  We consider stochastic differential games with $N$ nearly identical players,
linear-Gaussian dynamics, and infinite horizon discounted quadratic 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, proving that for small discount factors
quadratic-Gaussian solutions exist and are unique. Then, we prove the
convergence of such solutions to the unique quadratic-Gaussian solution of the
pair of Mean Field equations. We also discuss some singular limits, such as
vanishing noise, cheap control and vanishing discount.
