  We construct an analogue of Dyson Brownian motion in the Siegel half-space H
that we term Siegel Brownian motion. Given \beta in (0,\infty], a stochastic
flow for Z_t in H is introduced so that the law of the eigenvalues \lambda_t of
the cross ratio matrix R(Z_t,iI_n) is determined by the Ito differential
equation corresponds to stochastic gradient ascent of a function S. S turns out
to be the log volume of isospectral orbit in H and can be understood as a
Boltzmann entropy. In the limit \beta=\infty, the group orbits evolve by motion
by minus a half times mean curvature.
