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Eigenvalues of the Laguerre Process as Noncolliding Squared Bessel Processes
Konig, Wolfgang; O'Connell, Neil
HPLBRIMS200107
Keyword(s): Wishart and Laguerre ensembles and processes; eigenvalues as diffusions; noncolliding 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 ) = II_{i<j} ( x_{i}x _{j}) is harmonic with respect to p independent squared Bessel processes of dimension 2(np+ 1), and the eigenvalue process has the same law as the corresponding Doob htransform. 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 noncolliding 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 OrnsteinUhlenbeck processes) which are all amenable to the same htransform, and compute the corresponding transition densities and upper tail asymptotics for the first collision time. Notes: Wolfgang Konig, Fachbereich Mathematik, MA 75, room MA783 Technische Universitat Berlin Strasse des 17. Juni 136 D10623 Berlin Germany
8 Pages
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