Abstract:
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.

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:
We show that the monotonic independence introduced by Muraki can also be used to define a multiplicative convolution. We also find a method for the calculation of this convolution based on an appropriate form of the Cauchy transform. We discuss infinite divisibility in the multiplicative monotonic context as well.

Abstract:
Let G be a finite group acting by automorphism on a lattice A, and hence on the group algebra S=k[A]. The algebra of G-invariants in S is called an algebra of multiplicative invariants. We investigate when algebras of multiplicative invariants are semigroup algebras. In particular, we present an explicit version of a result of Farkas stating that multiplicative invariants of finite reflection groups are indeed semigroup algebras. On the other hand, multiplicative invariants arising from fixed point free actions are shown to never be semigroup algebras. In particular, this holds whenever G has odd prime order.

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:
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:
We study the multiplicative convolution for c-monotone independence. This convolution unifies the monotone, Boolean and orthogonal multiplicative convolutions. We characterize convolution semigroups for the c-monotone multiplicative convolution on the unit circle. We also prove that an infinitely divisible distribution can always be embedded in a convolution semigroup. We furthermore discuss the (non)-uniqueness of such embeddings including the monotone case. Finally connections to the multiplicative Boolean convolution are discussed.

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:
In this work we detail the application of a fast convolution algorithm computing high dimensional integrals to the context of multiplicative noise stochastic processes. The algorithm provides a numerical solution to the problem of characterizing conditional probability density functions at arbitrary time, and we applied it successfully to quadratic and piecewise linear diffusion processes. The ability in reproducing statistical features of financial return time series, such as thickness of the tails and scaling properties, makes this processes appealing for option pricing. Since exact analytical results are missing, we exploit the fast convolution as a numerical method alternative to the Monte Carlo simulation both in objective and risk neutral settings. In numerical sections we document how fast convolution outperforms Monte Carlo both in velocity and efficiency terms.

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.