We introduce and study here a renewal process de fined by means of a time-fractional relaxation equation with derivative order \alpha(t) varying with time t \geq 0. In particular, we use the operator introduced by Scarpi in the Seventies (see [23]) and later reformulated in the regularized Caputo sense in [4], inside the framework of the so-called general fractional calculus. The model obtained extends the well-known time-fractional Poisson process of fi xed order \alpha \in (0,1) and tries to overcome its limitation consisting in the constancy of the derivative order (and therefore of the memory degree of the interarrival times) with respect to time. The variable order renewal process is proved to fall outside the usual subordinated representation, since it can not be simply defi ned as a Poisson process with random time (as happens in the standard fractional case). Finally a related continuous-time random walk model is analysed and its limiting behavior established.

Renewal processes linked to fractional relaxation equations with variable order / Beghin, Luisa; Cristofaro, Lorenzo; Garrappa, Roberto. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - (2023), pp. 1-24. [10.1016/j.jmaa.2023.127795]

Renewal processes linked to fractional relaxation equations with variable order

Luisa Beghin
;
Lorenzo Cristofaro;
2023

Abstract

We introduce and study here a renewal process de fined by means of a time-fractional relaxation equation with derivative order \alpha(t) varying with time t \geq 0. In particular, we use the operator introduced by Scarpi in the Seventies (see [23]) and later reformulated in the regularized Caputo sense in [4], inside the framework of the so-called general fractional calculus. The model obtained extends the well-known time-fractional Poisson process of fi xed order \alpha \in (0,1) and tries to overcome its limitation consisting in the constancy of the derivative order (and therefore of the memory degree of the interarrival times) with respect to time. The variable order renewal process is proved to fall outside the usual subordinated representation, since it can not be simply defi ned as a Poisson process with random time (as happens in the standard fractional case). Finally a related continuous-time random walk model is analysed and its limiting behavior established.
2023
fractional relaxation equation; renewal processes; Scarpi derivative; general fractional calculus; Sonine pair
01 Pubblicazione su rivista::01a Articolo in rivista
Renewal processes linked to fractional relaxation equations with variable order / Beghin, Luisa; Cristofaro, Lorenzo; Garrappa, Roberto. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - (2023), pp. 1-24. [10.1016/j.jmaa.2023.127795]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1689528
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