Tempered relaxation equation and related generalized stable processes

Abstract

AbstractFractional relaxation equations, as well as relaxation functions time-changed by independent stochastic processes have been widely studied (see, for example, [21], [33] and [11]). We start here by proving that the upper-incomplete Gamma function satisfies the tempered-relaxation equation (of indexρ∈ (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 then-times Laplace transform of its density) which is indexed by the parameterρ: in the special case whereρ= 1, it reduces to the stable subordinator.Therefore the parameterρcan be seen as a measure of the local deviation from the temporal dependence structure displayed in the standard stable case.