@article{ECP1040,
author = {Wolfgang König and Neil O'Connell},
title = {Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {6},
year = {2001},
keywords = {Wishart and Laguerre ensembles and processes, eigenvalues as diffusions, non-colliding squared Bessel processes.},
abstract = {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.},
pages = {no. 11, 107-114},
issn = {1083-589X},
doi = {10.1214/ECP.v6-1040},
url = {http://ecp.ejpecp.org/article/view/1040}}