Time-nonlocal operators in probability: a brief summary

Ascione Giacomo, Scuola Superiore Meridionale

In recent years several models involving nonlocal operators in the time variable have been considered and some corresponding stochastic processes have been studied. Such models have been used, in particular, in the description of a wide range of anomalous diffusion. The aim of this talk is to briefly introduce some of these processes, highlighting the link with some anomalous diffusion phenomena that have been experimentally observed. In particular, we will focus, in the first part of the talk, on the link between a specific class of semi-Markov processes and some nonlocal operators, in particular from the point of view of the stochastic representation of the solutions of time-nonlocal abstract Cauchy problem. In the second part of the talk, we will present an example in which time (and space)-nonlocality emerges from a leader-follower system, due to the coupling of a parabolic partial differential equation with a system of ordinary differential equations. The introductory nature of such a talk, whose first aim is to stimulate interest in these kind of problems more than presenting some recent result, will be complemented by the further and deeper discussion of these themes in the other talks of the session.

Area: IS13 - Non-local operators in probability (Giacomo Ascione)

Keywords: Nonlocal operators, subordinators, anomalous diffusions

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