{"title":"Fractional Operators and Fractionally Integrated Random Fields on Zν","authors":"Vytautė Pilipauskaitė, D. Surgailis","doi":"10.3390/fractalfract8060353","DOIUrl":null,"url":null,"abstract":"We consider fractional integral operators (I−T)d,d∈(−1,1) acting on functions g:Zν→R,ν≥1, where T is the transition operator of a random walk on Zν. We obtain the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels τ(s;d),s∈Zν of (I−T)d. The asymptotic behavior of τ(s;d) as |s|→∞ is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on Zν solving the difference equation (I−T)dX=ε with white noise on the right-hand side is discussed and their scaling limits. Several examples, including fractional lattice Laplace and heat operators, are studied in detail.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":null,"pages":null},"PeriodicalIF":3.6000,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fractal and Fractional","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3390/fractalfract8060353","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider fractional integral operators (I−T)d,d∈(−1,1) acting on functions g:Zν→R,ν≥1, where T is the transition operator of a random walk on Zν. We obtain the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels τ(s;d),s∈Zν of (I−T)d. The asymptotic behavior of τ(s;d) as |s|→∞ is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on Zν solving the difference equation (I−T)dX=ε with white noise on the right-hand side is discussed and their scaling limits. Several examples, including fractional lattice Laplace and heat operators, are studied in detail.
期刊介绍：
Fractal and Fractional is an international, scientific, peer-reviewed, open access journal that focuses on the study of fractals and fractional calculus, as well as their applications across various fields of science and engineering. It is published monthly online by MDPI and offers a cutting-edge platform for research papers, reviews, and short notes in this specialized area. The journal, identified by ISSN 2504-3110, encourages scientists to submit their experimental and theoretical findings in great detail, with no limits on the length of manuscripts to ensure reproducibility. A key objective is to facilitate the publication of detailed research, including experimental procedures and calculations. "Fractal and Fractional" also stands out for its unique offerings: it warmly welcomes manuscripts related to research proposals and innovative ideas, and allows for the deposition of electronic files containing detailed calculations and experimental protocols as supplementary material.
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