关于控制单调均场博弈主方程的无条件适定性
Unconditional well-posedness of the master equation for monotone mean field games of controls
AI总结：
本文研究了均场博弈主方程的无条件适定性，通过N玩家纳什系统解的紧性证明了主方程的解存在性，适用于位移单调或拉斯-洛恩斯单调数据及共同噪声。
摘要：
我们建立了与一般类均场博弈控制相关主方程的第一个无条件适定性结果。我们的分析涵盖了位移单调或拉斯-洛恩斯单调数据的游戏，以及具有小时间跨度的游戏。所谓无条件，意味着所有假设仅在拉格朗日函数和终端成本的层面施加。特别是，我们不需要任何先验正则性或结构性假设，对额外的固定点映射不进行假设；相反，我们证明这些固定点映射是良好的，因为数据的正则性和单调性。我们的方法是自下而上的，不同于大多数先前结果依赖于广义特征方法。特别是，我们通过证明对应的N玩家纳什系统解的紧性（在适当的意义上）以及它们的子序列极限点必须是主方程的解，来构建主方程的经典解。紧性通过对于N玩家价值函数导数的均匀-in-N衰减估计获得。底层游戏由非退化异质布朗运动驱动，我们的结果允许存在强度恒定的共同噪声。
英文摘要：
We establish the first unconditional well-posedness result for the master equation associated with a general class of mean field games of controls. Our analysis covers games with displacement monotone or Lasry--Lions monotone data, as well as those with a small time horizon. By unconditional, we mean that all assumptions are imposed solely at the level of the Lagrangian and the terminal cost. In particular, we do not require any a priori regularity or structural assumptions on the additional fixed-point mappings arising from the control interactions; instead we show that these fixed-point mappings are well-behaved as a consequence of the regularity and the monotonicity of the data. Our approach is bottom-up in nature, unlike most previous results which rely on a generalized method of characteristics. In particular, we build a classical solution of the master equation by showing that the solutions of the corresponding $N$-player Nash systems are compact, in an appropriate sense, and that their subsequential limit points must be solutions to the master equation. Compactness is obtained via uniform-in-$N$ decay estimates for derivatives of the $N$-player value functions. The underlying games are driven by non-degenerate idiosyncratic Brownian noise, and our results allow for the presence of common noise with constant intensity.
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