Spectral distribution of the free jacobi process

In this paper, we describe the spectral distribution of the free Jacobi process associated with the parameter values θ = 1, λ = 1/2 and starting at the unit of the compressed probability space where it takes values. To proceed, we derive a time-dependent recurrence equation for its moments (actually valid for all parameter values) or equivalently a nonlinear partial differential equation (PDE) for its moment generating function. Then, we solve this PDE and expand the obtained solution around the origin. Doing so leads to an explicit formula for the moments, which shows that the free Jacobi process is distributed at any time t as 1/4 (2 + Y2t + Y*_2t), where Y is a free unitary Brownian motion. We recover this formula relying on enumeration techniques together with the following result: if a is a symmetric Bernoulli random variable which is free from {Y, Y*}, then the distributions of Y_2t and that of aY_taY*_t coincide. We close the exposition by investigating the spectral distribution of the free Jacobi process associated with the parameter values λ = 1, θ ε (0, 1).