Fractional relaxation equations, as well as relaxation functions time-changed by independent stochastic processes have been widely studied (see, for example, MAI, STAW and GAR). We start here by proving that the upper-incomplete Gamma function satisfies the tempered-relaxation equation (of index ρ∈(0,1)); thanks to this explicit form of the solution, we can then derive its spectral distribution, which extends the stable law. Accordingly, we define a new class of selfsimilar processes (by means of the n-times Laplace transform of its density) which is indexed by the parameter ρ: in the special case where ρ=1, it reduces to the stable subordinator. Therefore the parameter ρ can be seen as a measure of the local deviation from the temporal dependence structure displayed in the standard stable case.

TEMPERED RELAXATION EQUATION AND RELATED GENERALIZED STABLE PROCESSES / Beghin, L.; Gajda, J.. - In: FRACTIONAL CALCULUS & APPLIED ANALYSIS. - ISSN 1314-2224. - 23:5(2020), pp. 1248-1273.

TEMPERED RELAXATION EQUATION AND RELATED GENERALIZED STABLE PROCESSES

L. Beghin
Primo
;
2020

Abstract

Fractional relaxation equations, as well as relaxation functions time-changed by independent stochastic processes have been widely studied (see, for example, MAI, STAW and GAR). We start here by proving that the upper-incomplete Gamma function satisfies the tempered-relaxation equation (of index ρ∈(0,1)); thanks to this explicit form of the solution, we can then derive its spectral distribution, which extends the stable law. Accordingly, we define a new class of selfsimilar processes (by means of the n-times Laplace transform of its density) which is indexed by the parameter ρ: in the special case where ρ=1, it reduces to the stable subordinator. Therefore the parameter ρ can be seen as a measure of the local deviation from the temporal dependence structure displayed in the standard stable case.
2020
Incomplete gamma function; tempered derivative; stable laws; relaxation function; H-functions; self-similar processes.
01 Pubblicazione su rivista::01a Articolo in rivista
TEMPERED RELAXATION EQUATION AND RELATED GENERALIZED STABLE PROCESSES / Beghin, L.; Gajda, J.. - In: FRACTIONAL CALCULUS & APPLIED ANALYSIS. - ISSN 1314-2224. - 23:5(2020), pp. 1248-1273.
File allegati a questo prodotto
File Dimensione Formato  
Beghin_tempered-relaxation-equation_2020.pdf

Open Access dal 17/11/2020

Tipologia: Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 995.25 kB
Formato Adobe PDF
995.25 kB Adobe PDF

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1455781
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 5
  • ???jsp.display-item.citation.isi??? 4
social impact