Intertwining certain fractional derivatives

TL;DR

This paper establishes an intertwining relation between specific fractional derivatives and certain stable Lévy processes, offering new insights into their probabilistic structure and connections.

Contribution

It introduces a novel intertwining relation linking Riemann-Liouville operators of order between 1 and 2 with stable Lévy processes, using probabilistic and analytical methods.

Findings

Derived an intertwining relation for fractional derivatives and stable Lévy processes.

Connected fractional calculus with probabilistic identities in law.

Provided an alternative approach via self-similar Markov processes.

Abstract

We obtain an intertwining relation between some Riemann-Liouville operators of order a in (1,2) connecting through a certain multiplicative identity in law the one-dimensional marginals of reflected completely asymmetric a-stable L\'evy processes. An alternative approach based on recurrent extensions of positive self-similar Markov processes and exponential functionals of L\'evy processes is also discussed.