Charles BertucciCMAP, Pierre Louis LionsCdF, CEREMADE
{"title":"An approximation of the squared Wasserstein distance and an application to Hamilton-Jacobi equations","authors":"Charles BertucciCMAP, Pierre Louis LionsCdF, CEREMADE","doi":"arxiv-2409.11793","DOIUrl":null,"url":null,"abstract":"We provide a simple $C^{1,1}$ approximation of the squared Wasserstein\ndistance on R^d when one of the two measures is fixed. This approximation\nconverges locally uniformly. More importantly, at points where the differential\nof the squared Wasserstein distance exists, it attracts the differentials of\nthe approximations at nearby points. Our method relies on the Hilbertian\nlifting of PL Lions and on the regularization in Hilbert spaces of Lasry and\nLions. We then provide an application of this result by using it to establish a\ncomparison principle for an Hamilton-Jacobi equation on the set of probability\nmeasures.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"30 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.11793","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We provide a simple $C^{1,1}$ approximation of the squared Wasserstein
distance on R^d when one of the two measures is fixed. This approximation
converges locally uniformly. More importantly, at points where the differential
of the squared Wasserstein distance exists, it attracts the differentials of
the approximations at nearby points. Our method relies on the Hilbertian
lifting of PL Lions and on the regularization in Hilbert spaces of Lasry and
Lions. We then provide an application of this result by using it to establish a
comparison principle for an Hamilton-Jacobi equation on the set of probability
measures.