论文标题
带有choquard类型的奇异均值野外游戏
Ergodic Mean-Field Games with aggregation of Choquard-type
论文作者
论文摘要
我们考虑整个空间中的二阶均值平均游戏系统$ \ mathbb {r}^n $具有强制性潜力和汇总非本地耦合，该耦合根据Riesz的交互内核来定义。这些MFG系统描述了纳什均衡游戏的平衡，其中大量无法区分的理性参与者吸引了人口高度分布的地区。平衡求解了PDES系统，其中汉密尔顿 - 雅各比 - 贝尔曼方程与kolmogorov-fokker-planck方程相结合，用于质量分布。由于有吸引力的项的强度与扩散部分的行为之间的相互作用，我们将获得三种不同的制度，以实现MFG系统的经典解决方案的存在和不存在。通过POHOZAEV型身份，我们证明了对MFG系统的常规解决方案不存在，而在Hardy-Littlewood-sobolev-Supergilitical危害中没有潜力。另一方面，使用固定点参数，我们显示了至少小于给定阈值值的质量，至少对于较小的质量而言，经典解决方案的存在。在质量临界制度中，我们表明实际上该阈值可以为$+\ infty $。
We consider second-order ergodic Mean-Field Games systems in the whole space $\mathbb{R}^N$ with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. These MFG systems describe Nash equilibria of games with a large population of indistinguishable rational players attracted toward regions where the population is highly distributed. Equilibria solve a system of PDEs where an Hamilton-Jacobi-Bellman equation is combined with a Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for the existence and non existence of classical solutions to the MFG system. By means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential in the Hardy-Littlewood-Sobolev-supercritical regime. On the other hand, using a fixed point argument, we show existence of classical solutions in the Hardy-Littlewood-Sobolev-subcritical regime at least for masses smaller than a given threshold value. In the mass-subcritical regime we show that actually this threshold can be taken to be $+\infty$.
