On the spectral distribution of the free Jacobi process

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 $\lambda=1, \theta =1/2$. Firstly, we derive a time-dependent 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 $\lambda = 1, \theta =1/2$. 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 $(1/4)(2+Y_{2t}+Y_{2t}^{\star})$ 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 $\{Y, Y^{\star}\}$, then the distributions of $Y_{2t}$ and that of $aY_taY_t^{\star}$ coincide. We close the exposition by investigating the spectral distribution associated with general sets of parameter values.