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The Fractional Poisson Process and the Inverse Stable Subordinator
Ear,Nih grant R Eb +1 more
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TLDR
In this paper, it was shown that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional poisson process with Mittag-Leffler waiting times.Abstract:
The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve
a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a
traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a
fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional
diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered
fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.read more
Citations
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Fractional Poisson fields and Martingales
TL;DR: In this paper, a martingale characterization for the Fractional Poisson process on the plane is given, and the authors extend this result to Fractionally Poisson fields, obtaining some other characterizations.
Journal ArticleDOI
Complexity and the Fractional Calculus
TL;DR: In this article, the mean time interval between two consecutive critical events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that supports the hypothesis that the Mittag-Leffler function is a universal property of nature.
Journal ArticleDOI
On Mittag-Leffler distributions and related stochastic processes
TL;DR: A type-2 Mittag-Leffler process turns out to be the Siegmund dual to Neveu's CSBP block-counting process arising in sampling from P D ( e - t , 0 ) .
Journal ArticleDOI
Semi-Markov Models and Motion in Heterogeneous Media
Costantino Ricciuti,Bruno Toaldo +1 more
TL;DR: In this article, the authors studied continuous time random walks such that the holding time in each state has a distribution depending on the state itself, and provided integro-differential (backward and forward) equations of Volterra type, exhibiting a position dependent convolution kernel.
Journal ArticleDOI
Semi-Markov graph dynamics.
TL;DR: A model of graph (or network) dynamics based on a Markov chain on the space of possible graphs and a semi-Markov counting process of renewal type where the chain transitions occur at random time instants called epochs is outlined.
References
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Journal ArticleDOI
Correlation Structure of Time-Changed Lévy Processes
TL;DR: In this article, the correlation function for time-changed L evy processes has been studied in the context of continuous time random walks, where the second-order correlation function of a continuous-time random walk is defined.
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Applications of inverse tempered stable subordinators
TL;DR: This paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time-fractional diffusion equation, and its “folded” density solves a temperamental telegraph equation.
Journal ArticleDOI
Time-changed Poisson processes
TL;DR: In this article, the authors considered time-changed Poisson processes and derived the governing difference-differential equations (DDEs) for these processes, and derived a new governing partial differential equation for the tempered stable subordinator of index 0 β 1.
Journal ArticleDOI
Fractional Skellam processes with applications to finance
TL;DR: In this paper, the authors define fractional Skellam processes via the time changes in Poisson and Skekam processes by an inverse of a standard stable subordinator.
Posted Content
Inverse Tempered Stable Subordinators
TL;DR: In this paper, the first-hitting time of a tempered β-stable subordinator, also called inverse tempered stable (ITS) subordinator is considered, and the limiting form of the ITS density, as the space variable $x\rightarrow 0$, and its $k$-th order derivatives are obtained.