Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator

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 nonhomogeneous versions of such models, which can be defined by means of the so-called multistable subordinator (a jump process with nonstationary increments), denoted by $H:=H(t),t\ge 0$. 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 analog of the time-fractional Poisson process.

Keywords:

• Subordinators
• time-inhomogeneous processes
• multistable subordinators
• Bernstein functions
• fractional calculus
• Mittag-Leffler distribution

AMS Subject Classification (2010)::

• 60G51
• 60J75
• 26A33

Notes

1 Such a formula can be proved for k = 1 and then generalized to k > 1, by a standard use of the principle of induction.