On self-similar Bernstein functions and corresponding generalized fractional derivatives
Probability
2023-12-22 v1 Mathematical Physics
Classical Analysis and ODEs
math.MP
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).
Keywords
Cite
@article{arxiv.2106.07265,
title = {On self-similar Bernstein functions and corresponding generalized fractional derivatives},
author = {Peter Kern and Svenja Lage},
journal= {arXiv preprint arXiv:2106.07265},
year = {2023}
}