具有非单调相互作用的确定性均场博弈的长期行为

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Mathematical Analysis Pub Date : 2024-07-15 DOI:10.1137/23m1608100

Martino Bardi, Hicham Kouhkouh

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引用次数: 0

摘要

SIAM 数学分析期刊》，第 56 卷第 4 期，第 5079-5098 页，2024 年 8 月。摘要。我们考虑的是全欧几里得空间中的确定性均场博弈（MFGs），其代价函数相对于代理人的分布是连续的，并且在一个紧凑集合中达到其最小值。我们首先证明了具有这种代价的静态 MFG 有一个均衡点，并由此建立了具有相同代价的一阶 PDE 的遍历 MFG 系统的解。接下来，我们将讨论成本函数满足各种附加假设（但不包括经典的 Lasry-Lions 单调性条件）的有限视界 MFG 解的长期极限。相反，我们假设成本对于所有描述群体的度量都有相同的最小值集。我们证明了当博弈的视界趋于无穷大时，代理人的分布和价值函数收敛于遍历 MFG 系统的解，并将弱 KAM 理论的一些结果扩展到这一类 MFGs。

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Long-Time Behavior of Deterministic Mean Field Games with Nonmonotone Interactions

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5079-5098, August 2024.
Abstract. We consider deterministic mean field games (MFGs) in all Euclidean space with a cost functional continuous with respect to the distribution of the agents and attaining its minima in a compact set. We first show that the static MFG with such a cost has an equilibrium, and we build from it a solution of the ergodic MFG system of first order PDEs with the same cost. Next we address the long-time limit of the solutions to finite horizon MFGs with cost functional satisfying various additional assumptions, but not the classical Lasry–Lions monotonicity condition. Instead we assume that the cost has the same set of minima for all measures describing the population. We prove the convergence of the distribution of the agents and of the value function to a solution of the ergodic MFG system as the horizon of the game tends to infinity, extending to this class of MFGs some results of weak KAM theory.

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来源期刊

SIAM Journal on Mathematical Analysis 数学-应用数学

CiteScore

3.30

自引率

5.00%

发文量

175

审稿时长

12 months

期刊介绍： SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.