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 mean field games-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 mean field games), that the limit system is not necessarily of mean field games-type. A complete theory for such systems was developed by the authors in [28].

中文翻译：

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周期环境下均值后向博弈系统的同质化

我们研究在（粘度）后向均值场博弈系统的小粘度极限和周期性环境中的均质性。我们考虑分离的哈密顿量，并为具有（i）“平滑”耦合和一般初始和最终数据，以及（ii）具有“局部耦合”但准备好的数据的系统提供结果。该限制是一阶向前-向后系统。在非局部耦合的情况下，平均系统是平均场博弈类型的，在某些情况下它的位置很好。对于局部耦合的问题，假设正式获得的极限系统具有平滑的解且具有良好的初始和最终数据，则证明了均质化结果。还通过一个非常一般的示例（潜在的平均场博弈）表明，限额系统不一定是平均场博弈类型的。