论文标题
二阶本地最少的平均现场比赛
Second order local minimal-time Mean Field Games
论文作者
论文摘要
本文考虑了在平均野外游戏（MFG）模型中产生的抛物线PDE的前向后系统的系统，在该模型中，每个代理都控制轨迹的漂移，但要逃脱给定有限的域$ω$，以最小的预期时间逃脱给定的有限域$ω$。根据其他代理在其位置的密度，代理受到漂移的约束。有限时间范围的存在$ t $是通过固定点参数证明的，但是这个问题的自然设置在无限的时间范围内。需要估计来处理极限$ t \ to \ infty $，并且还研究了以这种方式获得的解决方案的渐近行为。这通过了MFG的经典抛物线参数和特定计算。代理密度和汉密尔顿 - jacobi-bellman方程的fokker - planck方程在值函数上显示了dirichlet边界条件，这是因为代理到达$ \ \partialΩ$，这一事实立即停止。给出了密度的初始基准，并且值函数的长时间限制的特征是固定问题的解决方案。
The paper considers a forward-backward system of parabolic PDEs arising in a Mean Field Game (MFG) model where every agent controls the drift of a trajectory subject to Brownian diffusion, trying to escape a given bounded domain $Ω$ in minimal expected time. Agents are constrained by a bound on the drift depending on the density of other agents at their location. Existence for a finite time horizon $T$ is proven via a fixed point argument, but the natural setting for this problem is in infinite time horizon. Estimates are needed to treat the limit $T\to\infty$, and the asymptotic behavior of the solution obtained in this way is also studied. This passes through classical parabolic arguments and specific computations for MFGs. Both the Fokker--Planck equation on the density of agents and the Hamilton--Jacobi--Bellman equation on the value function display Dirichlet boundary conditions as a consequence of the fact that agents stop as soon as they reach $\partialΩ$. The initial datum for the density is given, and the long-time limit of the value function is characterized as the solution of a stationary problem.
