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 $0<{\nu }_k\le 1$ 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.

References in corpus (3)

• The Space-Fractional Poisson Process
• Fractional pure birth processes
• A note on fractional linear pure birth and pure death processes in epidemic models

Cited by in corpus (9)

• Semi-Markov models and motion in heterogeneous media
• On Distributions of Certain State Dependent Fractional Point Processes
• Generalized Nonlinear Yule Models
• Convoluted Fractional Poisson Process
• Superposition of time-changed Poisson processes and their hitting times
• Time-inhomogeneous jump processes and variable order operators
• Generalized Fractional Birth Process
• Asymptotic results for families of power series distributions
• Asymptotic results for a multivariate version of the alternative fractional Poisson process