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Fractional stable random fields on the Sierpiński gasket

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  • Baudoin, Fabrice
  • Lacaux, Céline

Abstract

We define and study fractional stable random fields on the Sierpiński gasket. Such fields are formally defined as (−Δ)−sWK,α, where Δ is the Laplace operator on the gasket and WK,α is a stable random measure. Both Neumann and Dirichlet boundary conditions for Δ are considered. Sample paths regularity and scaling properties are obtained. The techniques we develop are general and extend to the more general setting of the Barlow fractional spaces.

Suggested Citation

  • Baudoin, Fabrice & Lacaux, Céline, 2024. "Fractional stable random fields on the Sierpiński gasket," Stochastic Processes and their Applications, Elsevier, vol. 178(C).
    • Handle: RePEc:eee:spapps:v:178:y:2024:i:c:s030441492400187x
    DOI: 10.1016/j.spa.2024.104481
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    References listed on IDEAS

    as
    1. Baudoin, Fabrice & Chen, Li, 2023. "Dirichlet fractional Gaussian fields on the Sierpinski gasket and their discrete graph approximations," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 593-616.
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    4. Ayache, Antoine & Roueff, François & Xiao, Yimin, 2009. "Linear fractional stable sheets: Wavelet expansion and sample path properties," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1168-1197, April.
    5. Antoine Ayache & Geoffrey Boutard, 2017. "Stationary Increments Harmonizable Stable Fields: Upper Estimates on Path Behaviour," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1369-1423, December.
    6. Arnold, Ludwig & Imkeller, Peter, 1996. "Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 19-54, March.
    7. Biermé, Hermine & Meerschaert, Mark M. & Scheffler, Hans-Peter, 2007. "Operator scaling stable random fields," Stochastic Processes and their Applications, Elsevier, vol. 117(3), pages 312-332, March.
    8. Biermé, Hermine & Lacaux, Céline, 2009. "Hölder regularity for operator scaling stable random fields," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2222-2248, July.
    9. Cambanis, Stamatis & Maejima, Makoto, 1989. "Two classes of self-similar stable processes with stationary increments," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 305-329, August.
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