TY - JOUR

T1 - Spectral distribution of the free jacobi process

AU - Demni, Nizar

AU - Hamdi, Tarek

AU - Hmidi, Taoufik

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2012

Y1 - 2012

N2 - In this paper, we describe the spectral distribution of the free Jacobi process associated with the parameter values θ = 1, λ = 1/2 and starting at the unit of the compressed probability space where it takes values. To proceed, we derive a time-dependent recurrence equation for its moments (actually valid for all parameter values) or equivalently a nonlinear partial differential equation (PDE) for its moment generating function. Then, we solve this PDE and expand the obtained solution around the origin. Doing so leads to an explicit formula for the moments, which shows that the free Jacobi process is distributed at any time t as 1/4 (2 + Y2t + Y*2t), where Y is a free unitary Brownian motion. We recover this formula relying on enumeration techniques together with the following result: if a is a symmetric Bernoulli random variable which is free from {Y, Y*}, then the distributions of Y2t and that of aYtaY*t coincide. We close the exposition by investigating the spectral distribution of the free Jacobi process associated with the parameter values λ = 1, θ ε (0, 1).

AB - In this paper, we describe the spectral distribution of the free Jacobi process associated with the parameter values θ = 1, λ = 1/2 and starting at the unit of the compressed probability space where it takes values. To proceed, we derive a time-dependent recurrence equation for its moments (actually valid for all parameter values) or equivalently a nonlinear partial differential equation (PDE) for its moment generating function. Then, we solve this PDE and expand the obtained solution around the origin. Doing so leads to an explicit formula for the moments, which shows that the free Jacobi process is distributed at any time t as 1/4 (2 + Y2t + Y*2t), where Y is a free unitary Brownian motion. We recover this formula relying on enumeration techniques together with the following result: if a is a symmetric Bernoulli random variable which is free from {Y, Y*}, then the distributions of Y2t and that of aYtaY*t coincide. We close the exposition by investigating the spectral distribution of the free Jacobi process associated with the parameter values λ = 1, θ ε (0, 1).

KW - Free Jacobi process

KW - Free unitary Brownian motion

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U2 - 10.1512/iumj.2012.61.5034

DO - 10.1512/iumj.2012.61.5034

M3 - Article

AN - SCOPUS:84880888181

VL - 61

SP - 1351

EP - 1368

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 3

ER -