Renewal processes linked to fractional relaxation equations with variable order

We introduce and study here a renewal process defined by means of a time-fractional relaxation equation with derivative order $\alpha(t)$ varying with time $t\geq0$. In particular, we use the operator introduced by Scarpi in the Seventies and later reformulated in the regularized Caputo sense in Garrappa et al. (2021), inside the framework of the so-called general fractional calculus. The model obtained extends the well-known time-fractional Poisson process of fixed order $\alpha \in (0,1)$ and tries to overcome its limitation consisting in the constancy of the derivative order (and therefore of the memory degree of the interarrival 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 analysed and its limiting behavior established.