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arXiv · 1912.12190

Tempered relaxation equation and related generalized stable processes

Luisa Beghin·Janusz Gajda
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Abstract

Fractional relaxation equations, as well as relaxation functions time-changed by independent stochastic processes have been widely studied (see, for example, \cite{MAI}, \cite{STAW} and \cite{GAR}). We start here by proving that the upper-incomplete Gamma function satisfies the tempered-relaxation equation (of index $ρ\in (0,1)$); 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.

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Luisa Beghin, Janusz Gajda. 2020-09-01. Tempered relaxation equation and related generalized stable processes. https://doi.org/10.1515/fca-2020-0063

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