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Viscosity solutions methods are used to pass to the limit in some penalization problems for rst order and second order, degenerate parabolic, Hamilton-Jacobi-Bellman equations. This characterizes the limit of the value functions of singularly perturbed optimal control problems for nonlinear deterministic systems and controlled degenerate di usions, respectively. The results cover also cases where the usual order reduction method does not give the correct limit, and di usion processes with fast state variables depending nonlinearly on the control. Some connections with ergodic control and periodic homogenization are discussed.

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Chapter 6. Controllable fast variables 6.1. Bounded-time controllability and ergodicity 6.2. Stabilization and a formula for the effective initial data 6.3. An explicit formula for the effective Hamiltonian and the limit differential game 6.4. Uniform convergence 6.5. The reduction order formula for the effective control problem Chapter 7. Nonresonant fast variables 7.1. Ergodicity 7.2. Stabilization 7.3. Uniform convergence Chapter 8. A counterexample to uniform convergence Chapter 9. Applications to homogenization 9.1. Periodic homogenization of 1st order H-J equations v vi CONTENTS 9.2. Periodic homogenization of 2nd order equations Bibliography 73

We consider N -person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Plank equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number N of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games [19]. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter. We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.