Spectral distribution of the free Jacobi process associated with one projection

Nizar Demni ¹   ; Taoufik Hmidi ¹

¹ IRMAR, Université de Rennes 1 Campus de Beaulieu 35042 Rennes Cedex, France

Colloquium Mathematicum, Tome 137 (2014) no. 2, pp. 271-296

Résumé

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 $\mu _t$.

Détail

DOI : 10.4064/cm137-2-11

Keywords: given orthogonal projection unitary brownian motion geq star non commutative probability space star free voiculescus sense study spectral distribution tpy star compressed space end focus spectral distribution unitary operator tsy star p whose moments related those via binomial type expansion already obtained demni indiana univ math connection stochastic calculus order derive partial differential equation herglotz transform exhibit flow psi cdot valued composition herglotz transform flow governed the initial stationary distributions infty enables compute weights which together binomial type expansion lead fatous theorem harmonic functions upper half plane shows absolutely continuous part related nontangential extension herglotz transform unit circle part paper combinatorics noncrossing partitions order analyze term corresponding exponential decay nt expansion nth moment