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

TL;DR

This paper studies fractional differential equations with random order, introduces processes governed by random times expressed via convolutions of stable-laws, and explores their applications to renewal, Cox processes, and diffusion equations.

Contribution

It presents a novel analysis of fractional differential equations with random order, linking them to processes with random time arguments expressed through convolutions of stable-laws.

Findings

The process $N( ilde{ au}_{ u_1, u_2}(t))$ is a renewal and Cox process.

Survival probability and generating function solve fractional relaxation equations of distributed order.

Provides an alternative approach to solving diffusion-type fractional equations using composition with Brownian motion.

Abstract

We analyze here different types of fractional differential equations, under the assumption that their fractional order $\nu \in (0,1] $ is random\ with probability density $n(\nu).$ 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 $n(\nu)$, it leads to a process with random time, defined as $N(% \widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}}(t)),t>0.$ The distribution of the random time argument $\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}}(t)$ can be expressed, for any fixed $t$, in terms of convolutions of stable-laws. The new process $N(\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}})$ is itself a renewal and can be shown to be a Cox process. Moreover we prove that the survival probability of $N(\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}})$, as well as its probability generating function, are solution to the so-called fractional relaxation equation of distributed order (see \cite{Vib}%). 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 $\widetilde{\mathcal{T}}_{\nu_{1,}\nu_{2}}$. We thus provide an alternative to the constructions presented in Mainardi and Pagnini \cite{mapagn} and in Chechkin et al. \cite{che1}, at least in the double-order case.