeter groups of type D. The construction is analogous to that of the $q$-Fock space (of type A) and the $(\alpha,q)$-Fock space (of type B)., Comment: 25 pages, 8 figures, major revision, following comments by a referee

The Euler characteristic of a semialgebraic set can be considered as a generalization of the cardinality of a finite set. An advantage of semialgebraic sets is that we can define "negative sets" to be the sets with negative Euler characteristics. Applying this idea to posets, we introduce the notion of semialgebraic posets. Using "negative posets", we establish Stanley's reciprocity theorems for order polynomials at the level of Euler characteristics. We also formulate the Euler characteristic reciprocities for chromatic and flow polynomials., Comment: 16 pages; flow polynomial reciprocity added; title changed

Let $\nu_{\alpha,q}$ be the probability and orthogonality measure for the $q$-Meixner-Pollaczek orthogonal polynomials, which has appeared in \cite{BEH15} as the distribution of the $(\alpha,q)$-Gaussian process (the Gaussian process of type B) over the $(\alpha,q)$-Fock space (the Fock space of type B). The main purpose of this paper is to find the radial Bargmann representation of $\nu_{\alpha,q}$. Our main results cover not only the representation of $q$-Gaussian distribution by \cite{LM95}, but also of $q^2$-Gaussian and symmetric free Meixner distributions on $\mathbb R$. In addition, non-trivial commutation relations satisfied by $(\alpha,q)$-operators are presented., Comment: 13 pages, minor changes have been made

In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu's sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model., Comment: 33 pages. Minor revisions

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 particular show that $|S|^r \,\text{sign}(S)$ is freely infinitely divisible for $r\geq2$. Therefore $S^n$ is freely infinitely divisible for every $n\in\mathbb N$. The results on free Poisson and semicircular random variables have a good correspondence with classical ID properties of powers of gamma and normal random variables., Comment: 24 pages, 24 figures. The statement of Theorem 3.5 is modified (weakened) because an error was found in the proof of a part of Theorem 3.5 in the published version

We will prove that: (1) A symmetric free L\'evy process is unimodal if and only if its free L\'evy measure is unimodal; (2) Every free L\'evy process with boundedly supported L\'evy measure is unimodal in sufficiently large time. (2) is completely different property from classical L\'evy processes. On the other hand, we find a free L\'evy process such that its marginal distribution is not unimodal for any time $s>0$ and its free L\'evy measure does not have a bounded support. Therefore, we conclude that the boundedness of the support of free L\'evy measure in (2) cannot be dropped. For the proof we will (almost) characterize the existence of atoms and the existence of continuous probability densities of marginal distributions of a free L\'evy process in terms of L\'evy--Khintchine representation., Comment: 20 pages. To appear in Annales de l'Institut Henri Poincar\'e (B) Probabilit\'es et Statistiques

In this article we construct a generalized Gaussian process coming from Coxeter groups of type B. It is given by creation and annihilation operators on an $(\alpha,q)$-Fock space, which satisfy the commutation relation $$ b_{\alpha,q}(x)b_{\alpha,q}^\ast(y)-qb_{\alpha,q}^\ast(y)b_{\alpha,q}(x)=\langle x, y\rangle I+\alpha\langle \overline{x}, y \rangle q^{2N}, $$ where $x,y$ are elements of a complex Hilbert space with a self-adjoint involution $x\mapsto\bar{x}$ and $N$ is the number operator with respect to the grading on the $(\alpha,q)$-Fock space. We give an estimate of the norms of creation operators. We show that the distribution of the operators $b_{\alpha,q}(x)+b_{\alpha,q}^\ast(x)$ with respect to the vacuum expectation becomes a generalized Gaussian distribution, in the sense that all mixed moments can be calculated from the second moments with the help of a combinatorial formula related with set partitions. Our generalized Gaussian distribution associates the orthogonal polynomials called the $q$-Meixner-Pollaczek polynomials, yielding the $q$-Hermite polynomials when $\alpha=0$ and free Meixner polynomials when $q=0$., Comment: 22 pages, 6 figures

We realize the Belinschi-Nica semigroup of homomorphisms as a free multiplicative subordination. This realization allows to define more general semigroups of homomorphisms with respect to free multiplicative convolution. For these semigroups we show that a differential equation holds, generalizing the complex Burgers equation. We give examples of free multiplicative subordination and find a relation to the Markov-Krein transform, Boolean stable laws and monotone stable laws. A similar idea works for additive subordination, and in particular we study the free additive subordination associated to the Cauchy distribution and show that it is a homomorphism with respect to monotone, Boolean and free additive convolutions., Comment: 21 pages, minor corrections, Complex Analysis and Operator Theory, First online: 16 October 2015

