SPECTRAL DISTRIBUTION OF THE FREE JACOBI PROCESS ASSOCIATED WITH ONE PROJECTION

Given an orthogonal projection P and a free unitary Brownian motion Y = (Y-t)(t >= 0) in a W*-non commutative probability space such that Y and P are *-free in Voiculescu's sense, we study the spectral distribution v(t) of J(t) = PYtPYt*P in the compressed space. To this end, we focus on the spectral distribution mu(t) of the unitary operator SYtSYt*, S = 2P - 1, whose moments are related to those of J(t) via a binomial-type expansion already obtained by Demni et al. [Indiana Univ. Math. J. 61 (2012)]. In this connection, we use free stochastic calculus in order to derive a partial differential equation for the Herglotz transform mu(t). Then, we exhibit a flow psi(t,.) valued in [-1, 1] such that the composition of the Herglotz transform with the flow is governed by both the ones of the initial and the stationary distributions mu(o) and mu(infinity). This enables us to compute the weights mu(t){1} and mu(t){-1} which together with the binomial-type expansion lead to v(t) {1} and v(t){0}. Fatou's theorem for harmonic functions in the upper half-plane shows that the absolutely continuous part of v(t) is related to the nontangential extension of the Herglotz transform of mu(t) to the unit circle. In the last part of the paper, we use combinatorics of noncrossing partitions in order to analyze the term corresponding to the exponential decay e(-nt) in the expansion of the nth moment of mu(t).