Fractional Poisson processes and related planar random motions
Summary (2 min read)
1 Introduction
- The distribution (1.4) coincides with formula (25) of Laskin (2003) , despite the latter being obtained as a solution to equation (1.1) with the fractional derivative defined in the Riemann-Liouville sense (instead of (1.3) ).
- The relationship (1.7) means that the fractional Poisson process ν has the following representation EQUATION and thus it can be considered as a homogeneous Poisson process stopped at a random time 2ν (t).
- This has been their first motivation of the research of this paper.
- In the standard case, when the process governing the changes of directions is the standard Poisson process, this model has been studied in Kolesnik and Orsingher (2005) .
2 A first form of the fractional Poisson process
- The authors construct the first type of fractional Poisson process (which they will denote by ν (t), t > 0) by replacing, in the differential equations governing the distribution of the classical Poisson process, the standard derivatives by the fractional derivatives, defined in (1.3), i.e. in the Dzerbayshan-Caputo sense.
- The authors note that the usual equality between the mean value and the variance does not hold for this model, while, for ν = 1, it can be checked that 1 (t) = Var 1 (t).
- The authors derive now an explicit representation of the solution to (2.1)-(2.2), which suggests an interesting probabilistic interpretation of the related process.
- The next result represents also an important link between the fractional Poisson process and the fractional diffusion equations.
3 Planar random motions changing direction at fractional Poisson times
- Let us consider a planar random motion described by a particle moving at finite velocity c and changing direction at Poisson time instants (for details on this process see Kolesnik and Orsingher (2005) ).
- At each Poisson event, the new direction is chosen with uniform law in [0, 2π] .
- The authors will examine here the same planar motion (taken at a Brownian time) when the process governing the changes of direction is replaced by a fractional Poisson process.
- In both cases, since the radius is the random variable c|B(t)| distributed on the positive real line, the support covers the whole plane.
Theorem 3.1
- Proof (3.7) Formula (3.7) coincides with the density of the n-times iterated Brownian motion EQUATION where the j 's are standard independent Brownian motions (i.e. with infinitesimal variance 1).
- In order to apply Theorem 2.1, the authors must suitably adapt the previous results, taking into account that, for ν = 1 4 , equation (2.12) reduces to EQUATION and this differs from (3.6) for the coefficient representing the infinitesimal variance.
4 Alternative forms of fractional Poisson processes
- For ν = 1 the distribution (4.1) coincides with that of the homogeneous Poisson process.
- Nevertheless some basic properties of the Poisson process are lost when considering (4.1): in particular the independence of increments on non-overlapping intervals.
- The distributions of the maximum and the minimum of this sequence can be written as follows.
- This coincides with the result of Remark 2.4 and thus the one-dimensional distributions of the first and third models coincide.
- Another representation of the distribution of the fractional Poisson process is given in the next theorem.
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Cites background or methods from "Fractional Poisson processes and re..."
...We will consider only the case γ 6= 0 as the case γ = 0 corresponds in fact to the case studied in [19, 26, 4] and others....
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...Let us here only recall the few relevant papers [10, 11, 40, 4, 3, 5, 20, 29, 32, 28] and see also the references cited therein....
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110 citations
Cites background or methods from "Fractional Poisson processes and re..."
...…(1.10) is similar to the following representation of the time-fractional Poisson process: Nν(t) d = N T2ν(t) , (1.11) where T2ν(t), t > 0, is a process whose one-dimensional distribution is obtained by folding the solution to the time-fractional diffusion equation [Beghin and Orsingher, 2009]....
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.... 1 Introduction FractionalPoissonprocessesstudiedsofarhavebeenobtained eitherbyconsidering renewal processeswith intertimesbetweeneventsrepresented byMittag–Lefflerdistributions [Mainardi et al.,2004,Beghin and Orsingher, 2009]or by replacing the time derivative in the equations governing the state probabilities with the fractional derivative in the sense of Caputo. In this paper we introduce a space-fractional Poisson pro...
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...ional Poisson process: Nν(t) =d N T2ν(t) , (1.11) where T2ν(t), t >0, is a process whose one-dimensional distribution is obtained by folding the solution to the time-fractional diffusion equation [Beghin and Orsingher, 2009]. Finally we can note that the probability generating function (1.7), for all u ∈ (0,1), can be represented as νGα(u,t)= Pr ˆ min 0≤k≤Nν(t) X1/α k ≥ 1−u ˙ , (1.12) where the X ks are i.i.d. uniformly...
