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Relaxation equations with stretched non-local operators: renewal and time-changed processes

Luisa Beghin, Nikolai Leonenko, Jayme Vaz

TL;DR

This work extends renewal theory by introducing a stretched non-local operator $\mathcal{D}^{(\alpha,\gamma)}_t$ that blends Caputo-type fractional dynamics with a time-stretching parameter. It derives a Kilbas-Saigo–based solution for a first-order relaxation equation, yielding a renewal process whose interarrival survival can be tuned to be infinite or finite by the sum $\alpha+\gamma$, and offers a time-change representation via a generalized inverse subordinator. The authors then develop second-order relaxation models, showing that, under suitable conditions, the resulting renewal processes are convex combinations (mixtures) of first-order distributions, and they provide corresponding stochastic representations as well as non-renewal time-change perspectives. The analysis leverages advanced special functions (Kilbas-Saigo, double gamma, Fibonacci polynomials) to obtain explicit forms, convergence results, and Laplace/Legendre-type transform representations, broadening the toolkit for modeling anomalous interarrival dynamics and time-changed processes in complex systems.

Abstract

We introduce and study renewal processes defined by means of extensions of the standard relaxation equation through ``stretched" non-local operators (of order $α$ and with parameter $γ$). In a first case we obtain a generalization of the fractional Poisson process, which displays either infinite or finite expected waiting times between arrivals, depending on the parameter $γ$. Therefore, the introduction in the operator of the non-homogeneous term driven by $γ$ allows us to regulate the transition between different regimes of our renewal process. We then consider a second-order relaxation-type equation involving the same operator, under different sets of conditions on the constants involved; for a particular choice of these constants, we prove that the corresponding renewal process is linked to the first one by convex combination of its distributions. We also discuss alternative models related to the same equations and their time-changed representation, in terms of the inverse of a non-decreasing process which generalizes the $α$-stable Lévy subordinator.

Relaxation equations with stretched non-local operators: renewal and time-changed processes

TL;DR

This work extends renewal theory by introducing a stretched non-local operator that blends Caputo-type fractional dynamics with a time-stretching parameter. It derives a Kilbas-Saigo–based solution for a first-order relaxation equation, yielding a renewal process whose interarrival survival can be tuned to be infinite or finite by the sum , and offers a time-change representation via a generalized inverse subordinator. The authors then develop second-order relaxation models, showing that, under suitable conditions, the resulting renewal processes are convex combinations (mixtures) of first-order distributions, and they provide corresponding stochastic representations as well as non-renewal time-change perspectives. The analysis leverages advanced special functions (Kilbas-Saigo, double gamma, Fibonacci polynomials) to obtain explicit forms, convergence results, and Laplace/Legendre-type transform representations, broadening the toolkit for modeling anomalous interarrival dynamics and time-changed processes in complex systems.

Abstract

We introduce and study renewal processes defined by means of extensions of the standard relaxation equation through ``stretched" non-local operators (of order and with parameter ). In a first case we obtain a generalization of the fractional Poisson process, which displays either infinite or finite expected waiting times between arrivals, depending on the parameter . Therefore, the introduction in the operator of the non-homogeneous term driven by allows us to regulate the transition between different regimes of our renewal process. We then consider a second-order relaxation-type equation involving the same operator, under different sets of conditions on the constants involved; for a particular choice of these constants, we prove that the corresponding renewal process is linked to the first one by convex combination of its distributions. We also discuss alternative models related to the same equations and their time-changed representation, in terms of the inverse of a non-decreasing process which generalizes the -stable Lévy subordinator.

Paper Structure

This paper contains 23 sections, 14 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The double gamma function
  4. Kilbas-Saigo function
  5. Fibonacci polynomials
  6. A first-order equation involving $\mathcal{D}^{(\alpha,\gamma)}_t$
  7. Preliminary results
  8. Particular case
  9. First renewal model
  10. Stochastic representation
  11. Alternative (non-renewal) approach
  12. A second-order equation involving $\mathcal{D}^{(\alpha,\gamma)}_t$
  13. Preliminary results
  14. Particular case
  15. Second-order renewal models
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\alpha \in (0,1)$ and $\alpha + \gamma > 0$. Then for $\alpha+\gamma \leq 1$, the expected value of the interarrival time $U^{(\alpha, \gamma)}$ of the process $\mathcal{N}_{\alpha,\gamma}$ is infinite, whereas, for $\alpha+\gamma>1$, it is finite and reads

Theorems & Definitions (38)

  • Remark 1
  • Definition 1
  • Theorem 1
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 3
  • ...and 28 more