Fractional Relaxation Equations and Brownian Crossing Probabilities of a Random Boundary

Beghin, Luisa

doi:10.1017/s0001867800005693

Cited by 9 publications

(9 citation statements)

References 26 publications

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“…where d β dt β is the Caputo fractional derivative of order β ∈ (0, 1). Note that, formally, (3) is the continuous version of (1), in the same way as (4) is the continuous version of (2). Here we prove that a similar but non equivalent scheme holds in discrete time.…”

Section: Angelica Pachon Federico Polito and Costantino Ricciutisupporting

confidence: 56%

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On discrete-time semi-Markov processes

Pachón¹,

Polito²,

Ricciuti³

2021

Discrete &Amp; Continuous Dynamical Systems - B

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In the last years, several authors studied a class of continuous-time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integrodifferential convolution equations of generalized fractional type. The aim of this paper is to develop a discrete-time counterpart of such a theory and to show relationships and differences with respect to the continuous time case. We present a class of discrete-time semi-Markov chains which can be constructed as time-changed Markov chains and we obtain the related governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling limits.

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Section: Angelica Pachon Federico Polito and Costantino Ricciutisupporting

confidence: 56%

“…which is also the Laplace transform of − d dt E(−λ i t α ). For analytical properties of the Mittag-Leffler function and its role in fractional calculus consult [57]; see also [4] and [19] for some applications on relaxation phenomena.…”

Section: Brief Review On Continuous-time Semi-markov Chainsmentioning

confidence: 99%

On discrete-time semi-Markov processes

Pachón¹,

Polito²,

Ricciuti³

2021

Discrete &Amp; Continuous Dynamical Systems - B

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show abstract

“…which is also the Laplace transform of − d dt E(−λ i t α ). For analytical properties of the Mittag-Leffler function and its role in fractional calculus consult [55]; see also [4] and [18] for some applications on relaxation phenomena.…”

Section: Preliminariesmentioning

confidence: 99%

On discrete-time semi-Markov processes

Pachón¹,

Polito²,

Ricciuti³

2018

Preprint

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In the last years many authors studied a class of continuous time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integro-differential convolution equations of generalized fractional type. The aim of this paper is to develop the discrete-time version of such a theory. We show that a class of discrete-time semi-Markov chains can be seen as time-changed Markov chains and we obtain governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling limits.

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“…1 is the so-called fractional relaxation equation (see e.g. Uchaikin 2002;Beghin 2012), we can call the second equation in Eq. 1 a generalized fractional birth equation (see e.g.…”

Section: Ams 2010 Subject Classifications 60g52 • 34a08 • 33e12 • 26amentioning

confidence: 99%

Generalized Fractional Nonlinear Birth Processes

Alipour

Beghin

Rostamy

2013

Methodol Comput Appl Probab

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We consider here generalized fractional versions of the difference-differential equation governing the classical nonlinear birth process. Orsingher and Polito (Bernoulli 16(3):858-881, 2010) defined a fractional birth process by replacing, in its governing equation, the first order time derivative with the Caputo fractional derivative of order υ ∈ (0, 1]. We study here a further generalization, obtained by adding in the equation some extra terms; as we shall see, this makes the expression of its solution much more complicated. Moreover we consider also the case υ ∈ (1, +∞ ), as well as υ ∈ (0, 1], using correspondingly two different definitions of fractional derivative: we apply the fractional Caputo derivative and the right-sided fractional Riemann-Liouville derivative on ℝ+, for υ ∈ (0, 1] and υ ∈ (1, +∞ ), respectively. For the two cases, we obtain the exact solutions and prove that they coincide with the distribution of some subordinated stochastic processes, whose random time argument is represented by a stable subordinator (for υ ∈ (1, +∞ )) or its inverse (for υ ∈ (0, 1]). © 2013 Springer Science+Business Media New York

show abstract

Fractional Relaxation Equations and Brownian Crossing Probabilities of a Random Boundary

Cited by 9 publications

References 26 publications

On discrete-time semi-Markov processes

On discrete-time semi-Markov processes

On discrete-time semi-Markov processes

Generalized Fractional Nonlinear Birth Processes

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