  We prove the sets of polynomials on configuration spaces are cores of
Dirichlet forms describing interacting Brownian motion in infinite dimensions.
Typical examples of these stochastic dynamics are Dyson's Brownian motion and
Airy interacting Brownian motion. Both particle systems have logarithmic
interaction potentials, and naturally arise from random matrix theory. The
results of the present paper will be used in a forth coming paper to prove the
identity of the infinite-dimensional stochastic dynamics related to the random
matrix theories constructed by apparently different methods: the method of
space-time correlation functions and that of stochastic analysis.
