  We study the long-time behavior of the dynamics of interacting planar
Brow-nian particles, confined by an external field and subject to a singular
pair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs
measure. For a special inverse temperature, it matches the Coulomb gas known as
the complex Ginibre ensemble. The difficulty comes from the interaction which
is not convex, in contrast with the case of one-dimensional log-gases
associated with the Dyson Brownian Motion. Despite the fact that the invariant
law is neither product nor log-concave, we show that the system is well-posed
for any inverse temperature and that Poincar{\'e} inequalities are available.
Moreover the second moment dynamics turns out to be a nice Cox -- Ingersoll --
Ross process in which the dependency over the number of particles leads to
identify two natural regimes related to the behavior of the noise and the speed
of the dynamics.
