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 consider a pair of probability measures $\mu,\nu$ on the unit circle such that $\Sigma_{\lambda}(\eta_{\nu}(z))=z/\eta_{\mu}(z)$. We prove that the same type of equation holds for any $t\geq 0$ when we replace $\nu$ by $\nu\boxtimes\lambda_t$ and $\mu$ by $\mathbb{M}_t(\mu)$, where $\lambda_t$ is the free multiplicative analogue of the normal distribution on the unit circle of $\mathbb{C}$ and $\mathbb{M}_t$ is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite divisibility of the measures associated with subordination functions in multiplicative free Brownian motion and multiplicative free convolution semigroups. We use the modified $\mathcal{S}$-transform introduced by Raj Rao and Speicher to deal with the case that $\nu$ has mean zero. The same type of the result holds for convolutions on the positive real line. We also obtain some regularity properties for the free multiplicative analogue of the normal distributions.

Abstract:
Let $P$ be a symmetric generalised laplacian on $R^{2n+1}$. It is known that $P$ generates semigroups of measures $\mu_{t}$ on the Heisenberg group $H^{n}$ and $\nu_{t}$ on the Abelian group $R^{2n+1}$. Recall that the underlying manifold of the Heisenberg group is $R^{2n+1}$. Suppose that the negative defined function $\psi(\xi)=-\hat{P}(\xi)$ satisfies some weight conditions and $|D^{\alpha}\psi(\xi)| \leq c_{\alpha}\psi(\xi)(1+\|\xi\|)^{-|\alpha|}, \xi \in R^{2n+1}.$ We show that the semigroup $\mu_{t}$ is a kind of perturbation of the semigroup $\nu_{t}$. More precisely, we give pointwise estimates for the difference between the densities of $\mu_{t}$ and $\nu_{t}$ and we show that it is small with respect to $t$ and $x$. As a consequence we get a description of the asymptotic behaviour at origin of densities of a semigroup of measures which is analogon of the symmetrized gamma (gamma-variance) semigroup on the Heisenberg group.

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:
Semiuniform semigroups provide a natural setting for the convolution of generalized finite measures on semigroups. A semiuniform semigroup is said to be ambitable if each uniformly bounded uniformly equicontinuous set of functions on the semigroup is contained in an ambit. In the convolution algebras constructed over ambitable semigroups, topological centres have a tractable characterization.

Abstract:
Recently, Bercovici has introduced multiplicative convolutions based on Muraki's monotone independence and shown that these convolution of probability measures correspond to the composition of some function of their Cauchy transforms. We provide a new proof of this fact based on the combinatorics of moments. We also give a new characterisation of the probability measures that can be embedded into continuous monotone convolution semigroups of probability measures on the unit circle and briefly discuss a relation to Galton-Watson processes.

Abstract:
Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C_0-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.

Abstract:
In this paper,the method of partial groupization is applied to the study of limit behavioursof probability measures on Abelian semigroups.First the existence of the limit for the i.i.d.random variable convolution power sequence is discussed,and a result of Kawada-Ito type oncompact Abelian semigroups is given.Then,a strong principle of Kloss convergence on com-pact Abelian semigroups is set up.This principle was once obtained by Maksimov on finitgroups and compact groups.

Abstract:
We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R}^d$ under the assumption that its L\'{e}vy--Khintchine exponent is regularly varying of index between $0$ and $2$.

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.