  We solve infinite-dimensional stochastic differential equations (ISDEs)
describing an infinite number of Brownian particles interacting via
two-dimensional Coulomb potentials. The equilibrium states of the associated
unlabeled stochastic dynamics are the Ginibre random point field and Dyson's
measures, which appear in random matrix theory. To solve the ISDEs we establish
an integration by parts formula for these measures. Because the long-range
effect of two-dimensional Coulomb potentials is quite strong, the properties of
Brownian particles interacting with two-dimensional Coulomb potentials are
remarkably different from those of Brownian particles interacting with Ruelle's
class interaction potentials. As an example, we prove that the interacting
Brownian particles associated with the Ginibre random point field satisfy
plural ISDEs.
