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On Self-Similar Bernstein Functions and Corresponding Generalized Fractional Derivatives

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  • Peter Kern

    (Heinrich-Heine-University Düsseldorf)

  • Svenja Lage

    (Heinrich-Heine-University Düsseldorf)

Abstract

We use the theory of Bernstein functions to analyze power law tail behavior with log-periodic perturbations which corresponds to self-similarity of the Bernstein functions. Such tail behavior appears in the context of semistable Lévy processes. The Bernstein approach enables us to solve some open questions concerning semi-fractional derivatives recently introduced in Kern et al. (Fract Calc Appl Anal 22(2):326–357, 2019) by means of the generators of certain semistable Lévy processes. In particular, it is shown that semi-fractional derivatives can be seen as generalized fractional derivatives in the sense of Kochubei (Integr Equ Oper Theory 71:583–600, 2011) and generalized fractional derivatives can be constructed by means of arbitrary Bernstein functions vanishing at the origin.

Suggested Citation

  • Peter Kern & Svenja Lage, 2023. "On Self-Similar Bernstein Functions and Corresponding Generalized Fractional Derivatives," Journal of Theoretical Probability, Springer, vol. 36(1), pages 348-371, March.
    • Handle: RePEc:spr:jotpro:v:36:y:2023:i:1:d:10.1007_s10959-022-01166-0
    DOI: 10.1007/s10959-022-01166-0
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    References listed on IDEAS

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    1. Nadjib Bouzar, 2008. "The semi-Sibuya distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(2), pages 459-464, June.
    2. Tomasz J. Kozubowski & Krzysztof Podgórski, 2018. "A generalized Sibuya distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 855-887, August.
    3. Peter Kern & Mark M. Meerschaert & Yimin Xiao, 2018. "Asymptotic Behavior of Semistable Lévy Exponents and Applications to Fractal Path Properties," Journal of Theoretical Probability, Springer, vol. 31(1), pages 598-617, March.
    4. Christoph, Gerd & Schreiber, Karina, 2000. "Scaled Sibuya distribution and discrete self-decomposability," Statistics & Probability Letters, Elsevier, vol. 48(2), pages 181-187, June.
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