  We consider mean field game systems in time-horizon $(0,T)$, where the
individual cost functional depends locally on the density distribution of the
agents, and the Hamiltonian is locally uniformly convex. We show that, even if
the coupling cost functions are mildly non-monotone, then the system is still
well posed due to the effect of individual noise. The rate of anti-monotonicity
(i.e. the aggregation rate of the cost functions) which can be afforded depends
on the intensity of the diffusion and on global bounds of solutions. We give
applications to either the case of globally Lipschitz Hamiltonians or the case
of quadratic Hamiltonians and couplings having mild growth.
  Under similar conditions, we investigate the long time behavior of solutions
and we give a complete description of the ergodic and long term properties of
the system. In particular we prove: (i) the turnpike property of solutions in
the finite (long) horizon $(0,T)$, (ii) the convergence of the system in
$(0,T)$ towards the system in $(0,\infty)$, (iii) the vanishing discount limit
of the infinite horizon problem and the long time convergence towards the
ergodic stationary solution.
  This way we extend previous results which were known only for the case of
monotone and smoothing couplings; our approach is self-contained and does not
need the use of the linearized system or of the master equation.
