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Multifractional Random Systems on Fractal Domains

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Modern Mathematical Tools and Techniques in Capturing Complexity

Part of the book series: Understanding Complex Systems ((UCS))

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Summary

In this paper, the effect of the domain geometry on the local regularity/singularity properties of the solution to the trace of a multifractional pseudodifferential equation on a fractal domain is studied. The singularity spectrum of the Gaussian solution to this type of models is trivial due to regularity assumptions on the variable order of its fractional derivatives. The theory of reproducing kernel Hilbert spaces (RKHSs) and generalized random fields is applied in this study. Specifically, the associated family of RKHSs is isomorphically identified with the trace on a compact fractal domain of a multifractional Sobolev space. The fractal defect modifies the variable order of weak-sense factional derivatives of the functions in these spaces. In the Gaussian case, random fields on fractal domains having sample paths with variable local Hölder exponent are introduced in this framework.

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Authors and Affiliations

  1. Faculty of Sciences, University of Granada, Campus Fuente Nueva s/n, 18071, Granada, Spain

    José Miguel Angulo & María Dolores Ruiz-Medina

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  1. José Miguel Angulo
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  2. María Dolores Ruiz-Medina
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Editors and Affiliations

  1. Departamento de Estadística e I. O., Facultad de Matemáticas, Universidad Complutense de Madrid, 28040, Madrid, Spain

    Leandro Pardo

  2. Department of Mathematics and Statistics, McMaster University Hamilton, L8S 4K1, Ontario, Canada

    Narayanaswamy Balakrishnan

  3. Departamento de Estadística e I. O. y, D.M., Universidad de Oviedo, C/ Calvo Sotelo, s/n, 33071, Oviedo, Spain

    María Ángeles Gil

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Angulo, J.M., Ruiz-Medina, M.D. (2011). Multifractional Random Systems on Fractal Domains. In: Pardo, L., Balakrishnan, N., Gil, M.Á. (eds) Modern Mathematical Tools and Techniques in Capturing Complexity. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20853-9_25

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  • DOI: https://doi.org/10.1007/978-3-642-20853-9_25

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  • Online ISBN: 978-3-642-20853-9

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