Fractional Poisson process with random drift

Beghin, Luisa; D’Ovidio, Mirko

doi:10.1214/ejp.v19-3258

Cited by 11 publications

(25 citation statements)

References 14 publications

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“…For the sake of clarity and completeness, in the Appendix A we report a brief construction of the fractional nonlinear birth processes. Here we limit ourselves to refer to some of the papers present in the literature such as Laskin [26], Beghin and Orsingher [5], Mainardi et al [27], Politi et al [36], Cahoy and Polito [10], Meerschaert et al [29], Garra et al [17], Beghin and D'Ovidio [4], Orsingher and Polito [33]. The fractional nonlinear birth process was studied in Orsingher and Polito [32,34].…”

Section: Introductionmentioning

confidence: 99%

Generalized Nonlinear Yule Models

2016

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With the aim of considering models related to random graphs growth exhibiting persistent memory, we propose a fractional nonlinear modification of the classical Yule model often studied in the context of macroevolution. Here the model is analyzed and interpreted in the framework of the development of networks such as the World Wide Web. Nonlinearity is introduced by replacing the linear birth process governing the growth of the in-links of each specific webpage with a fractional nonlinear birth process with completely general birth rates.Among the main results we derive the explicit distribution of the number of in-links of a webpage chosen uniformly at random recognizing the contribution to the asymptotics and the finite time correction. The mean value of the latter distribution is also calculated explicitly in the most general case. Furthermore, in order to show the usefulness of our results, we particularize them in the case of specific birth rates giving rise to a saturating behaviour, a property that is often observed in nature. The further specialization to the non-fractional case allows us to extend the Yule model accounting for a nonlinear growth.

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Section: Introductionmentioning

confidence: 99%

Generalized Nonlinear Yule Models

2016

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“…Now we give some expressions of the state probabilities {{p η,ν k (t) : k ≥ 0} : t ≥ 0} in (2). We start with an implicit expression which generalizes (3.19) in [2] (note that we use the notation ∂ λ i in place of ∂ ∂ λ i ). The most explicit formulas are given in Proposition 3.5.…”

Section: Results For the Processes In Definitions 11 And 12mentioning

confidence: 99%

Multivariate fractional Poisson processes and compound sums

Beghin

Macci

2016

Adv. Appl. Probab.

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In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (nonfractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes.(see e.g. [18], page 17).We start with the simplest case, i.e. the multivariate version of the space-time fractional Poisson process in [15]. In particular we consider the time-change approach in terms of the stable subordinator and of its inverse (see (3.18), together with (3.1), in [2]; see also [22]). So we introduce the following notation: for ν ∈ (0, 1), let {A ν (t) : t ≥ 0} be the stable subordinator and let {L ν (t) : t ≥ 0} be its inverse, i.e. L ν (t) := inf{z ≥ 0 : A ν (z) ≥ t}.

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“…N(t) + at, a, t ≥ 0. The latter has been studied in [6], where also the case of a general Lévy process subordinated by it has been analyzed.…”

Section: Lemmamentioning

confidence: 99%

Fractional diffusion-type equations with exponential and logarithmic differential operators

Beghin

2018

Stochastic Processes and their Applications

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We deal with some extensions of the space-fractional diffusion equation, which is satisfied by the density\ud of a stable process (see Mainardi et al. (2001)): the first equation considered here is obtained by adding an\ud exponential differential (or shift) operator expressed in terms of the Riesz–Feller derivative. We prove that\ud this produces a random component in the time-argument of the corresponding stable process, which is\ud represented by the so-called Poisson process with drift. Analogously, if we add, to the space-fractional\ud diffusion equation, a logarithmic differential operator involving the Riesz-derivative, we obtain, as a\ud solution, the transition semigroup of a stable process subordinated by an independent gamma subordinator\ud with drift. Finally, we show that an extension of the space-fractional diffusion equation, containing both the\ud fractional shift operator and the Feller integral, is satisfied by the transition density of the process obtained\ud by time-changing the stable process with an independent linear birth process with drift

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Fractional Poisson process with random drift

Cited by 11 publications

References 14 publications

Generalized Nonlinear Yule Models

Generalized Nonlinear Yule Models

Multivariate fractional Poisson processes and compound sums

Fractional diffusion-type equations with exponential and logarithmic differential operators

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