Random-time processes governed by differential equations of fractional distributed order

Abstract

We analyze here different types of fractional differential equations, under the assumption that their fractional order ν∈(0,1] is random with probability density n(ν). We start by considering the fractional extension of the recursive equation governing the homogeneous Poisson process N(t), t>0. We prove that, for a particular (discrete) choice of n(ν), it leads to a process with random time, defined as N(T∼ν1,ν2(t)),t>0. The distribution of the random time argument T∼ν1,ν2(t) can be expressed, for any fixed t, in terms of convolutions of stable-laws. The new process N(T∼ν1,ν2) is itself a renewal and can be shown to be a Cox process. Moreover we prove that the survival probability of N(T∼ν1,ν2), as well as its probability generating function, are solution to the so-called fractional relaxation equation of distributed order (see [16]).In view of the previous results it is natural to consider diffusion-type fractional equations of distributed order. We present here an approach to their solutions in terms of composition of the Brownian motion B(t), t>0 with the random time T∼ν1,ν2. We thus provide an alternative to the constructions presented in Mainardi and Pagnini [19] and in Chechkin et al. [6], at least in the double-order case.