On self-similar Bernstein functions and corresponding generalized fractional derivatives
arXiv:2106.07265 · doi:10.1007/s10959-022-01166-0
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 {\it Fract. Calc. Appl. Anal.} {\bf 22}(2), pp. 326--357, by means of the generator 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 ({\it Integr. Equ. Oper. Theory} {\bf 71}, pp. 583--600).