Mean field games of controls: Propagation of monotonicities
Mean field games of controls: Propagation of monotonicities
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DOI：
10.3934/puqr.2022015
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发表时间：
2022-05
期刊：
影响因子：
--
通讯作者：
Chenchen Mou;Jianfeng Zhang
中科院分区：
文献类型：
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作者：
Chenchen Mou;Jianfeng Zhang
控制的平均场博弈理论考虑了一类通过状态和控制的联合分布进行交互的平均场博弈。众所周知，对于标准平均场对策，一定的单调性条件对于保证平均场平衡点的唯一性，进而保证主方程的全局适定性是至关重要的。在文献中，单调性条件可以是Lasry-Lions单调性、位移单调性或反单调性条件。本文研究了控制的平均域对策的这三种单调性条件，并给出了它们沿带共同噪声的主方程解的传播。特别地，我们将位移单调性推广到半单调性，其传播结果是新的，即使对于标准平均场对策也是如此。这是向控制平均场博弈主方程的全局适定性理论迈出的第一步。
The theory of Mean Field Game of Controls considers a class of mean field games where the interaction is through the joint distribution of the state and control. It is well known that, for standard mean field games, certain monotonicity conditions are crucial to guarantee the uniqueness of mean field equilibria and then the global wellposedness for master equations. In the literature the monotonicity condition could be the Lasry–Lions monotonicity, the displacement monotonicity, or the anti-monotonicity conditions. In this paper, we investigate these three types of monotonicity conditions for Mean Field Games of Controls and show their propagation along the solutions to the master equations with common noises. In particular, we extend the displacement monotonicity to semi-monotonicity, whose propagation result is new even for standard mean field games. This is the first step towards the global wellposedness theory for master equations of Mean Field Games of Controls.