The Derivation of Ergodic Mean Field Game Equations for Several Populations of Players

This note contains a detailed derivation of the equations of the recent mean field games theory (abbr. MFG), developed by M. Huang, P. E. Caines, and R. P. Malhame on one hand and by J.-M. Lasry and P.-L. Lions on the other, associated with a class of stochastic differential games, where the players belong to several populations, each of which consisting of a large number of similar and indistinguishable individuals, in the context of periodic diffusions and long-time-average (or ergodic) costs. After introducing a system of N Hamilton-Jacobi-Bellman (abbr. HJB) and N Kolmogorov-Fokker-Planck (abbr. KFP) equations for an N-player game belonging to such a class of games, the system of MFG equations (consisting of as many HJB equations, and of as many KFP equations as is the number of populations) is derived by letting the number of the members of each population go to infinity. For the sake of clarity and for reader's convenience, the case of a single population of players, as formulated in the work of J.-M. Lasry and P.-L. Lions, is presented first. The note slightly improves the results in this case too, by dealing with more general dynamics and costs.