Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator. (English) Zbl 1488.60124
Summary: 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 \(H:=H(t)\), \(t\geq 0\). 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.
MSC:
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 60G51 | Processes with independent increments; Lévy processes |
| 60J76 | Jump processes on general state spaces |
Keywords:
subordinators; time-inhomogeneous processes; multistable subordinators; Bernstein functions; fractional calculus; Mittag-Leffler distributionReferences:
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