We develop the counterpart of weak KAM theory for potential mean field games. This allows to describe the long time behavior of time-dependent potential mean field game systems. Our main result is the existence of a limit, as time tends to infinity, of the value function of an optimal control problem stated in the space of measures. In addition, we show a mean field limit for the ergodic constant associated with the corresponding Hamilton-Jacobi equation.

中文翻译：

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潜在MFG的弱KAM理论

我们为潜在的平均场博弈开发了弱 KAM 理论的对应物。这允许描述依赖于时间的潜在平均场博弈系统的长时间行为。我们的主要结果是存在一个极限，随着时间趋于无穷大，在度量空间中陈述的最优控制问题的价值函数存在一个极限。此外，我们还展示了与对应的 Hamilton-Jacobi 方程相关联的遍历常数的平均场极限。