EIGENVALUES OF THE LAGUERRE PROCESS AS NON-COLLIDING SQUARED BESSEL PROCESSES

Let A(t) be a np 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 )= Q i<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 ofA(t) are real Brownian motions, (M(t))t0 is the Wishart process considered by Bru [Br91]. There it is shown that the eigenvalues of M(t) evolve according to a certain diusion 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 rst collision time.