控制平均场博弈的主方程：弱解概念的统一

中文摘要 AI 辅助

本文研究了控制平均场博弈的主方程，仅假设数据在测度变量上满足Lipschitz连续性。据此，我们提出了一个较弱的解概念，称为主方程的弱解。我们在Lasry-Lions单调性和位移$\lambda$-单调性条件下，分别建立了此类解的全局适定性。论证依赖于对关联的Pontryagin前向-后向随机微分方程系统的分析，尤其是测度变量上的稳定性。最后，我们回顾了文献中已有的几种非光滑解概念，即好解、弱粘性解、Lipschitz解、单调解，并考察了它们与我们弱解的关系。我们证明在适当假设下，所有这些解按定义都是等价的，因此本文为控制平均场博弈导出的主方程提供了弱解概念的统一。

英文摘要

In this manuscript, we study the master equation for mean field game of controls, assuming only that the data are Lipschitz continuous in the measure variable. Accordingly, we propose a weaker notion of solution called weak solutions for the master equation. We establish the global well-posedness of the master equation for such solution under the Lasry-Lions monotonicity and displacement $λ$-monotonicity conditions, respectively. The arguments rely on the analysis of the associated Pontryagin forward-backward stochastic differential equation system, especially the stability in the measure variable. Finally, we review several existing notions of non-smooth solution from the literature, namely good solutions, weak viscosity solutions, Lipschitz solutions, monotone solutions, and examine their relationship with our weak solution. We show that under appropriate assumptions all of them are equivalent by their definitions, and thus our manuscript provides a unification of weak solution notions for the master equations derived from mean field game of controls.

发表机构

• Shandong University（山东大学）
• City University of Hong Kong（香港城市大学）
• Dalian University of Technology（大连理工大学）

机构由 AI 辅助整理，请以论文原文为准。