The aim of this paper is the analysis of the fractional Poisson process where the state probabilities $p_k^{\nu_k}(t)$, $t\ge 0$, are governed by time-fractional equations of order $0<\nu_k\leq 1$ depending on the number $k$ of events occurred up to time $t$. We are able to obtain explicitely the Laplace transform of $p_k^{\nu_k}(t)$ and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on $\nu_k$ differs from that constructed from the fractional state equations (in the case $\nu_k = \nu$, for all $k$, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally we consider the fractional birth process governed by equations with state-dependent fractionality.

### State-dependent Fractional Point Processes

#### Abstract

The aim of this paper is the analysis of the fractional Poisson process where the state probabilities $p_k^{\nu_k}(t)$, $t\ge 0$, are governed by time-fractional equations of order $0<\nu_k\leq 1$ depending on the number $k$ of events occurred up to time $t$. We are able to obtain explicitely the Laplace transform of $p_k^{\nu_k}(t)$ and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on $\nu_k$ differs from that constructed from the fractional state equations (in the case $\nu_k = \nu$, for all $k$, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally we consider the fractional birth process governed by equations with state-dependent fractionality.
##### Scheda breve Scheda completa Scheda completa (DC)
2015
52
1
18
36
http://arxiv.org/pdf/1303.6699v2
R. Garra; E. Orsingher; F. Polito
File in questo prodotto:
File
published.pdf

Accesso riservato

Tipo di file: PDF EDITORIALE
Dimensione 165.61 kB
Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/142465