940 articles – 1212 references  [version française]
 HAL: hal-00704985, version 1
 arXiv: 1204.6227
 On the spectral distribution of the free Jacobi process
 (2012-04-27)
 In this paper, we are interested in the free Jacobi process starting at the unit of the compressed probability space where it takes values and associated with the parameter values $\lambda=1, \theta =1/2$. Firstly, we derive a time-dependent recurrence equation for the moments of the process (valid for any starting point and all parameter values). Secondly, we transform this equation to a nonlinear partial differential one for the moment generating function that we solve when $\lambda = 1, \theta =1/2$. The obtained solution together with tricky computations lead to an explicit expression of the moments which shows that the free Jacobi process is distributed at any time $t$ as $(1/4)(2+Y_{2t}+Y_{2t}^{\star})$ where $Y$ is a free unitary Brownian motion. This expression is recovered relying on enumeration techniques after proving that if $a$ is a symmetric Bernoulli random variable which is free from $\{Y, Y^{\star}\}$, then the distributions of $Y_{2t}$ and that of $aY_taY_t^{\star}$ coincide.
 1: Institut de Recherche Mathématique de Rennes (IRMAR) CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
 Research team: Théorie ergodiqueEquations aux dérivées partielles
 Subject : Mathematics/Spectral Theory