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...…obtained either by considering renewal processes with intertimes between events represented by Mittag–Leffler distributions [Mainardi et al., 2004, Beghin and Orsingher, 2009] or by replacing the time derivative in the equations governing the state probabilities with the fractional derivative in…...
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92 citations
73 citations
Cites background or methods or result from "Fractional Poisson processes and re..."
...As already proved in [2] the Laplace transform of Gν = Gν(u, t) is given by L Gν(u, t); s = sν−1 sν −λ(u− 1) (2....
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...[3] Beghin, L., Orsingher, E. (2009), Moving randomly amid scattered obstacles, Stochastics, in press....
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...[18] Orsingher, E., Beghin, L. (2009), Fractional diffusion equations and processes with randomly-varying time, Annals of Probability, 37 (1), 206-249....
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...9) In [2] the following subordinating relation has been proved: p k (t) = ∫ +∞...
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...10) of [2], which was obtained by a different approach....
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References
15,898 citations
"Fractional Poisson processes and re..." refers background in this paper
...(2.13) The condition (2.13) is due to the application of the Laplace transform of the fractional derivative (see Podlubny (1999), formula (2.253), p.106)....
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...(1.2) We apply in (1.1) the definition of fractional derivative in the sense of Dzerbayshan-Caputo, that is, for m ∈ N, dν d tν u(t) = ( 1 Γ(m−ν) ∫ t 0 1 (t−s)1+ν−m dm dsm u(s)ds, for m− 1 ν m dm d tm u(t), for ν = m , (1.3) (see Podlubny (1999), p.78)....
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...An alternative form of the distribution is given in terms of the generalized Mittag-Leffler function, which is defined (see Podlubny (1999), p.17) as Eν ,µ(x) = ∞∑ k=0 xk Γ(νk+µ) , ν ,µ > 0, x ∈ R. (1.5) For small values of k we have meaningful expression: for example, for k = 0, we get that Pr…...
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7,096 citations
"Fractional Poisson processes and re..." refers background in this paper
...For a general reference on fractional calculus see Samko et al. (1993)....
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3,962 citations
"Fractional Poisson processes and re..." refers background in this paper
...(2.13) The condition (2.13) is due to the application of the Laplace transform of the fractional derivative (see Podlubny (1999), formula (2.253), p.106)....
[...]
...(1.2) We apply in (1.1) the definition of fractional derivative in the sense of Dzerbayshan-Caputo, that is, for m ∈ N, dν d tν u(t) = ( 1 Γ(m−ν) ∫ t 0 1 (t−s)1+ν−m dm dsm u(s)ds, for m− 1 ν m dm d tm u(t), for ν = m , (1.3) (see Podlubny (1999), p.78)....
[...]
...An alternative form of the distribution is given in terms of the generalized Mittag-Leffler function, which is defined (see Podlubny (1999), p.17) as Eν ,µ(x) = ∞∑ k=0 xk Γ(νk+µ) , ν ,µ > 0, x ∈ R. (1.5) For small values of k we have meaningful expression: for example, for k = 0, we get that Pr…...
[...]
1,123 citations
302 citations
"Fractional Poisson processes and re..." refers background or methods in this paper
...1 Introduction Attempts to construct fractional versions of the Poisson process have been undertaken by Repin and Saichev (2000), Jumarie (2001) and Laskin (2003)....
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...The distribution (1.4) coincides with formula (25) of Laskin (2003), despite the latter being obtained as a solution to equation (1.1) with the fractional derivative defined in the Riemann-Liouville sense (instead of (1.3))....
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...The additional impulse term appearing in equation (25) of Laskin (2003) justifies this coincidence....
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Q1. What are the contributions mentioned in the paper "Fractional poisson processes and related planar random motions" ?
The authors present three different fractional versions of the Poisson process and some related results concerning the distribution of order statistics and the compound Poisson process. For this process, denoted by Nν ( t ), t > 0, the authors obtain an interesting probabilistic representation in terms of a composition of the standard Poisson process with a random time, of the form Nν ( t ) = N ( T2ν ( t ) ), t > 0. The time argument T2ν ( t ), t > 0, is itself a random process whose distribution is related to the fractional diffusion equation. The authors also construct a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by the fractional Poisson processNν. For this model the authors 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 ν ∈ ( 0,1 ] the authors show that the random position has a Brownian behavior ( for ν = 1/2 ) or a cylindrical-wave structure ( for ν = 1 ).