Abstract:
In this paper, we study the supports of measures in multiplicative free semigroups on the positive real line and on the unit circle. We provide formulas for the density of the absolutely continuous parts of measures in these semigroups. The descriptions rely on the characterizations of the images of the upper half-plane and the unit disc under certain subordination functions. These subordination functions are $\eta$-transforms of infinitely divisible measures with respect to multiplicative free convolution. The characterizations also help us study the regularity properties of these measures. One of the main results is that the number of components in the support of measures in the semigroups is a decreasing function of the semigroup parameter.

Abstract:
Given a probability measure $\mu$ on the real line, there exists a semigroup $\mu_t$ with real parameter $t>1$ which interpolates the discrete semigroup of measures $\mu_n$ obtained by iterating its free convolution. It was shown in \cite{[BB2004]} that it is impossible that $\mu_t$ has no mass in an interval whose endpoints are atoms. We extend this result to semigroups related to multiplicative free convolution. The proofs use subordination results.

Abstract:
Using the combinatorics of non-crossing partitions, we construct a conditionally free analogue of the Voiculescu's S-transform. The result is applied to analytical description of conditionally free multiplicative convolution and characterization of infinite divisibility.

Abstract:
Let M denote the space of Borel probability measures on the real line. For every nonnegative t we consider the transformation $\mathbb B_t : M \to M$ defined for any given element in M by taking succesively the the (1+t) power with respect to free additive convolution and then the 1/(1+t) power with respect to Boolean convolution of the given element. We show that the family of maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of composition and that, quite surprisingly, every $\mathbb B_t$ is a homomorphism for the operation of free multiplicative convolution. We prove that for t=1 the transformation $\mathbb B_1$ coincides with the canonical bijection $\mathbb B : M \to M_{inf-div}$ discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here M_{inf-div} stands for the set of probability distributions in M which are infinitely divisible with respect to free additive convolution. As a consequence, we have that $\mathbb B_t(\mu)$ is infinitely divisible with respect to free additive convolution for any for every $\mu$ in M and every t greater than or equal to one. On the other hand we put into evidence a relation between the transformations $\mathbb B_t$ and the free Brownian motion; indeed, Theorem 4 of the paper gives an interpretation of the transformations $\mathbb B_t$ as a way of re-casting the free Brownian motion, where the resulting process becomes multiplicative with respect to free multiplicative convolution, and always reaches infinite divisibility with respect to free additive convolution by the time t=1.

Abstract:
In classical probability the law of large numbers for the multiplicative convolution follows directly from the law for the additive convolution. In free probability this is not the case. The free additive law was proved by D. Voiculescu in 1986 for probability measures with bounded support and extended to all probability measures with first moment by J. M. Lindsay and V. Pata in 1997, while the free multiplicative law was proved only recently by G. Tucci in 2010. In this paper we extend Tucci's result to measures with unbounded support while at the same time giving a more elementary proof for the case of bounded support. In contrast to the classical multiplicative convolution case, the limit measure for the free multiplicative law of large numbers is not a Dirac measure, unless the original measure is a Dirac measure. We also show that the mean value of \ln x is additive with respect to the free multiplicative convolution while the variance of \ln x is not in general additive. Furthermore we study the two parameter family (\mu_{\alpha,\beta})_{\alpha,\beta \ge 0} of measures on (0,\infty) for which the S-transform is given by S_{\mu_{\alpha,\beta}}(z) = (-z)^\beta (1+z)^{-\alpha}, 0 < z < 1.

Abstract:
Let $\boxplus$, $\boxtimes$ and $\uplus$ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure $\mu$ on $[0,\infty)$ with finite second moment, we find the scaling limit of $(\mu^{\boxtimes N})^{\boxplus N}$ as $N$ goes to infinity. The $\mathcal{R}$--transform of the limit distribution can be represented by the Lambert's $W$ function. We also find similar limit theorem by replacing the free additive convolution with the boolean convolution.

Abstract:
This paper investigates homomorphisms \`a la Bercovici-Pata between additive and multiplicative convolutions. We also consider their matricial versions which are associated with measures on the space of Hermitian matrices and on the unitary group. The previous results combined with a matricial model of Benaych-Georges and Cabanal-Duvillard allows us to define and study the large N limit of a new matricial model on the unitary group for free multiplicative L\'evy processes.

Abstract:
We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of $\mathbb{C}$ with the free multiplicative analogues of the normal distribution on the unit circle. This description relies on a characterization of the image of the unit disc under the subordination function, which also allows us to prove some regularity properties of the measures obtained in this way. As an application, we give a new proof for Biane's classic result on the densities of the free multiplicative analogue of the normal distributions. We obtain analogue results for probability measures on $\mathbb{R}^+$. Finally, we describe the density of the free multiplicative analogue of the normal distributions as an example and prove unimodality and some symmetry properties of these measures.

Abstract:
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 infinity; we derive an upper bound of the indicator in terms of Jacobi parameters. This upper bound is achieved only by free Meixner distributions. We also prove Bozejko's conjecture which says the Boolean power of a probability measure mu by 0 < t < 1 is freely infinitely divisible if mu is so. In the other half of this paper, we introduce an analogous composition semigroup for multiplicative convolutions and define free divisibility indicators for these convolutions. Moreover, we prove that a probability measure on the unit circle is freely infinitely divisible concerning the multiplicative free convolution if and only if the indicator is not less than one. We also prove how the multiplicative divisibility indicator changes under free and Boolean powers and then the multiplicative analogue of Bozejko's conjecture. We include an appendix, where the Cauchy distributions and point measures are shown to be the only fixed points of the Boolean-to-free Bercovici-Pata bijection.

Abstract:
Free probability and random matrix theory has shown to be a fruitful combination in many fields of research, such as digital communications, nuclear physics and mathematical finance. The link between free probability and eigenvalue distributions of random matrices will be strengthened further in this paper. It will be shown how the concept of multiplicative free convolution can be used to express known results for eigenvalue distributions of a type of random matrices called Information-Plus-Noise matrices. The result is proved in a free probability framework, and some new results, useful for problems related to free probability, are presented in this context. The connection between free probability and estimators for covariance matrices is also made through the notion of free deconvolution.