卡诺环面上平均场博弈主方程的适定性
Well-posedness of the mean field game master equation on Carnot tori
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中文总结 AI 辅助
研究卡诺环面上二阶平均场博弈主方程的适定性，通过分析与之等价的退化平均场博弈系统，利用两类线性退化抛物方程及一类线性退化耦合系统解的正则性，推导主方程解的存在性。
中文摘要 AI 辅助
我们研究卡诺环面上二阶平均场博弈的主方程，即一般参与者只能沿着由生成卡诺群的向量场族给出的允许轨迹做周期性移动。作为次黎曼流形的例子，卡诺群是一类具有分层李代数结构的非交换群。为获得主方程的适定性，我们通过研究一个退化平均场博弈系统来分析其解的性质，该系统与主方程存在等价刻画。本文主要利用两类线性退化抛物方程和一类线性退化耦合系统解的正则性性质来推导主方程解的存在性。本文的研究受到文献[19CDLL,24MMM]的启发。
英文摘要
We study the master equation for a second-order mean field game on Carnot tori, which means the generic player can move periodically only along admissible trajectories given by the family of vector fields generating the Carnot group. As examples of sub-Riemannian manifolds, Carnot groups represent a type of non-commutative groups characterized by stratified Lie algebra structures. In order to obtain the well-posedness of the master equation, we analyze the properties of its solution by investigating a degenerate mean field game system for which there exists an equivalent characterization with the master equation. The main part of this paper lies in leveraging the regularity properties of solutions to two classes of linear degenerate parabolic equations and a class of linear degenerate coupled systems to derive the existence of solutions to the master equation. The research in this paper is motivated by \cite{19CDLL,24MMM}.
