Eigenvalues of the Laguerre Process as Non-Colliding Squared Bessel Processes
Wolfgang König, BRIMS, HP Labs
Neil O'Connell, BRIMS, HP Labs
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)=prodi<j(xi-xj) 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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