  We present general theorems solving the long-standing problem of the
existence and pathwise uniqueness of strong solutions of infinite-dimensional
stochastic differential equations (ISDEs) called interacting Brownian motions.
These ISDEs describe the dynamics of infinite-many Brownian particles moving in
$ \mathbb{R}^d $ with free potential $ \Phi $ and mutual interaction potential
$ \Psi $.
  We apply the theorems to essentially all interaction potentials of Ruelle's
class such as the Lennard-Jones 6-12 potential and Riesz potentials, and to
logarithmic potentials appearing in random matrix theory. We solve ISDEs of the
Ginibre interacting Brownian motion and the sine$_{\beta}$ interacting Brownian
motion with $ \beta = 1,2,4$. We also use the theorems in separate papers for
the Airy and Bessel interacting Brownian motions. One of the critical points
for proving the general theorems is to establish a new formulation of solutions
of ISDEs in terms of tail $ \sigma $-fields of labeled path spaces consisting
of trajectories of infinitely many particles. These formulations are equivalent
to the original notions of solutions of ISDEs, and more feasible to treat in
infinite dimensions.
