Zν 上的分数算子和分数积分随机场

IF 3.6 2区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Fractal and Fractional Pub Date : 2024-06-13 DOI:10.3390/fractalfract8060353

Vytautė Pilipauskaitė, D. Surgailis

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引用次数: 0

摘要

我们考虑作用于函数 g:Zν→R,ν≥1 的分数积分算子 (I-T)d,d∈(-1,1)，其中 T 是 Zν 上随机行走的过渡算子。我们得到了 (I-T)d 的核τ(s;d),s∈Zν存在性、可逆性和平方可求和性的充分必要条件。τ(s;d)随着 |s|→∞ 的渐近行为是根据随机游走的局部极限定理确定的。讨论了 Zν 上一类求解右侧白噪声差分方程 (I-T)dX=ε 的分数积分随机场 X 及其缩放极限。详细研究了几个例子，包括分数格拉普拉斯算子和热算子。

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Fractional Operators and Fractionally Integrated Random Fields on Zν

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.

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来源期刊

Fractal and Fractional MATHEMATICS, INTERDISCIPLINARY APPLICATIONS-

CiteScore

4.60

自引率

18.50%

发文量

632

审稿时长

11 weeks

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.