On self-similar Bernstein functions and corresponding generalized fractional derivatives

TL;DR

This paper explores the use of Bernstein functions to analyze self-similar properties and tail behaviors in semistable Lévy processes, providing new insights into semi-fractional derivatives as generalized fractional derivatives.

Contribution

It introduces a Bernstein function approach to solve open questions about semi-fractional derivatives and links them to generalized fractional derivatives.

Findings

Semi-fractional derivatives are shown to be generalized fractional derivatives.

The Bernstein approach clarifies the structure of tail behaviors in semistable Lévy processes.

Open questions about semi-fractional derivatives are addressed and resolved.

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 L\'evy processes. The Bernstein approach enables us to solve some open questions concerning semi-fractional derivatives recently introduced in {\it Fract. Calc. Appl. Anal.} {\bf 22}(2), pp. 326--357, by means of the generator of certain semistable L\'evy processes. In particular it is shown that semi-fractional derivatives can be seen as generalized fractional derivatives in the sense of Kochubei ({\it Integr. Equ. Oper. Theory} {\bf 71}, pp. 583--600).