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 = PY_tP _tP in the compressed space. To this end, we focus on the spectral distribution µt of the unitary operator SYtSYt⋆, S = 2P - 1, whose moments are related to those of Jt 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 µt. Then, we exhibit a flow (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 µ0 and µ∞. This enables us to compute the weights µt{1} and µt{-1} which together with the binomial-type expansion lead to νt{1} and νt{0}. Fatou’s theorem for harmonic functions in the upper half-plane shows that the absolutely continuous part of νt is related to the nontangential extension of the Herglotz transform of µt 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 µt.