Spectral distribution of the free Jacobi process with equal rank projections

Abstract

The free Jacobi process is the radial part of the compression of the free unitary Brownian motion by two free orthogonal projections in a non commutative probability space. In this paper, we derive spectral properties of the free Jacobi process associated with projections having the same rank $\alpha \in (0,1)$. To start with, we determine the characteristic curves of the partial differential equation satisfied by the moment generating function of its spectral distribution. Doing so leads for any fixed time $t >0$ to an expression of this function in a neighborhood of the origin, therefore extends our previous results valid for $\alpha = 1/2$. Moreover, the obtained characteristic curves are encoded by an $\alpha$-deformation of the compositional inverse of the $\chi$-transform of the spectral distribution of the free unitary Brownian motion. In this respect, we study mapping properties of this deformation and use the saddle point method to prove that the compositional inverse of a $\alpha$-deformation of the $\chi$-transform of the free unitary Brownian motion is analytic in the open unit disc (for large enough time $t$). The last part of the paper is devoted to a dynamical version of a recent identity pointed out by T. Kunisky in \cite{Kun}. Actually, this identity relates the stationary distributions of the free Jacobi processes corresponding to the sets of parameters $(\alpha, \alpha)$ and $(1/2,\alpha)$ respectively and we explain how it follows from the Nica-Speicher semi-group. Our dynamical version then relates the partial differential equations of the Cauchy-Stieltjes transforms of the densities of the finite-time spectral distributions. It also raises the problem of whether a dynamical analogue of the Nica-Speicher semi-group exists when the compressing projection has rank $1/2$.