"Noncolliding Brownian motions and Harish-Chandra formula",
( with Makoto Katori)
Elect. Comm. in Probab. 8(2003), 112-121.
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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.
