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Spectral distribution of the free Jacobi process associated with one projection

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Demni N., Hmidi T.

Colloquium Mathematicum

2014

tom 137

nr 2

271-296

Given an orthogonal projection P and a free unitary Brownian motion $Y = (Yₜ)_{t≥0}$ in a W*-non commutative probability space such that Y and P are *-free in Voiculescu's sense, we study the spectral distribution νₜ of Jₜ = PYₜPYₜ*P 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 ψ(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.

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1 Colloquium Mathematicum

1 Demni N.

1 Hmidi T.

1 2014

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Spectral distribution of the free Jacobi process associated with one projection