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Fractal Fract
Special Issues
Interplay between Fractional Differential Equations and Stochastic Processes
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Interplay between Fractional Differential Equations and Stochastic Processes

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: closed (31 October 2022) | Viewed by 4897

Special Issue Editor

Dr. Costantino Ricciuti

E-Mail Website
Guest Editor
Department of Statistical Sciences, University Sapienza, Rome, Italy
Interests: probability and stochastic processes

Special Issue Information

Dear Colleagues,

The link between probability theory and fractional differential equations has been a considerably robust research topic for several years.

Time-changed stochastic processes, fractional differential equations, anomalous diffusions, and continuous-time random walks are all interrelated according to well-established theories. Many applications have been developed in statistical physics, economics, and applied sciences.

Recent studies have deepened the interplay between fractional equations and stochastic processes. We could consider, for example, time–fractional equations where the order of fractionality is space dependent; one of the most remarkable cases is a fractional heat equation modelling diffusion in a heterogeneous environment.

However, we could also mention fractional equations governing Markov processes time-changed by multivariate subordinators, which include models of diffusions in anisotropic media.

One of the most studied models is the fractional Poisson process. Some of its extensions have recently been proposed, e.g., its time-inhomogeneous versions, discrete-time counterparts (known as fractional Bernoulli processes), as well as a multiparametric version (called fractional Poisson field).

This Special Issue aims to collect recent viewpoints on the above topics. Some examples of possible research themes include, but are not limited to, the following topics:

  • Probabilistic interpretation and properties of solutions for (possibly new) fractional differential equations;
  • New types of time-changed processes and continuous-time random walks;
  • Deepening on anomalous diffusions and related problems in statistical mechanics;
  • Applications in various fields, including physics, economics, and engineering.

Dr. Costantino Ricciuti
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional differential equations
  • fractional derivatives and integrals
  • fractional laplacian
  • fractional poisson processes
  • levy processes and subordinators
  • semi-markov processes
  • anomalous diffusion

Published Papers (3 papers)

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Research

20 pages, 365 KiB  
Article
Some Compound Fractional Poisson Processes
by Mostafizar Khandakar and Kuldeep Kumar Kataria
Fractal Fract. 2023, 7(1), 15; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7010015 - 25 Dec 2022
Cited by 3 | Viewed by 975
Abstract
In this paper, we introduce and study fractional versions of the Bell–Touchard process, the Poisson-logarithmic process and the generalized Pólya–Aeppli process. The state probabilities of these compound fractional Poisson processes solve a system of fractional differential equations that involves the Caputo fractional derivative [...] Read more.
In this paper, we introduce and study fractional versions of the Bell–Touchard process, the Poisson-logarithmic process and the generalized Pólya–Aeppli process. The state probabilities of these compound fractional Poisson processes solve a system of fractional differential equations that involves the Caputo fractional derivative of order 0<β<1. It is shown that these processes are limiting cases of a recently introduced process, namely, the generalized counting process. We obtain the mean, variance, covariance, long-range dependence property, etc., for these processes. Further, we obtain several equivalent forms of the one-dimensional distribution of fractional versions of these processes. Full article
(This article belongs to the Special Issue Interplay between Fractional Differential Equations and Stochastic Processes)
20 pages, 380 KiB  
Article
Tychonoff Solutions of the Time-Fractional Heat Equation
by Giacomo Ascione
Fractal Fract. 2022, 6(6), 292; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6060292 - 27 May 2022
Cited by 1 | Viewed by 1362
Abstract
In the literature, one can find several applications of the time-fractional heat equation, particularly in the context of time-changed stochastic processes. Stochastic representation results for such an equation can be used to provide a Monte Carlo simulation method, upon proving that the solution [...] Read more.
In the literature, one can find several applications of the time-fractional heat equation, particularly in the context of time-changed stochastic processes. Stochastic representation results for such an equation can be used to provide a Monte Carlo simulation method, upon proving that the solution is actually unique. In the classical case, however, this is not true if we do not consider any additional assumption, showing, thus, that the Monte Carlo simulation method identifies only a particular solution. In this paper, we consider the problem of the uniqueness of the solutions of the time-fractional heat equation with initial data. Precisely, under suitable assumptions about the regularity of the initial datum, we prove that such an equation admits an infinity of classical solutions. The proof mimics the construction of the Tychonoff solutions of the classical heat equation. As a consequence, one has to add some addtional conditions to the time-fractional Cauchy problem to ensure the uniqueness of the solution. Full article
(This article belongs to the Special Issue Interplay between Fractional Differential Equations and Stochastic Processes)
11 pages, 309 KiB  
Article
A Generalization of Multifractional Brownian Motion
by Neha Gupta, Arun Kumar and Nikolai Leonenko
Fractal Fract. 2022, 6(2), 74; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6020074 - 30 Jan 2022
Cited by 1 | Viewed by 1952
Abstract
In this article, some properties of multifractional Brownian motion (MFBM) are discussed. It is shown that it has persistence of signs long range dependence (LRD) and persistence of magnitudes LRD properties. A generalization called here nth order multifractional Brownian motion (n [...] Read more.
In this article, some properties of multifractional Brownian motion (MFBM) are discussed. It is shown that it has persistence of signs long range dependence (LRD) and persistence of magnitudes LRD properties. A generalization called here nth order multifractional Brownian motion (n-MFBM) that allows to take the functional parameter H(t) values in the range (n−1,n) is discussed. Two representations of the n-MFBM are given and their relationship with each other is obtained. Full article
(This article belongs to the Special Issue Interplay between Fractional Differential Equations and Stochastic Processes)
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