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

Let A(t) be a nxp 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) = Pi(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 [Br91]. 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.