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2001 Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes
Wolfgang König, Neil O'Connell
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Electron. Commun. Probab. 6: 107-114 (2001). DOI: 10.1214/ECP.v6-1040


Let $A(t)$ be an $n\times p$ matrix with independent standard complex Brownian entries and set $M(t)=A(t)^*A(t)$. This is a process version of the Laguerre ensemble and as such we shall refer to it as the Laguerre process;. The purpose of this note is to remark that, assuming $n > p$, the eigenvalues of $M(t)$ evolve like $p$ independent squared Bessel processes of dimension $2(n-p+1)$, conditioned (in the sense of Doob) never to collide. More precisely, the function $h(x)=\prod_{i < j}(x_i-x_j)$ is harmonic with respect to $p$ independent squared Bessel processes of dimension $2(n-p+1)$, and the eigenvalue process has the same law as the corresponding Doob $h$-transform. In the case where the entries of $A(t)$ are real Brownian motions, $(M(t))_{t > 0}$ is the Wishart process considered by Bru (1991). There it is shown that the eigenvalues of $M(t)$ evolve according to a certain diffusion process, the generator of which is given explicitly. An interpretation in terms of non-colliding processes does not seem to be possible in this case. We also identify a class of processes (including Brownian motion, squared Bessel processes and generalised Ornstein-Uhlenbeck processes) which are all amenable to the same $h$-transform, and compute the corresponding transition densities and upper tail asymptotics for the first collision time.


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Wolfgang König. Neil O'Connell. "Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes." Electron. Commun. Probab. 6 107 - 114, 2001.


Accepted: 31 August 2001; Published: 2001
First available in Project Euclid: 19 April 2016

zbMATH: 1011.15012
MathSciNet: MR1871699
Digital Object Identifier: 10.1214/ECP.v6-1040

Primary: 15A52
Secondary: 60J65, 62E10


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