"Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems",
As an extension of the theory of Dyson's Brownian motion models for the standard
Gaussian random-matrix ensembles, we report a systematic study of hermitian
matrix-valued processes and their eigenvalue processes associated with the
chiral and nonstandard random-matrix ensembles. In addition to the noncolliding
Brownian motions, we introduce a one-parameter family of temporally homogeneous
noncolliding systems of the Bessel processes and a two-parameter family of
temporally inhomogeneous noncolliding systems of Yor's generalized meanders
and show that all of the ten classes of eigenvalue statistics in the
Altland-Zirnbauer classification are realized as particle distributions
in the special cases of these diffusion particle systems. As a corollary
of each equivalence in distribution of a temporally inhomogeneous eigenvalue
process and a noncolliding diffusion process, a stochastic-calculus proof of
a version of the Harish-Chandra (Itzykson-Zuber) formula of integral over
unitary group is established.
