A Generalization of the Space-Fractional Poisson Process and its Connection to some L\'evy Processes

TL;DR

This paper generalizes the space-fractional Poisson process by extending the difference operator, relating it to Lévy processes via subordination, and introduces a new model involving the Prabhakar derivative in time.

Contribution

It extends the space-fractional Poisson process with a more general difference operator and connects it to Lévy processes through explicit subordination and Lévy measures.

Findings

Established a relation between the generalized process and Lévy subordination.

Derived explicit Lévy measures for the subordinators.

Introduced a model with Prabhakar derivative acting in time.

Abstract

This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing L\'evy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.