## Electronic Journal of Probability

### Fractional Poisson processes and related planar random motions

#### Abstract

We present three different fractional versions of the Poisson process and some related results concerning the distribution of order statistics and the compound Poisson process. The main version is constructed by considering the difference-differential equation governing the distribution of the standard Poisson process, $N(t),t>0$, and by replacing the time-derivative with the fractional Dzerbayshan-Caputo derivative of order $\nu\in(0,1]$. For this process, denoted by $\mathcal{N}_\nu(t),t>0,$ we obtain an interesting probabilistic representation in terms of a composition of the standard Poisson process with a random time, of the form $\mathcal{N}_\nu(t)= N(\mathcal{T}_{2\nu}(t)),$ $t>0$. The time argument $\mathcal{T}_{2\nu }(t),t>0$, is itself a random process whose distribution is related to the fractional diffusion equation. We also construct a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by the fractional Poisson process $\mathcal{N}_\nu.$ For this model we obtain the distributions of the random vector representing the position at time $t$, under the condition of a fixed number of events and in the unconditional case. For some specific values of $\nu\in(0,1]$ we show that the random position has a Brownian behavior (for $\nu =1/2$) or a cylindrical-wave structure (for $\nu =1$).

#### Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 61, 1790-1826.

Dates
Accepted: 25 August 2009
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464819522

Digital Object Identifier
doi:10.1214/EJP.v14-675

Mathematical Reviews number (MathSciNet)
MR2535014

Zentralblatt MATH identifier
1190.60028

Subjects
Primary: 60G55: Point processes
Secondary: 26A33: Fractional derivatives and integrals

Rights

#### Citation

Beghin, Luisa; Orsingher, Enzo. Fractional Poisson processes and related planar random motions. Electron. J. Probab. 14 (2009), paper no. 61, 1790--1826. doi:10.1214/EJP.v14-675. https://projecteuclid.org/euclid.ejp/1464819522

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