Tempered Relaxation Equation and Related Generalized Stable Processes

Fractional relaxation equations, as well as relaxation functions timechanged by independent stochastic processes have been widely studied (see, for example, [21], [33] and [11]). We start here by proving that the upperincomplete 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. MSC 2010 : Primary 26A33; Secondary 34A08, 33B20, 60G52, 60G18