The Space-Fractional Poisson Process

TL;DR

This paper introduces the space-fractional Poisson process, deriving explicit distributions and generating functions, and compares it with the time-fractional Poisson process, culminating in a more general space-time fractional model.

Contribution

The paper explicitly derives the distributions and generating functions for the space-fractional Poisson process and introduces a more general space-time fractional Poisson process.

Findings

Explicit distributions $p_k^eta(t)$ derived

Probability generating functions expressed as distributions of minima of uniforms

Comparison with time-fractional Poisson process conducted

Abstract

In this paper we introduce the space-fractional Poisson process whose state probabilities $p_k^\alpha(t)$, $t>0$, $\alpha \in (0,1]$, are governed by the equations $(\mathrm d/\mathrm dt)p_k(t) = -\lambda^\alpha (1-B)p_k^\alpha(t)$, where $(1-B)^\alpha$ is the fractional difference operator found in the study of time series analysis. We explicitly obtain the distributions $p_k^\alpha(t)$, the probability generating functions $G_\alpha(u,t)$, which are also expressed as distributions of the minimum of i.i.d.\ uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space-time fractional Poisson process of which we give the explicit distribution.