Maha Mosaad A Alghamdi, Nikolai Leonenko, Andriy Olenko
{"title":"具有循环长程依赖性的随机 FPDE 的多尺度极限定理","authors":"Maha Mosaad A Alghamdi, Nikolai Leonenko, Andriy Olenko","doi":"arxiv-2409.09215","DOIUrl":null,"url":null,"abstract":"The paper studies solutions of stochastic partial differential equations with\nrandom initial conditions. First, it overviews some of the known results on\nscaled solutions of such equations and provides several explicit motivating\nexamples. Then, it proves multiscaling limit theorems for renormalized\nsolutions for the case of initial conditions subordinated to the random\nprocesses with cyclic long-range dependence. Two cases of stochastic partial\ndifferential equations are examined. The spectral and covariance\nrepresentations for the corresponding limit random fields are derived.\nAdditionally, it is discussed why analogous results are not valid for\nsubordinated cases with Hermite ranks greater than 1. Numerical examples that\nillustrate the obtained theoretical results are presented.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"46 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Multiscaling limit theorems for stochastic FPDE with cyclic long-range dependence\",\"authors\":\"Maha Mosaad A Alghamdi, Nikolai Leonenko, Andriy Olenko\",\"doi\":\"arxiv-2409.09215\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The paper studies solutions of stochastic partial differential equations with\\nrandom initial conditions. First, it overviews some of the known results on\\nscaled solutions of such equations and provides several explicit motivating\\nexamples. Then, it proves multiscaling limit theorems for renormalized\\nsolutions for the case of initial conditions subordinated to the random\\nprocesses with cyclic long-range dependence. Two cases of stochastic partial\\ndifferential equations are examined. The spectral and covariance\\nrepresentations for the corresponding limit random fields are derived.\\nAdditionally, it is discussed why analogous results are not valid for\\nsubordinated cases with Hermite ranks greater than 1. Numerical examples that\\nillustrate the obtained theoretical results are presented.\",\"PeriodicalId\":501245,\"journal\":{\"name\":\"arXiv - MATH - Probability\",\"volume\":\"46 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-13\",\"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-2409.09215\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09215","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Multiscaling limit theorems for stochastic FPDE with cyclic long-range dependence
The paper studies solutions of stochastic partial differential equations with
random initial conditions. First, it overviews some of the known results on
scaled solutions of such equations and provides several explicit motivating
examples. Then, it proves multiscaling limit theorems for renormalized
solutions for the case of initial conditions subordinated to the random
processes with cyclic long-range dependence. Two cases of stochastic partial
differential equations are examined. The spectral and covariance
representations for the corresponding limit random fields are derived.
Additionally, it is discussed why analogous results are not valid for
subordinated cases with Hermite ranks greater than 1. Numerical examples that
illustrate the obtained theoretical results are presented.