  We investigate the construction of diffusions consisting of infinitely
numerous Brownian particles moving in $\mathbb{R}^d$ and interacting via
logarithmic functions (two-dimensional Coulomb potentials). These potentials
are very strong and act over a long range in nature. The associated equilibrium
states are no longer Gibbs measures. We present general results for the
construction of such diffusions and, as applications thereof, construct two
typical interacting Brownian motions with logarithmic interaction potentials,
namely the Dyson model in infinite dimensions and Ginibre interacting Brownian
motions. The former is a particle system in $\mathbb{R}$, while the latter is
in $\mathbb{R}^2$. Both models are translation and rotation invariant in space,
and as such, are prototypes of dimensions $d=1,2$, respectively. The
equilibrium states of the former diffusion model are determinantal or Pfaffian
random point fields with sine kernels. They appear in the thermodynamical
limits of the spectrum of the ensembles of Gaussian random matrices such as
GOE, GUE and GSE. The equilibrium states of the latter diffusion model are the
thermodynamical limits of the spectrum of the ensemble of complex non-Hermitian
Gaussian random matrices known as the Ginibre ensemble.
