Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator

TL;DR

This paper introduces non-homogeneous fractional Poisson processes using multistable subordinators, providing explicit distributions and governing equations for these new models, extending classical fractional Poisson processes to non-stationary settings.

Contribution

It develops a novel framework for inhomogeneous fractional Poisson processes based on multistable subordinators, including explicit distributions and governing equations.

Findings

Derived explicit distribution for the non-homogeneous Poisson process with multistable subordinator.

Established governing equations for the inhomogeneous fractional Poisson process.

Extended classical fractional Poisson models to non-stationary, inhomogeneous cases.

Abstract

The space-fractional and the time-fractional Poisson processes are two well-known models of fractional evolution. They can be constructed as standard Poisson processes with the time variable replaced by a stable subordinator and its inverse, respectively. The aim of this paper is to study non-homogeneous versions of such models, which can be defined by means of the so-called multistable subordinator (a jump process with non-stationary increments), denoted by H. Firstly, we consider the Poisson process time-changed by H and we obtain its explicit distribution and governing equation. Then, by using the right-continuous inverse of H, we define an inhomogeneous analogue of the time-fractional Poisson process.