{"title":"Isotropic random tangential vector fields on spheres","authors":"Tianshi Lu","doi":"10.1016/j.spl.2024.110172","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we characterized isotropic random tangential vector fields on <span><math><mi>d</mi></math></span>-spheres for <span><math><mrow><mi>d</mi><mo>≥</mo><mn>1</mn></mrow></math></span> by the cross-covariance, and derived their Karhunen–Loève expansion. The tangential vector field can be decomposed into a curl-free part and a divergence-free part by the Helmholtz–Hodge decomposition. We proved that the two parts can be correlated on a 2-sphere, while they must be uncorrelated on a <span><math><mi>d</mi></math></span>-sphere for <span><math><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. On a 3-sphere, the divergence-free part can be further decomposed into two isotropic flows.</p></div>","PeriodicalId":49475,"journal":{"name":"Statistics & Probability Letters","volume":"213 ","pages":"Article 110172"},"PeriodicalIF":0.7000,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Statistics & Probability Letters","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S016771522400141X","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/6/3 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we characterized isotropic random tangential vector fields on -spheres for by the cross-covariance, and derived their Karhunen–Loève expansion. The tangential vector field can be decomposed into a curl-free part and a divergence-free part by the Helmholtz–Hodge decomposition. We proved that the two parts can be correlated on a 2-sphere, while they must be uncorrelated on a -sphere for . On a 3-sphere, the divergence-free part can be further decomposed into two isotropic flows.
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