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Motivated by quantum information theory, we introduce a dynamical random state built out of the sum of $k \geq 2$ independent unitary Brownian motions. In the large size limit, its spectral distribution equals, up to a normalising factor, that of the free Jacobi process associated with a single self-adjoint projection with trace $1/k$. Using free stochastic calculus, we extend this equality to the radial part of the free average of $k$ free unitary Brownian motions and to the free Jacobi process associated with two self-adjoint projections with trace $1/k$, provided the initial distributions coincide. In the single projection case, we derive a binomial-type expansion of the moments of the free Jacobi process which extends to any $k \geq 3$ the one derived in \cite {DHH} in the special case $k=2$. Doing so give rise to a non normal (except for $k=2$) operator arising from the splitting of a self-adjoint projection into the convex sum of $k$ unitary operators. This binomial expansion is then used to derive a pde for the moment generating function of this non normal operator and for which we determine the corresponding characteristic curves., Comment: The characteristic curves are determined

The monograph is devoted to the study of stochastic area functionals of Brownian motions and of the associated heat kernels on Lie groups and Riemannian manifolds. It is essentially self-contained and as such can serve as a textbook on the theory of Brownian motions and horizontal Brownian motions on manifolds. Emphasis is put on concrete examples which allows us to concretely illustrate the rich and deep interactions between stochastic calculus, Riemannian and sub-Riemannian geometry, the theory of complex and quaternionic symmetric spaces and random matrices., Comment: 295 pages. This survey. written as a textbook, puts in a broader perspective and unifies several works of the authors. It also contains some new and original results

We relate Gruet formula for the heat kernel on real hyperbolic spaces to the commonly used one derived from Millson induction. The bridge between both formulas is settled by Yor result on the joint distribution of a Brownian motion and of its exponential functional at fixed time. This result allows further to relate Gruet formula with real parameter to the heat kernel of the hyperbolic Jacobi operator and to derive a new integral representation for the heat kernel of the Maass Laplacian. When applied to harmonic AN groups, Yor result yields also new a integral representation of their corresponding heat kernels which does not distinguish the parity of the dimension of the center of the Lie group N., Comment: typos corrected

The Lamperti correspondence gives a prominent role to two random time changes: the exponential functional of a L\'evy process drifting to $\infty$ and its inverse, the clock of the corresponding positive self-similar process. We describe here asymptotical properties of these clocks in large time, extending the results of Yor and Zani.

We revisit the description provided by Ph. Biane of the spectral measure of the free unitary Brownian motion. We actually construct for any $t \in (0,4)$ a Jordan curve $\gamma_t$ around the origin, not intersecting the semi-axis $[1,\infty[$ and whose image under some meromorphic function $h_t$ lies in the circle. Our construction is naturally suggested by a residue-type integral representation of the moments and $h_t$ is up to a M\"obius transformation the main ingredient used in the original proof. Once we did, the spectral measure is described as the push-forward of a complex measure under a local diffeomorphism yielding its absolute-continuity and its support. Our approach has the merit to be an easy yet technical exercise from real analysis.

We define and study a multidimensional process that generalizes the eigenvalues of matrix Jacobi processes on the one hand and whose stationary distribution is given by the beta Jacobi ensemble on the other hand., Comment: the paper is accepted for publication in APAM

We stduy radial Dunkl processes associated with dihedral systems: we derive the semi group, the generalized Bessel function, the Dunkl-Hermite polynomials. Then we give a skew product decomposition by means of independent Bessel processes and we compute the tail distribution of the first hitting time of the boundary of Weyl chamber., Comment: improvement of the skew product decomposition

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 pages