AI 中文总结
研究测度空间上函数，提出垂直与水平导数等分析工具，应用于平均场博弈主方程和一阶哈密顿 - 雅可比 - 贝尔曼方程，建立主方程单调解性质，刻画平均场最优控制问题值函数。
AI 中文摘要
本文提出了用于研究定义在测度空间上函数的分析工具，并给出两个应用：概率测度集上的平均场博弈主方程和一阶哈密顿 - 雅可比 - 贝尔曼方程。第一部分引入了此类函数的几种可微性概念——垂直（或平坦）导数和水平导数，以及相关的凸性、单调性和次可微性概念。第二部分关注主方程，建立了单调解的唯一性和稳定性，并讨论了存在性。第三部分研究平均场最优控制问题，并将其值函数刻画为相关哈密顿 - 雅可比 - 贝尔曼方程的唯一粘性解。本文旨在让不熟悉测度空间的读者也能读懂。
英文摘要
This work presents analytical tools for studying functions defined on spaces of measures, together with two applications: mean field game master equations and first order Hamilton-Jacobi-Bellman equations on the set of probability measures. The first part introduces several notions of differentiability for such functions - vertical (or flat) and horizontal derivatives - as well as associated notions of convexity, monotonicity and sub-differentiability. The second part is concerned with master equations, for which uniqueness and stability of monotone solutions are established, and existence is discussed. The third part studies mean field optimal control problems and characterizes their value functions as the unique viscosity solutions of the associated Hamilton-Jacobi-Bellman equations. This work is intended to be accessible to readers unfamiliar with spaces of measures.
