Tempered relaxation equation and related generalized stable processes

TL;DR

This paper introduces a new class of self-similar processes based on the tempered relaxation equation, extending stable laws and providing explicit solutions involving the upper-incomplete Gamma function, with implications for understanding temporal dependence.

Contribution

It proves that the upper-incomplete Gamma function satisfies the tempered-relaxation equation and defines a new class of processes indexed by a parameter measuring deviation from stable laws.

Findings

Explicit solution involving the upper-incomplete Gamma function

Extension of stable laws through spectral distribution

Introduction of a new self-similar process class

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 $\rho \in (0,1)$); 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 $\rho $: in the special case where $\rho =1$, it reduces to the stable subordinator. Therefore the parameter $\rho $ can be seen as a measure of the local deviation from the temporal dependence structure displayed in the standard stable case.