On Self-Similar Bernstein Functions and Corresponding Generalized Fractional Derivatives

Peter Kern () and Svenja Lage ()
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Peter Kern: Heinrich-Heine-University Düsseldorf
Svenja Lage: Heinrich-Heine-University Düsseldorf

Journal of Theoretical Probability, 2023, vol. 36, issue 1, 348-371

Abstract: 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.

Keywords: Power law tails; Log-periodic behavior; Laplace exponent; Bernstein functions; Self-similarity; Discrete scale invariance; Semistable Lévy process; Semi-fractional derivative; Semi-fractional diffusion; Sonine kernel; Sibuya distribution; Space–time duality; Primary 26A33; Secondary 35R11; 44A10; 60E07; 60G22; 60G51; 60G52 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-022-01166-0

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