论文标题
海森伯格集团的一阶平均野外游戏：周期性和非周期性案例
First order Mean Field Games in the Heisenberg group: periodic and non periodic case
论文作者
论文摘要
在本文中，我们在海森伯格集团〜$ \ he^1 $中研究了进化的一阶平均野外游戏；每个代理只能沿着“水平”轨迹移动，这些轨迹是根据生成〜$ \ he^1 $的向量场给出的，而成本的动力学部分仅取决于水平速度。在梯度期内，哈密顿量不是强制性的，并且连续性方程中一阶术语的系数可能在无穷大时具有二次增长。本文的主要结果是两个：前者是建立平均野外游戏系统的弱解决方案，而后者则在两个均值时期的均值范围内代表了均值的均值。不同的方法。为了获得这些结果，我们证明了一些具有自身兴趣的属性：二阶fokker-planck方程的唯一性结果和解决方案解决方案的概率表示。
In this paper we study evolutive first order Mean Field Games in the Heisenberg group~$\He^1$; each agent can move only along "horizontal" trajectories which are given in terms of the vector fields generating~$\He^1$ and the kinetic part of the cost depends only on the horizontal velocity. The Hamiltonian is not coercive in the gradient term and the coefficients of the first order term in the continuity equation may have a quadratic growth at infinity.The main results of this paper are two: the former is to establish the existence of a weak solution to the Mean Field Game system while the latter is to represent this solution following the Lagrangian formulation of the Mean Field Games.We shall tackle both the Heisenberg-periodic and the non periodic case following two different approaches. To get these results, we prove some properties which have their own interest: uniqueness results for a second order Fokker-Planck equation and a probabilistic representation of the solution to the continuity equation.\end{abstract}
