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 language | English |
|---|---|
| Pages (from-to) | 112-121 |
| Number of pages | 10 |
| Journal | Electronic Communications in Probability |
| Volume | 8 |
| DOIs | |
| Publication status | Published - 2003 Jan 1 |
| Externally published | Yes |
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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 journal › Article
}
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 -