论文标题
潜在平均野外游戏的主方程的薄弱解决方案
Weak solutions to the master equation of potential mean field games
论文作者
论文摘要
这项工作的目的是将弱解决方案的概念引入潜在的平均野外游戏的主方程，并证明在相当一般的假设下存在的存在和唯一性。值得注意的是，这是在没有任何单调性限制的情况下实现的。关键点是要以保守的意义来解释主方程，然后适应源自汉密尔顿 - 雅各比 - 贝尔曼方程的双曲线系统的无限尺寸设置。在这里，主方程确实被认为是概率度量空间上的无限维度系统，并正式写作是汉密尔顿 - 雅各比 - 贝尔曼方程的衍生物，与均值野外游戏上方的平均场控制问题相关联。为了使分析更容易，我们假设系数是周期性的，它允许通过其傅立叶系数表示概率度量。然后，大多数分析都在于重写主方程和对均方根控制问题的相应汉密尔顿 - 雅各布 - 贝尔曼方程，作为傅立叶系数本身设置的部分微分方程。最后，我们建立了函数的存在和独特性，这些功能是量度参数中排量半循环的函数，并以适当的广义意义上的汉密尔顿 - 雅各比 - 贝尔曼方程来求解汉密尔顿 - 雅各比 - 贝尔曼方程，并且随后，我们获得了适当的弱理性的函数的存在和独特性，并以适当的弱弱的弱度和较弱的单向嘴唇lipschitz inforphitz in eque sore solfection sollose sollection。作为另一个新结果，我们还证明，相关平均场控制问题的最佳轨迹对于几乎每个起点都是唯一的，对于概率度量的空间进行了适当的概率度量。
The purpose of this work is to introduce a notion of weak solution to the master equation of a potential mean field game and to prove that existence and uniqueness hold under quite general assumptions. Remarkably, this is achieved without any monotonicity constraint on the coefficients. The key point is to interpret the master equation in a conservative sense and then to adapt to the infinite dimensional setting earlier arguments for hyperbolic systems deriving from a Hamilton-Jacobi-Bellman equation. Here, the master equation is indeed regarded as an infinite dimensional system set on the space of probability measures and is formally written as the derivative of the Hamilton-Jacobi-Bellman equation associated with the mean field control problem lying above the mean field game. To make the analysis easier, we assume that the coefficients are periodic, which allows to represent probability measures through their Fourier coefficients. Most of the analysis then consists in rewriting the master equation and the corresponding Hamilton-Jacobi-Bellman equation for the mean field control problem as partial differential equations set on the Fourier coefficients themselves. In the end, we establish existence and uniqueness of functions that are displacement semi-concave in the measure argument and that solve the Hamilton-Jacobi-Bellman equation in a suitable generalized sense and, subsequently, we get existence and uniqueness of functions that solve the master equation in an appropriate weak sense and that satisfy a weak one-sided Lipschitz inequality. As another new result, we also prove that the optimal trajectories of the associated mean field control problem are unique for almost every starting point, for a suitable probability measure on the space of probability measures.
