Homogenization of the backward-forward meanfield games systems in periodic environments

We study the homogenization properties in the small viscosity limit and in periodic environments of the (viscous) backward-forward mean-field games system. We consider separated Hamiltonians and provide results for systems with (i) "smoothing" coupling and general initial and terminal data, and (ii) with "local coupling" but well-prepared data.The limit is a first-order forward-backward system. In the nonlocal coupling case, the averaged system is of mfg-type, which is well-posed in some cases. For the problems with local coupling, the homogenization result is proved assuming that the formally obtained limit system has smooth solutions with well prepared initial and terminal data. It is also shown, using a very general example (potential mfg), that the limit system is not necessarily of mfg-type.