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Eigenvalues of the Laguerre Process as Non-colliding Squared Bessel Processes
Konig, Wolfgang; O'Connell, Neil
Keyword(s): Wishart and Laguerre ensembles and processes; eigenvalues as diffusions; non-colliding squared Bessel processes
Abstract: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- 1, 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 ) = IIi<j ( xi-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. Notes: Wolfgang Konig, Fachbereich Mathematik, MA 7-5, room MA783 Technische Universitat Berlin Strasse des 17. Juni 136 D-10623 Berlin Germany
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