Speaker: Fabrizio Cinque, Sapienza University of Rome, Italy

Time : 15:00 - 16.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom :  A zoom meeitng link will appear here, one hour before the talk

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords:  fractional Cauchy problems, Dzherbashyan--Caputo derivative, Convolutional derivative operators, Telegraph processes, Laplace and Fourier transforms

Abstract: We study Dzherbashyan–Caputo-fractional Cauchy problems related to equations with derivatives of order νk, for k non-negative integer and ν > 0. The explicit solution is expressed in terms of Mittag-Leffler-type functions and, introducing some additional hypotheses, it reduces into a linear combination of Mittag-Leffler functions with common fractional order ν. We establish a probabilistic relationship, involving the inverse of stable subordinator, between the solutions of differential problems with order αν and ν, for α ∈ (0, 1). Then, we use the described method to solve fractional differential equations arising by fractionalizing the time-operator of the partial differential equations related to the probability law of planar random motions with finite velocities.

The time-changing result is then presented in a more general framework concerning Cauchy problems with fractional differential equations in both space and time variables. Here the solution is expressed in terms of a “stochastic composition” of the solutions to two simpler problems. These Cauchy sub-problems respectively concern the space and the time differential operator involved in the main equation. We provide some probabilistic applications, where the solution can be interpreted as the transition density of a time-changed process

Biography: Currently working as an Expert in the Bank of Italy, since November 2021. I received my M.Sc in 2020 and PhD in Statistical Sciences (Methodological curriculum) in 2024 from Sapienza University of Rome, where I have been tutoring for several times in the graduated courses of stochastic processes and statistical inference. The core areas of my research are random motions with finite velocities (also known as telegraph processes or continuous time random walks), point processes, pseudo-processes, Airy functions and fractional calculus for stochastic processes.

Bibliography