无限维的平均场对策

IF 2.3 1区数学 Q1 MATHEMATICS

Journal de Mathematiques Pures et Appliquees Pub Date : 2025-08-28 DOI:10.1016/j.matpur.2025.103780

Salvatore Federico , Fausto Gozzi , Andrzej Święch

{"title":"无限维的平均场对策","authors":"Salvatore Federico , Fausto Gozzi , Andrzej Święch","doi":"10.1016/j.matpur.2025.103780","DOIUrl":null,"url":null,"abstract":"<div><div>We study a Mean Field Games (MFG) system in a real, separable infinite dimensional Hilbert space. The system consists of a second order parabolic type equation, called Hamilton-Jacobi-Bellman <span><span>(<strong>HJB</strong>)</span></span> equation in the paper, coupled with a nonlinear Fokker-Planck <span><span>(<strong>FP</strong>)</span></span> equation. Both equations contain a Kolmogorov operator. Solutions to the HJB equation are interpreted in the mild solution sense and solutions to the FP equation are interpreted in an appropriate weak sense. We prove well-posedness of the considered MFG system under certain conditions. The existence of a solution to the MFG system is proved using Tikhonov's fixed point theorem in a proper space. Uniqueness of solutions is obtained under typical separability and Lasry-Lions type monotonicity conditions.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103780"},"PeriodicalIF":2.3000,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On mean field games in infinite dimension\",\"authors\":\"Salvatore Federico , Fausto Gozzi , Andrzej Święch\",\"doi\":\"10.1016/j.matpur.2025.103780\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We study a Mean Field Games (MFG) system in a real, separable infinite dimensional Hilbert space. The system consists of a second order parabolic type equation, called Hamilton-Jacobi-Bellman <span><span>(<strong>HJB</strong>)</span></span> equation in the paper, coupled with a nonlinear Fokker-Planck <span><span>(<strong>FP</strong>)</span></span> equation. Both equations contain a Kolmogorov operator. Solutions to the HJB equation are interpreted in the mild solution sense and solutions to the FP equation are interpreted in an appropriate weak sense. We prove well-posedness of the considered MFG system under certain conditions. The existence of a solution to the MFG system is proved using Tikhonov's fixed point theorem in a proper space. Uniqueness of solutions is obtained under typical separability and Lasry-Lions type monotonicity conditions.</div></div>\",\"PeriodicalId\":51071,\"journal\":{\"name\":\"Journal de Mathematiques Pures et Appliquees\",\"volume\":\"205 \",\"pages\":\"Article 103780\"},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2025-08-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal de Mathematiques Pures et Appliquees\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021782425001242\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782425001242","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

研究了可分离无限维实数Hilbert空间中的平均场对策系统。该系统由一个二阶抛物型方程（本文称为Hamilton-Jacobi-Bellman （HJB）方程）和一个非线性Fokker-Planck （FP）方程组成。两个方程都包含一个Kolmogorov算子。HJB方程的解在弱解意义上得到解释，FP方程的解在弱解意义上得到解释。在一定条件下证明了所考虑的MFG系统的适定性。利用Tikhonov不动点定理，证明了MFG系统解的存在性。在典型可分性和Lasry-Lions型单调性条件下，得到了解的唯一性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

On mean field games in infinite dimension

We study a Mean Field Games (MFG) system in a real, separable infinite dimensional Hilbert space. The system consists of a second order parabolic type equation, called Hamilton-Jacobi-Bellman (HJB) equation in the paper, coupled with a nonlinear Fokker-Planck (FP) equation. Both equations contain a Kolmogorov operator. Solutions to the HJB equation are interpreted in the mild solution sense and solutions to the FP equation are interpreted in an appropriate weak sense. We prove well-posedness of the considered MFG system under certain conditions. The existence of a solution to the MFG system is proved using Tikhonov's fixed point theorem in a proper space. Uniqueness of solutions is obtained under typical separability and Lasry-Lions type monotonicity conditions.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Journal de Mathematiques Pures et Appliquees 数学-数学

CiteScore

4.30

自引率

0.00%

发文量

审稿时长

6 months

期刊介绍： Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.