Renewal processes linked to fractional relaxation equations with variable order

Abstract

We introduce and study here a renewal process defined by means of a time-fractional relaxation equation with derivative order α(t) varying with time t≥0. In particular, we use the operator introduced by Scarpi in the seventies [23] and later reformulated in the regularized Caputo sense in [5], inside the framework of the so-called general fractional calculus. The obtained model extends the well-known time-fractional Poisson process of fixed order α∈(0,1) and tries to overcome its limitation consisting in the constancy of the derivative order (and therefore of the memory degree of the inter-arrival times) with respect to time. The variable order renewal process is proved to fall outside the usual subordinated representation, since it can not be simply defined as a Poisson process with random time (as happens in the standard fractional case). Finally a related continuous-time random walk model is analyzed and its limiting behavior established.