  We introduce an elliptic extension of Dyson's Brownian motion model, which is
a temporally inhomogeneous diffusion process of noncolliding particles defined
on a circle. Using elliptic determinant evaluations related to the reduced
affine root system of types $A$, we give determinantal martingale
representation (DMR) for the process, when it is started at the configuration
with equidistant spacing on the circle. DMR proves that the process is
determinantal and the spatio-temporal correlation kernel is determined. By
taking temporally homogeneous limits of the present elliptic determinantal
process, trigonometric and hyperbolic versions of noncolliding diffusion
processes are studied.