We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed., Comment: 27 pages, 7 figures, AMS LaTeX

We study Boolean stable laws, $\mathbf{b}_{\alpha,\rho}$, with stability index $\alpha$ and asymmetry parameter $\rho$. We show that the classical scale mixture of $\mathbf{b}_{\alpha,\rho}$ coincides with a free mixture and also a monotone mixture of $\mathbf{b}_{\alpha,\rho}$. For this purpose we define the multiplicative monotone convolution of probability measures, one is supported on the positive real line and the other is arbitrary. We prove that any scale mixture of $\mathbf{b}_{\alpha,\rho}$ is both classically and freely infinitely divisible for $\alpha\leq1/2$ and also for some $\alpha>1/2$. Furthermore, we show the multiplicative infinite divisibility of $\mathbf{b}_{\alpha,1}$ with respect classical, free and monotone convolutions. Scale mixtures of Boolean stable laws include some generalized beta distributions of second kind, which turn out to be both classically and freely infinitely divisible. One of them appears as a limit distribution in multiplicative free laws of large numbers studied by Tucci, Haagerup and M\"oller. We use a representation of $\mathbf{b}_{\alpha,1}$ as the free multiplicative convolution of a free Bessel law and a free stable law to prove a conjecture of Hinz and M{\l}otkowski regarding the existence of the free Bessel laws as probability measures. The proof depends on the fact that $\mathbf{b}_{\alpha,1}$ has free divisibility indicator 0 for $1/2<\alpha$., Comment: 37 pages

We investigate analytical properties of free stable distributions and discover many connections with their classical counterparts. Our main result is an explicit formula for the Mellin transform, which leads to explicit series representations for the characteristic function and for the density of a free stable distribution. All of these formulas bear close resemblance to the corresponding expressions for classical stable distributions. As further applications of our results, we give an alternative proof of the duality law due to Biane and a new factorization of a classical stable random variable into an independent (in the classical sense) product of a free stable random variable and a power of a Gamma(2) random variable., Comment: 12 pages, final version of the preprint

We give a complete list of the Lebesgue-Jordan decomposition of Boolean and monotone stable distributions and a complete list of the mode of them. They are not always unimodal., Comment: 12pp and 1 figure

We show that any freely selfdecomposable probability law is unimodal. This is the free probabilistic analog of Yamazato's result in [Ann. Probab. 6 (1978), 523-531]., Comment: 17 pages; Accepted version

We investigate operator-valued monotone independence, a noncommutative version of independence for conditional expectation. First we introduce operator-valued monotone cumulants to clarify the whole theory and show the moment-cumulant formula. As an application, one can obtain an easy proof of Central Limit Theorem for operator-valued case. Moreover, we prove a generalization of Muraki's formula for the sum of independent random variables and a relation between generating functions of moments and cumulants., Comment: Proof of Theorem 3.4 is explained

We prove that many of beta, beta prime, gamma, inverse gamma, Student t- and ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability density functions. Moreover, we show that the Gaussian, ultraspherical and many of Student t-distributions have free divisibility indicator 1., Comment: 37 pages, 6 figures, slightly different from the published version

We prove that symmetric Meixner distributions, whose probability densities are proportional to $|\Gamma(t+ix)|^2$, are freely infinitely divisible for $0

We completely determine the free infinite divisibility for the Boolean stable law which is parametrized by a stability index $\alpha$ and an asymmetry coefficient $\rho$. We prove that the Boolean stable law is freely infinitely divisible if and only if one of the following conditions holds: $0<\alpha\leq\frac{1}{2}$; $\frac{1}{2}<\alpha\leq\frac{2}{3}$ and $2-\frac{1}{\alpha}\leq\rho \leq \frac{1}{\alpha}-1$; $\alpha=1,~\rho=\frac{1}{2}$. Positive Boolean stable laws corresponding to $\rho =1$ and $\alpha \leq \frac{1}{2}$ have completely monotonic densities and they are both freely and classically infinitely divisible. We also show that continuous Boolean convolutions of positive Boolean stable laws with different stability indices are also freely and classically infinitely divisible. Boolean stable laws, free stable laws and continuous Boolean convolutions of positive Boolean stable laws are non-trivial examples whose free divisibility indicators are infinity. We also find that the free multiplicative convolution of Boolean stable laws is again a Boolean stable law., Comment: 13 pages; minor changes