{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T04:39:16Z","timestamp":1787027956488,"version":"build-2736575974"},"reference-count":38,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,14]],"date-time":"2020-02-14T00:00:00Z","timestamp":1581638400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["ARC 160DP101366"],"award-info":[{"award-number":["ARC 160DP101366"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper investigates solutions of hyperbolic diffusion equations in     R 3     with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere     S 2     are studied. All assumptions are formulated in terms of the angular power spectrum or the spectral measure of the random initial conditions. Approximations to the exact solutions are given. Upper bounds for the mean-square convergence rates of the approximation fields are obtained. The smoothness properties of the exact solution and its approximation are also investigated. It is demonstrated that the H\u00f6lder-type continuity of the solution depends on the decay of the angular power spectrum. Conditions on the spectral measure of initial conditions that guarantee short- or long-range dependence of the solutions are given. Numerical studies are presented to verify the theoretical findings.<\/jats:p>","DOI":"10.3390\/e22020217","type":"journal-article","created":{"date-parts":[[2020,2,18]],"date-time":"2020-02-18T10:10:25Z","timestamp":1582020625000},"page":"217","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Spherically Restricted Random Hyperbolic Diffusion"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1784-5344","authenticated-orcid":false,"given":"Philip","family":"Broadbridge","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, La Trobe University, Melbourne VIC 3086, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7078-0116","authenticated-orcid":false,"given":"Alexander","family":"Kolesnik","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Computer Science, Academy Street 5, 2028 Kishinev, Moldova"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1932-4091","authenticated-orcid":false,"given":"Nikolai","family":"Leonenko","sequence":"additional","affiliation":[{"name":"School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0917-7000","authenticated-orcid":false,"given":"Andriy","family":"Olenko","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, La Trobe University, Melbourne VIC 3086, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3712-8449","authenticated-orcid":false,"given":"Dareen","family":"Omari","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, La Trobe University, Melbourne VIC 3086, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s00477-007-0157-7","article-title":"Spatiotemporal Random Fields Associated with Stochastic Fractional Helmholtz and Heat Equations","volume":"22","author":"Angulo","year":"2008","journal-title":"Stoch. 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