The space-fractional Poisson process
Citations
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Cites background from "The space-fractional Poisson proces..."
...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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Additional excerpts
...We recall that this process was presented in [25] by referring to the fractional difference operator (1 − B) , for α ∈ (0, 1], and it is called a space-fractional Poisson process....
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...[21] for the case ν ∈ (0, 1) and [25] for the case ν ∈ (1, ∞))....
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Cites background or methods from "The space-fractional Poisson proces..."
...Note that for γ = 0, δ = 1, the generating function (3.37) reduces to that of the space-time fractional Poisson process [Orsingher and Polito, 2012], while for γ = 0, ν = 1, δ ∈ (0,1), we obtain that of the tempered space-time fractional Poisson process....
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...An interesting special case related to these equations is that regarding the so-called space-fractional Poisson process [Orsingher and Polito, 2012]....
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...We saw from the above analysis (see Remark 2.3 and the subordination result) that the difference operator (3.24) is in practice connected with the Prabhakar derivative....
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...Note that these generalize the difference-differential equations governing the state probabilities of a space-fractional Poisson process [Orsingher and Polito, 2012] i.e. d dt pk(t) = −λ α(1− B)αpk(t), pk(0) = δk,0, k ≥ 0, t ≥ 0, (3.6) and of course that of the classical homogeneous process for α→…...
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...The Laplace transform of the process ηV ν ,δ t , t ≥ 0, can be written as Eexp −µ ηV ν ,δ t = exp −t h η+µν δ −ηδ i , t ≥ 0, µ > 0....
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References
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Additional excerpts
...1 Introduction Fractional Poisson processes studied so far have been 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 the sense of Caputo....
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...By means of Laplace transforms, some simple manipulations lead to νGα(u, t) = Eν(−λ α tν(1− u)α), |u| ≤ 1, (2.27) where Eν(x) is the Mittag–Leffler function [Kilbas et al., 2006]....
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...The space-fractional Poisson process has probability generating function Gα(u, t) = Eu Nα(t) = e−λ α(1−u)α t , |u| ≤ 1, (1.4) and can be compared with its time-fractional counterpart νG(u, t) = Eu Nν (t) = Eν (−λ(1− u)t ν) , |u| ≤ 1, (1.5) where Eν(x) = ∞ ∑ r=0 x r Γ(ν r + 1) , ν > 0, (1.6) is the one-parameter Mittag–Leffler function....
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...rivative operator (see Kilbas et al. [2006]). By means of Laplace transforms, some simple manipulations lead to νGα(u,t) = Eν(−λ αtν(1−u)α), |u| ≤ 1, (2.27) where Eν(x) is the Mittag–Leffler function [Kilbas et al., 2006]. In turn, by expanding the above probability generating function we have that pα,ν k (t)= (−1)k k! X∞ r=0 (−λαtν)r Γ(ν r +1 ) Γ(αr +1) α − , k ≥ 0, α∈ (0,1], ν∈ (0,1]. (2.28) When α= 1 these probabi...
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2,766 citations
"The space-fractional Poisson proces..." refers methods in this paper
.... In this paper we introduce a space-fractional Poisson process by means of the fractional difference operator ∆α= (1−B)α, α∈ (0,1], (1.1) which often appears in the study of long memory time series [Tsay, 2005]. The operator (1.1) implies a dependence of the state probabilities pα k (t) from all probabilities pα j (t), j <k. For α= 1 we recover the classical homogeneous Poisson process and the state pro...
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"The space-fractional Poisson proces..." refers background or methods in this paper
...…(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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