{"title":"McKean-Vlasov SDEs在全局Lipschitz条件下的局部同胚性质","authors":"Xianjin Cheng, Zhenxin Liu","doi":"10.1016/j.matpur.2023.10.007","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we establish the locally diffeomorphic property of the solution to McKean-Vlasov stochastic differential equations<span> defined in the Euclidean space. Our approach is built upon the insightful ideas put forth by Kunita. We observe that although the coefficients satisfy the global Lipschitz condition and some suitable regularity condition, the solution in general does not satisfy the globally homeomorphic property at any time except the initial time, which sets McKean-Vlasov stochastic differential equations apart significantly from classical stochastic differential equations. Finally, we provide an example to complement our results.</span></p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"180 ","pages":"Pages 97-121"},"PeriodicalIF":2.1000,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The locally homeomorphic property of McKean-Vlasov SDEs under the global Lipschitz condition\",\"authors\":\"Xianjin Cheng, Zhenxin Liu\",\"doi\":\"10.1016/j.matpur.2023.10.007\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we establish the locally diffeomorphic property of the solution to McKean-Vlasov stochastic differential equations<span> defined in the Euclidean space. Our approach is built upon the insightful ideas put forth by Kunita. We observe that although the coefficients satisfy the global Lipschitz condition and some suitable regularity condition, the solution in general does not satisfy the globally homeomorphic property at any time except the initial time, which sets McKean-Vlasov stochastic differential equations apart significantly from classical stochastic differential equations. Finally, we provide an example to complement our results.</span></p></div>\",\"PeriodicalId\":51071,\"journal\":{\"name\":\"Journal de Mathematiques Pures et Appliquees\",\"volume\":\"180 \",\"pages\":\"Pages 97-121\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2023-11-07\",\"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/S0021782423001484\",\"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/S0021782423001484","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The locally homeomorphic property of McKean-Vlasov SDEs under the global Lipschitz condition
In this paper, we establish the locally diffeomorphic property of the solution to McKean-Vlasov stochastic differential equations defined in the Euclidean space. Our approach is built upon the insightful ideas put forth by Kunita. We observe that although the coefficients satisfy the global Lipschitz condition and some suitable regularity condition, the solution in general does not satisfy the globally homeomorphic property at any time except the initial time, which sets McKean-Vlasov stochastic differential equations apart significantly from classical stochastic differential equations. Finally, we provide an example to complement our results.
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.