Free Martingale polynomials for stationary Jacobi processes

We generalize a previous result concerning free martingale polynomials for the stationary free Jacobi process of parameters $\lambda \in ]0.1], \theta = 1/2$. Hopelessly, apart from the case $\lambda = 1$, the polynomials we derive are no longer orthogonal with respect to the spectral measure. As a matter of fact, we use the multiplicative renormalization to write down the corresponding orthogonality measure.Comment: page number : 1

First hitting time of the boundary of a wedge of angle π/4\pi/4 by a radial Dunkl process

In this paper, we derive an integral representation for the density of the reciprocal of the first hitting time of the boundary of a wedge of angle $\pi/4$ by a radial Dunkl process with equal multiplicity values. Not only this representation readily yields the non negativity of the density, but also provides an analogue of Dufresne's result on the distribution of the first hitting time of zero by a Bessel process and a generalization of the Vakeroudis-Yor's identity satisfied by the first exit time from a wedge by a planar Brownian motion. We also use a result due to Spitzer on the angular part of the planar Brownian motion to prove a representation of the tail distribution of its first exit time from a dihedral wedge through the square wave function.Comment: Title is changed, many corrections, new result

Generalized Bessel function of Type D

We write down the generalized Bessel function associated with the root system of type $D$ by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type $D$.Comment: This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Radial Dunkl Processes : Existence and uniqueness, Hitting time, Beta Processes and Random Matrices

We begin with the study of some properties of the radial Dunkl process associated to a reduced root system $R$. It is shown that this diffusion is the unique strong solution for all $t \geq 0$ of a SDE with singular drift. Then, we study $T_0$, the first hitting time of the positive Weyl chamber : we prove, via stochastic calculus, a result already obtained by Chybiryakov on the finiteness of $T_0$. The second and new part deals with the law of $T_0$ for which we compute the tail distribution, as well as some insight via stochastic calculus on how root systems are connected with eigenvalues of standard matrix-valued processes. This gives rise to the so-called $\beta$-processes. The ultraspherical $\beta$-Jacobi case still involves a reduced root system while the general case is closely connected to a non reduced one. This process lives in a convex bounded domain known as principal Weyl alcove and the strong uniqueness result remains valid. The last part deals with the first hitting time of the alcove's boundary and the semi group density which enables us to answer some open questions.Comment: 33 page

On generalized Cauchy-Stieltjes transforms of some Beta distributions

We express generalized Cauchy-Stieltjes transforms of some particular Beta distributions (of ultraspherical type generating functions for orthogonal polynomials) as a powered Cauchy-Stieltjes transform of some measure. For suitable values of the power parameter, the latter measure turns out to be a probability measure and its density is written down using Markov transforms. The discarded values give a negative answer to a deformed free probability unless a restriction on the power parameter is made. A particular symmetric distribution interpolating between Wigner and arcsine distributions is obtained. Its moments are expressed through a terminating hypergeometric series interpolating between Catalan and shifed Catalan numbers. for small values of the power parameter, the free cumulants are computed. Interesting opne problems related to a deformed representation theory of the infinite symmetric group and to a deformed Bozejko's convolution are discussed