Noncolliding brownian motions and Harish-Chandra formula

Makoto Katori, Hideki Tanemura

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We consider a system of noncolliding Brownian motions introduced in our previous paper, in which the noncolliding condition is imposed in a �nite 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 → ∞ 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.

Original languageEnglish
Pages (from-to)112-121
Number of pages10
JournalElectronic Communications in Probability
Volume8
DOIs
Publication statusPublished - 2003 Jan 1
Externally publishedYes

Fingerprint

Brownian motion
Diffusion Process
Eigenvalues of a Hermitian matrix
Brownian Bridge
Transition Density
Unitary group
Hermitian matrix
Transition Probability
Probability Density
Identically distributed
Corollary
Eigenvalue
Converge
Interval
Model
Eigenvalues
Diffusion process

Keywords

  • Dyson’s Brownian motion
  • Harish-Chandra formula
  • Imhof’s relation
  • Random matrices

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Noncolliding brownian motions and Harish-Chandra formula. / Katori, Makoto; Tanemura, Hideki.

In: Electronic Communications in Probability, Vol. 8, 01.01.2003, p. 112-121.

Research output: Contribution to journalArticle

@article{50f862c9840f4c68b702dd50cfbdd6a0,
title = "Noncolliding brownian motions and Harish-Chandra formula",
abstract = "We consider a system of noncolliding Brownian motions introduced in our previous paper, in which the noncolliding condition is imposed in a �nite 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 → ∞ 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.",
keywords = "Dyson’s Brownian motion, Harish-Chandra formula, Imhof’s relation, Random matrices",
author = "Makoto Katori and Hideki Tanemura",
year = "2003",
month = "1",
day = "1",
doi = "10.1214/ECP.v8-1076",
language = "English",
volume = "8",
pages = "112--121",
journal = "Electronic Communications in Probability",
issn = "1083-589X",
publisher = "Institute of Mathematical Statistics",

}

TY - JOUR

T1 - Noncolliding brownian motions and Harish-Chandra formula

AU - Katori, Makoto

AU - Tanemura, Hideki

PY - 2003/1/1

Y1 - 2003/1/1

N2 - We consider a system of noncolliding Brownian motions introduced in our previous paper, in which the noncolliding condition is imposed in a �nite 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 → ∞ 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.

AB - We consider a system of noncolliding Brownian motions introduced in our previous paper, in which the noncolliding condition is imposed in a �nite 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 → ∞ 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.

KW - Dyson’s Brownian motion

KW - Harish-Chandra formula

KW - Imhof’s relation

KW - Random matrices

UR - http://www.scopus.com/inward/record.url?scp=3042616103&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3042616103&partnerID=8YFLogxK

U2 - 10.1214/ECP.v8-1076

DO - 10.1214/ECP.v8-1076

M3 - Article

AN - SCOPUS:3042616103

VL - 8

SP - 112

EP - 121

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

SN - 1083-589X

ER -