{"title":"Fine Properties of Geodesics and Geodesic \\(\\lambda \\)-Convexity for the Hellinger–Kantorovich Distance","authors":"Matthias Liero, Alexander Mielke, Giuseppe Savaré","doi":"10.1007/s00205-023-01941-1","DOIUrl":null,"url":null,"abstract":"<div><p>We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger–Kantorovich problem (<span>\\(\\textsf{H}\\!\\!\\textsf{K}\\)</span>), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton–Jacobi equation arising in the dual dynamic formulation of <span>\\(\\textsf{H}\\!\\!\\textsf{K}\\)</span>, which are sufficiently strong to construct a characteristic transport-growth flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of <span>\\(\\textsf{H}\\!\\!\\textsf{K}\\)</span> geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic <span>\\(\\lambda \\)</span>-convexity with respect to the Hellinger–Kantorovich distance. Examples of geodesically convex functionals are provided.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6000,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-023-01941-1.pdf","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-023-01941-1","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 2
Abstract
We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger–Kantorovich problem (\(\textsf{H}\!\!\textsf{K}\)), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton–Jacobi equation arising in the dual dynamic formulation of \(\textsf{H}\!\!\textsf{K}\), which are sufficiently strong to construct a characteristic transport-growth flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of \(\textsf{H}\!\!\textsf{K}\) geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic \(\lambda \)-convexity with respect to the Hellinger–Kantorovich distance. Examples of geodesically convex functionals are provided.
期刊介绍:
The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.