{"title":"球面上的各向同性随机切向矢量场","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":"{\"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}","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
摘要
在本文中，我们通过交叉协方差描述了 d≥1 时 d 球上各向同性随机切向矢量场的特征，并推导出了它们的卡尔胡宁-洛夫展开（Karhunen-Loève expansion）。切向矢量场可以通过亥姆霍兹-霍奇分解分解为无卷曲部分和无发散部分。我们证明了这两部分在 2 球面上可以相关，而在 d≥3 的 d 球面上必须不相关。在 3 球面上，无发散部分可以进一步分解为两个各向同性流。
Isotropic random tangential vector fields on spheres
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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