In this paper, we define and study free Jacobi processes of parameters λ > 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not depend on time) and derive, for λ ∈]0, 1] and 1/θ ≥ λ + 1 a free SDE analogous to the classical one. In the second part, we generalize this result under an additional condition. To proceed, we set a recurrence formula for the moments of the process using free stochastic calculus. This will also be used to compute the p. d. e. satisfied by the Cauchy transform of the free Jacobi's law.

In this paper, we are interested in the free Jacobi process starting at the unit of the compressed probability space where it takes values and associated with the parameter values λ = 1, θ = 1/2. Firstly, we derive a timedependent recurrence equation for the moments of the process (valid for any starting point and all parameter values). Secondly, we transform this equation to a nonlinear partial differential one for the moment generating function that we solve when λ = 1, θ = 1/2. The obtained solution together with tricky computations lead to an explicit expression of the moments which shows that the free Jacobi process is distributed at any time t as (1/4)where Y is a free unitary Brownian motion. This expression is recovered relying on enumeration techniques after proving that if a is a symmetric Bernoulli random variable which is free from {Y, Y ⋆ }, then the distributions of Y 2t and that of aYtaY ⋆ t coincide. We close the exposition by investigating the spectral distribution associated with general sets of parameter values: λ = 1, θ ∈ (0, 1) and λ ∈ (0, 2), θ = 1/2.

In this paper, we study complex Wishart processes or the so-called Laguerre processes (Xt) t≥0 . We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semi-group. We also give absolute-continuity relations between different indices. Finally, we compute the density function of the so-called generalized Hartman-Watson law as well as the law of T0 := inf{t, det(Xt) = 0} when the size of the matrix is 2.