Electronic Communications in Probability

Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes

Wolfgang König and Neil O'Connell

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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.

Article information

Electron. Commun. Probab., Volume 6 (2001), paper no. 11, 107-114.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52
Secondary: 60J65: Brownian motion [See also 58J65] 62E10: Characterization and structure theory

Wishart and Laguerre ensembles and processes eigenvalues as diffusions non-colliding squared Bessel processes

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König, Wolfgang; O'Connell, Neil. Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes. Electron. Commun. Probab. 6 (2001), paper no. 11, 107--114. doi:10.1214/ECP.v6-1040. https://projecteuclid.org/euclid.ecp/1461097556

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