Time dependent mean-field games in the superquadratic case
Time dependent mean-field games in the superquadratic case
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超二次案例中的时间相关平均场博弈
DOI：
10.1051/cocv/2015029
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
H'ector S'anchez
中科院分区：
文献类型：
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作者：
D. Gomes;Edgard A. Pimentel;H'ector S'anchez
我们研究了具有超二次哈密顿算子和幂依赖于测度的时间依赖平均场博弈。这些问题构成了重大的数学挑战，因为在次二次情况下使用的关键技术没有扩展到超二次设置。由于超二次结构的哈密顿量，Lipschitz估计的Hamilton-Jacobi方程的解决方案，通过一套新的技术。这些探索的抛物线性质的问题，通过非线性伴随方法。将Hamilton-Jacobi方程的Lipschitz正则性与Fokker-Planck方程解的多项式估计相结合，证明了该方程的适定性.经典解的存在性，然后可以证明的条件下，只依赖于增长的哈密顿量和尺寸。我们的结果也增加了目前的理解超二次Hamilton-Jacobi方程。
We investigate time-dependent mean-field games with superquadratic Hamiltonians and a power dependence on the measure. Such problems pose substantial mathematical challenges as the key techniques used in the subquadratic case do not extend to the superquadratic setting. Because of the superquadratic structure of the Hamiltonian, Lipschitz estimates for the solutions of the Hamilton-Jacobi equation are obtained through a novel set of techniques. These explore the parabolic nature of the problem through the non-linear adjoint method. Well-posedness is proved by combining Lipschitz regularity for the Hamilton-Jacobi equation with polynomial estimates for solutions of the Fokker-Planck equation. Existence of classical solutions can then be proved under conditions depending only on the growth of the Hamiltonian and the dimension. Our results also add to the current understanding of superquadratic Hamilton-Jacobi equations.