  We consider stochastic differential games with $N$ players, linear-Gaussian
dynamics in arbitrary state-space dimension, and long-time-average cost with
quadratic running cost. Admissible controls are feedbacks for which the system
is ergodic. We first study the existence of affine Nash equilibria by means of
an associated system of $N$ Hamilton-Jacobi-Bellman and $N$
Kolmogorov-Fokker-Planck partial differential equations. We give necessary and
sufficient conditions for the existence and uniqueness of quadratic-Gaussian
solutions in terms of the solvability of suitable algebraic Riccati and
Sylvester equations. Under a symmetry condition on the running costs and for
nearly identical players we study the large population limit, $N$ tending to
infinity, and find a unique quadratic-Gaussian solution of the pair of Mean
Field Game HJB-KFP equations. Examples of explicit solutions are given, in
particular for consensus problems.
