Noncolliding Brownian motions and Harish-Chandra formula
Makoto Katori, Chuo university
Hideki Tanemura, Chiba university
Abstract
We consider a system of noncolliding Brownian motions
introduced in our previous paper,
in which the noncolliding condition is imposed
in a finite time interval $(0,T]$.
This is a temporally inhomogeneous diffusion process
whose transition probability density depends on a value of $T$,
and in the limit $T to infty$
it converges to a temporally homogeneous diffusion process
called Dyson's model of Brownian motions.
It is known that the distribution of particle positions in Dyson's model
coincides with that of eigenvalues of a Hermitian matrix-valued
process, whose entries are independent Brownian motions.
In the present paper we construct such a Hermitian matrix-valued process,
whose entries are sums of
Brownian motions and Brownian bridges given
independently of each other,
that its eigenvalues are identically distributed
with the particle positions of
our temporally inhomogeneous system of noncolliding Brownian motions.
As a corollary of this identification we derive the
Harish-Chandra formula for an integral over the unitary group.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona.
The preferred way is to send a scanned (jpeg or pdf) copy of the signed copyright form to the managing editor Philippe Carmona at ejpecpme@math.univ-nantes.fr.
If a paper has several authors, the corresponding author signs the copyright form
on behalf of all the authors.