Probabilistic Method in Fractional Calculus

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 (20 November 2021) | Viewed by 5203

Special Issue Editors

Statistics Department, Warwick University, CV4 7AL Coventry, UK
Interests: probability; stochastic processes; fractional calculus; optimal control and games; mathematical physics
Rényi Institute of Mathematics, 1053 Budapest, Hungary
Interests: mathematical physics; stochastic and functional analysis; fractional calculus

Special Issue Information

Dear Colleagues,

The theory of fractional differential equations was initiated in the 19th century with the works of Riemann and Liouville, who introduced the basic objects of the theory, fractional integrals and fractional derivatives, now referred to usually as the Riemann–Liouville (RL) fractional integrals and fractional derivatives. Many versions of these definitions have appeared in the literature since then, including Gruenwald–Letnikov derivatives, Caputo derivatives and their multi-dimensional analogs. Remaining in a “sleepy mode” for an extended period (as it was poorly supported by the application), the theory of fractional equations started flourishing in recent decades, because it has been finally found to be extremely important for immense amounts of models in practically any domain of natural sciences, as well as in modelling social and economic behaviour. Together with new areas of applications, lots of new links with other domains of mathematics have been found and successfully exploited, both giving new insights into “fractional theory” and in the related domains.

Specifically distinguished are the strong links with probability and stochastic processes providing one of the strongest drives for the present research in the field. This link was initiated by physicists who promoted the theory of continuous time random walks (CTRW) that yield natural discrete approximations to fractional evolutions and at the same time describe many random physical processes (e.g., Hamiltonian chaos, Levy flights, anomalous diffusions). This development led to a variety of new insights linking theory with semi-Markov processes, Levy processes, stochastic optimal stopping theory, random time change, boundary value-problems for pseudo-differential equations and related questions about the behaviour of random processes near the boundary. A probabilistic approach to the study of fractional differential equations leads to the application of powerful Monte Carlo simulations to build effective numeric algorithms.

This Special Issue is aiming to highlight research where probability meets analysis in the study of various fractional phenomena, including various applications to natural processes, for instance, anomalous transport and anomalous diffusion, interacting particles, stochastic control and games, fractional stochastic processes and the scaling phenomenon in physics and biology, fractional stochastic equations and numeric algorithms.

Prof. Dr. Vassili N. Kolokoltsov
Dr. Jozsef Lorinczi
Guest Editors

Manuscript Submission Information

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Keywords

  • fractional differential equations (FDE)
  • fractional stochastic differential equations
  • fractional stochastic processes
  • fractional anomalous transport and diffusions
  • fractional stochastic control and games
  • continuous time random walks and scaling phenomena

Published Papers (4 papers)

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Research

11 pages, 300 KiB  
Article
Cauchy Processes, Dissipative Benjamin–Ono Dynamics and Fat-Tail Decaying Solitons
by Max-Olivier Hongler
Fractal Fract. 2022, 6(1), 15; https://doi.org/10.3390/fractalfract6010015 - 29 Dec 2021
Viewed by 778
Abstract
In this paper, a dissipative version of the Benjamin–Ono dynamics is shown to faithfully model the collective evolution of swarms of scalar Cauchy stochastic agents obeying a follow-the-leader interaction rule. Due to the Hilbert transform, the swarm dynamic is described by nonlinear and [...] Read more.
In this paper, a dissipative version of the Benjamin–Ono dynamics is shown to faithfully model the collective evolution of swarms of scalar Cauchy stochastic agents obeying a follow-the-leader interaction rule. Due to the Hilbert transform, the swarm dynamic is described by nonlinear and non-local dynamics that can be solved exactly. From the mutual interactions emerges a fat-tail soliton that can be obtained in a closed analytic form. The soliton median evolves nonlinearly with time. This behaviour can be clearly understood from the interaction of mutual agents. Full article
(This article belongs to the Special Issue Probabilistic Method in Fractional Calculus)
11 pages, 804 KiB  
Article
Random Times for Markov Processes with Killing
by Yuri G. Kondratiev and José Luís da Silva
Fractal Fract. 2021, 5(4), 254; https://doi.org/10.3390/fractalfract5040254 - 04 Dec 2021
Viewed by 1684
Abstract
We consider random time changes in Markov processes with killing potentials. We study how random time changes may be introduced in these Markov processes with killing potential and how these changes may influence their time behavior. As applications, we study the parabolic Anderson [...] Read more.
We consider random time changes in Markov processes with killing potentials. We study how random time changes may be introduced in these Markov processes with killing potential and how these changes may influence their time behavior. As applications, we study the parabolic Anderson problem, the non-local Schrödinger operators as well as the generalized Anderson problem. Full article
(This article belongs to the Special Issue Probabilistic Method in Fractional Calculus)
12 pages, 288 KiB  
Article
Nonexistence of Global Positive Solutions for p-Laplacian Equations with Non-Linear Memory
by Mokhtar Kirane, Ahmad Z. Fino and Sebti Kerbal
Fractal Fract. 2021, 5(4), 189; https://doi.org/10.3390/fractalfract5040189 - 28 Oct 2021
Viewed by 1059
Abstract
The Cauchy problem in Rd,d1, for a non-local in time p-Laplacian equations is considered. The nonexistence of nontrivial global weak solutions by using the test function method is obtained. Full article
(This article belongs to the Special Issue Probabilistic Method in Fractional Calculus)
16 pages, 313 KiB  
Article
Abstract Fractional Monotone Approximation with Applications
by George A. Anastassiou
Fractal Fract. 2021, 5(4), 158; https://doi.org/10.3390/fractalfract5040158 - 09 Oct 2021
Viewed by 1077
Abstract
Here we extended our earlier fractional monotone approximation theory to abstract fractional monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let [...] Read more.
Here we extended our earlier fractional monotone approximation theory to abstract fractional monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let fCp1,1, p0 and let L be a linear abstract left or right fractional differential operator such that Lf0 over 0,1 or 1,0, respectively. We can find a sequence of polynomials Qn of degree n such that LQn0 over 0,1 or 1,0, respectively. Additionally f is approximated quantitatively with rates uniformly by Qn with the use of first modulus of continuity of fp. Full article
(This article belongs to the Special Issue Probabilistic Method in Fractional Calculus)
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