Here, we study the existence and the convergence of solutions for the vanishing discount MFG problem with a quadratic Hamiltonian. We give conditions under which the discounted problem has a unique classical solution and prove convergence of the vanishing-discount limit to a unique solution up to constants. Then, we establish refined asymptotics for the limit. When those conditions do not hold, the limit problem may not have a unique solution and its solutions may not be smooth, as we illustrate in an elementary example. Finally, we investigate the stability of regular weak solutions and address the selection problem. Using ideas from Aubry-Mather theory, we establish a selection criterion for the limit.

中文翻译：

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一阶平稳Mean-field博弈的选择问题

在这里，我们研究了带有二次哈密顿量的消失的折扣MFG问题的解的存在性和收敛性。我们给出了折现问题具有唯一经典解决方案的条件，并证明了消失贴现限额收敛到常数的唯一解决方案。然后，我们为极限建立精炼渐近线。如我们在一个基本示例中所示，当这些条件不成立时，极限问题可能不会具有唯一的解决方案，并且其解决方案可能不会很顺利。最后，我们研究了常规弱解的稳定性并解决了选择问题。利用Aubry-Mather理论的思想，我们为极限建立了选择标准。