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 $\nu \in (0,1]$ 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 $N($ The distribution of the random time argument ${\tilde{\mathcal{T}}}_{{\nu }_{1,}{\nu }_2}(t)$ can be expressed, for any fixed $t$, in terms of convolutions of stable-laws. The new process $N({\tilde{\mathcal{T}}}_{{\nu }_{1,}{\nu }_2})$ is itself a renewal and can be shown to be a Cox process. Moreover we prove that the survival probability of $N({\tilde{\mathcal{T}}}_{{\nu }_{1,}{\nu }_2})$, 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 ${\tilde{\mathcal{T}}}_{{\nu }_{1,}{\nu }_2}$. 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.

26 pages

References in corpus (5)

• Fractional diffusion equations and processes with randomly varying time
• Fractional pure birth processes
• On a fractional linear birth--death process
• Reaction-diffusion systems and nonlinear waves
• Renewal processes of Mittag-Leffler and Wright type

Cited by in corpus (10)

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• Anomalous diffusion originated by two Markovian hopping-trap mechanisms
• Lévy mixing related to distributed order calculus, subordinators and slow diffusions
• Mixed fractional Risk Process
• Convoluted Fractional Poisson Process
• Correlation structure of time-changed fractional Brownian motion
• On the Long-Range Dependence of Mixed Fractional Poisson Process
• Generalized Fractional Counting Process
• Fractional Skellam Process of Order $k$
• Space-fractional versions of the negative binomial and Polya-type processes