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 . The distribution of the random time argument can be expressed, for any fixed t, in terms of convolutions of stable-laws. The new process is itself a renewal and can be shown to be a Cox process. Moreover we prove that the survival probability of , 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 . 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. [Copyright &y& Elsevier]