  Circular Brownian motion models of random matrices were introduced by Dyson
and describe the parametric eigenparameter correlations of unitary random
matrices. For symmetric unitary, self-dual quaternion unitary and an analogue
of antisymmetric hermitian matrix initial conditions, Brownian dynamics toward
the unitary symmetry is analyzed. The dynamical correlation functions of
arbitrary number of Brownian particles at arbitrary number of times are shown
to be written in the forms of quaternion determinants, similarly as in the case
of hermitian random matrix models.
