Fractional Poisson processes and related planar random motions

By Luisa Beghin, Enzo Orsingher. 2009

228 citations recorded by the source corpus.

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 $\nu =1$).

Subfields: Applied Mathematics, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/d384c9da750cc4e525feeffaf80530fb8714ae9a

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Citations in the corpus

A fractional counting process and its connection with the Poisson process (2015, background): “(2.7) and (2.8) of Beghin and Orsingher, 2009) E [ N νλ (t) ] = λ tν Γ(ν + 1) , Var [ N νλ (t) ] = 2(λ tν)2 Γ(2ν + 1) − (λ t ν)2 (Γ(ν + 1))2 + λ tν Γ(ν + 1) .”
Poisson-type processes governed by fractional and higher-order recursive differential equations (2009, methodology): “Moreover, as remarked in [2] and [15], the fractional Poisson process Nν(t), t > 0 represents a renewal process with interarrival times Uj distributed according to the following density, for j = 1, 2, .”
The fractional non-homogeneous Poisson process (2016, methodology): “Among the others, the process defined above was studied in Beghin and Orsingher (2009, 2010), Meerschaert et al. (2011), Politi et al. (2011). In particular, Beghin and Orsingher (2009, 2010) developed the renewal approach to FHPP and proved that its one-dimensional distributions coincide with the s”
A practical guide to Prabhakar fractional calculus (2020, background): “epin and A. I. Saichev [127]. Many (mostly independent) contributions investigated the feature of this new renewal process following the path paved by the previously mentioned pivotal works, see e.g. [3,4,90,103,122] and references therein. To this regard, it is worth stressing that the Prabhakar fu”
A Note on Hadamard Fractional Differential Equations with Varying Coefficients and Their Applications in Probability (2017, methodology): “The relationship of this kind of distributions with fractional calculus, first considered by Beghin and Orsingher in [3], is given by the fact that the probability generating function G(u, t) = ∞ ∑ k=0 ukP{Nα(t) = k} = Eα,1(ut) Eα,1(t) , α ∈ (0, 1], (21) satisfies the fractional differential equatio”
The space-fractional Poisson process (2011, methodology): “28) When α = 1 these probabilities easily reduce to those of a fractional Poisson process (see Beghin and Orsingher [2009]): p k (t) = (−1)k k! ∞ ∑”
On a fractional linear birth-death process (2011, background): “39) A well-known property for a fractional Poisson process Nν(t), t > 0, of degree 0< ν ≤ 1 and parameter λ > 0 is that [2] Pr{Nν(t) = 0}=Eν,1(−tλ) = Pr(Tν ≥ t), (2.”