Dear Colleagues,

We would like to invite you to the following SPASS seminars,

Fully nonlocal operators emerging from semi-Markov processes

by Giacomo Ascione (Scuola Superiore Meridionale),

TUE 2.12.2025 at 14:00 CET in Sala Riunioni, Dipartimento di Matematica, UNIPI.

A link for online participants will be shared on the website https://sites.google.com/unipi.it/spass 

Best,

Francesco Grotto on behalf of the organizers

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Abstract: In the context of anomalous diffusions, time/space-nonlocal operators play a quite prominent role. For instance, the usual fractional Laplacian can be easily recognized as the generator of an isotropic stable Lèvy motion, that, still preserving the Markov property, represent a first (actually pure jump) example of anomalous diffusion, in the sense that the mean square displacement of the described particles (or, in terms of the process, its variance) is not linear in time (actually it is infinite). On the other hand, if we consider the heat equation with a fractional derivative operator in time, a stochastic representation of the solution of such an equation can be given in terms of a Brownian motion time-changed by means of the first passage time process of an independent stable subordinator. This process, however, clearly loses the Markov property, due to the addition of a further dependence on the current sojourn time. In both case, we are describing (at least in terms of the one-dimensional distributions) the behaviour of a time-changed Brownian motion, where the time-change is indeed independent of the process itself. The same can be done if we apply a time-change via an inverse subordinator to an independent isotropic stable Lèvy motion, (obtaining) uncoupled non-local operators in both time and space. However, if we remove the independence assumption, we cannot expect to get uncoupled behaviours in time and space. This is the case, for instance, of a Brownian motion time-changed first with an independent stable subordinator and then with its own inverse. As the second time-change is clearly dependent of the first one, we expect the two operators (in time and space) to be coupled in some way: precisely, in such a case, the Kolmogorov-type equation one gets is governed by a fractional power of the whole heat operator. In this talk, I will present some well-posedness results for fully nonlocal operators obtained by means of Bochner subordination of heat-like operators in terms of a family of semi-Markov processes obtained by means of non-independent time-changes. The general theory will be set in the framework of Feller semigroups on the space of continuous functions vanishing at infinity. Nevertheless, if there is time, a quite specific case (i.e., a Black-Sholes model for intraday options) that requires refined arguments will be discussed and the role of the regularity of the initial data will be addressed. Finally, we will stress the following expected result: despite the appearance of constancy intervals and the lack of Markov property, if we start from a Brownian motion the newly obtained process exhibits a normal diffusive behaviour, instead of an anomalous one.