论文标题
一维一阶平均现场游戏和计划问题的规律性和长时间行为
Regularity and long time behavior of one-dimensional first-order mean field games and the planning problem
论文作者
论文摘要
我们研究了一维，本地，一阶平均野外游戏系统和计划问题的规律性和长时间行为，假设高等教育的汉密尔顿人具有超线性增长的哈密顿式，则没有分离，严格单调对密度的依赖性。我们通过获得两个规律性结果来改善现有文献。首先是经典解决方案的存在，而无需承担附近密度附近的成本函数。第二个结果是弱解决方案的内部平滑度，而无需假设成本函数的爆炸，也不需要将初始密度远离零。我们还表征了解决方案的长时间行为，证明了它们以指数的收敛速率满足收费公路的特性，并且它们会收敛到无限地平线系统的溶液。我们的方法取决于系统的椭圆结构和位移凸度估计。特别是，我们应用位移凸方法来获得整体和局部a在密度上的先验下限。
We study the regularity and long time behavior of the one-dimensional, local, first-order mean field games system and the planning problem, assuming a Hamiltonian of superlinear growth, with a non-separated, strictly monotone dependence on the density. We improve upon the existing literature by obtaining two regularity results. The first is the existence of classical solutions without the need to assume blow-up of the cost function near small densities. The second result is the interior smoothness of weak solutions without the need to assume neither blow-up of the cost function nor that the initial density be bounded away from zero. We also characterize the long time behavior of the solutions, proving that they satisfy the turnpike property with an exponential rate of convergence, and that they converge to the solution of the infinite horizon system. Our approach relies on the elliptic structure of the system and displacement convexity estimates. In particular, we apply displacement convexity methods to obtain both global and local a priori lower bounds on the density.
