  The Dirichlet forms related to various infinite systems of interacting
Brownian motions are studied. For a given random point field $ \mu $, there
exist two natural infinite-volume Dirichlet forms $
(\mathcal{E}^{\mathsf{upr}},\mathcal{D}^{\mathsf{upr}})$ and
$(\mathcal{E}^{\mathsf{lwr}},\mathcal{D}^{\mathsf{lwr}})$ on $ L^2(\mathsf{S}
,\mu ) $ describing interacting Brownian motions each with unlabeled
equilibrium state $ \mu $. The former is a decreasing limit of a scheme of such
finite-volume Dirichlet forms, and the latter is an increasing limit of another
scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an
extension of the former. We present a sufficient condition such that these two
Dirichlet forms are the same. In the first main theorem (Theorem 3.1) the
Markovian semi-group given by
$(\mathcal{E}^{\mathsf{lwr}},\mathcal{D}^{\mathsf{lwr}})$ is associated with a
natural infinite-dimensional stochastic differential equation (ISDE). In the
second main theorem (Theorem 3.2), we prove that these Dirichlet forms coincide
with each other by using the uniqueness of {\ws}s of ISDE. We apply Theorem 3.1
to stochastic dynamics arising from random matrix theory such as the sine,
Bessel, and Ginibre interacting Brownian motions and interacting Brownian
motions with Ruelle's class interaction potentials, and Theorem 3.2 to the
sine$ _2$ interacting Brownian motion and interacting Brownian motions with
Ruelle's class interaction potentials of $ C_0^3 $-class.
