Probabilistic Solutions and Stochastic Representation of Fractional Differential Equations

Dear Colleagues,

Fractional differential equations have been shown to be an important tool in probability due to their strict connection with some classes of processes that are widely used for modeling purposes. With this connection, different stochastic representation results have arisen, which are helpful in a twofold way: on one hand, one can use fractional equations (and both analytical and numerical methods for them) to determine some characteristics of the considered process; on the other hand, fractional equations gain a more visualizable interpretation (that was already useful in the classical case) that can help to determine some properties of their solutions. This interplay is mirrored in the modeling context as fractional differential equations are used to represent macroscopic behavior whose microscopic explanation is given by the involved stochastic processes. For these reasons, finding probabilistic solutions to (possibly generalized) fractional differential equations improves both our knowledge on the equation and the involved process, thus leading to an improvement of the modeling techniques involving fractional operators. Potential topics include but are not limited to:

• Stochastic representation results for solutions of time/space-fractional ordinary and partial differential equations;
• Probabilistic solutions to distributed-order time/space-fractional ordinary and partial differential equations;
• Generalized fractional calculus, time-nonlocal equations, and stochastic processes;
• Bernstein functions of elliptic operators and generators of jump processes;
• Fractional order stochastic (partial) differential equations;
• Deterministic and stochastic modeling involving fractional order operators;
• Numerical methods for fractional (partial) differential equations.

Dr. Giacomo Ascione
Dr. Alessandra Meoli
Prof. Dr. Enrica Pirozzi
Guest Editors

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• stochastic processes
• fractional order differential operators
• distributed order differential operators
• fractional calculus
• generalized fractional calculus
• time/space-nonlocal equations
• bernstein functions
• jump processes
• semi-markov processes
• stochastic modeling
• numerical methods for fractional differential equations
• stochastic simulation