Probabilistic Method in Fractional Calculus

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

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• 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