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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 firststudy the existence of affine Nash equilibria by means of an associated systemof $N$ Hamilton-Jacobi-Bellman and $N$ Kolmogorov-Fokker-Planck partialdifferential equations, proving that for small discount factorsquadratic-Gaussian solutions exist and are unique. Then, we prove theconvergence of such solutions to the unique quadratic-Gaussian solution of thepair of Mean Field equations. We also discuss some singular limits, such asvanishing noise, cheap control and vanishing discount.

Linear-Quadratic $N$-person and Mean-Field Games: Infinite Horizon Games with Discounted Cost and Singular Limits