Spectral distribution of the free Jacobi process associated with one projection
Volume 137 / 2014
Abstract
Given an orthogonal projection $P$ and a free unitary Brownian motion $Y = (Y_t)_{t \geq 0}$ in a $W^{\star }$-non commutative probability space such that $Y$ and $P$ are $\star $-free in Voiculescu's sense, we study the spectral distribution $\nu _t$ of $J_t = PY_tPY_t^{\star }P$ in the compressed space. To this end, we focus on the spectral distribution $\mu _t$ of the unitary operator $SY_tSY_t^{\star }$, $S = 2P-1$, 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 $\psi (t, \cdot )$ 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 $\mu _0$ and $\mu _{\infty }$. This enables us to compute the weights $\mu _t\{1\}$ and $\mu _t\{-1\}$ which together with the binomial-type expansion lead to $\nu _t\{1\}$ and $\nu _t\{0\}$. Fatou's theorem for harmonic functions in the upper half-plane shows that the absolutely continuous part of $\nu _t$ 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 $n$th moment of $\mu _t$.
