M. D’Ovidio : Fractional Boundary Value Problems: Results and Applications

Date: Friday, 19 July, 2024 - 16:00 to 17:00 CEST
Title : Fractional Boundary Value Problems: Results and Applications
Speaker : Mirko D’Ovidio, Dept. of Basic and Applied Sciences for Engeenering / Sapienza University of Rome

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy
Organizers : Pavan Pranjivan Mehta* and Arran Fernandez**
SISSA, International School of Advanced Studies, Italy
** Eastern Mediterranean University, Northern Cyprus

Keywords: Sticky Brownian motions, nonlocal dynamic conditions, time changes, metric graph

Abstract

Sticky diffusion processes spend finite time (and finite mean time) on a lower-dimensional boundary. Once the process hits the boundary, then it starts again after a random amount of time. While on the boundary it can stay or move according to dynamics that are different from those in the interior. Such processes may be characterized by a time-derivative appearing in the boundary condition for the governing problem. We use time changes obtained by right-inverses of suitable processes in order to describe fractional (or nonlocal in general) sticky conditions and the associated boundary behaviours. We obtain that fractional boundary value problems (involving fractional dynamic boundary conditions) lead to sticky diffusions spending an infinite mean time (and finite time) on a lower-dimensional boundary. Such a behaviour can be associated with a trap effect in the macroscopic point of view.

For the nonocal time boundary conditions, we first discuss the apparently simple case of the half line with boundary of zero Lebesgue measure. In this case we present some applications concerned with motions on metric graphs. Such results turn out to be instructive for the general case of boundary with positive (finite) Borel measures. In this regard, we provide some results on open, connected and non-empty sets with smooth boundaries and describe possible applications involving motions on irregular domains, fractals for instance.

We briefly discuss also nonlocal space conditions on the boundary and the associated processes. The underlying dynamics can be related with the stochastic resetting.

The talk is based on the works listed below in the References.

Biography

Mirko D’Ovidio is associate professor in Probability and Mathematical Statistics at Sapienza University of Rome. He works on the connections between stochastic processes and PDEs, time changes and boundary value problems, nonlocal operators and irregular domains.

Bibliography

[1] M. D’Ovidio. Fractional Boundary Value Problems. Fract. Calc. Appl. Anal. 25 (2022), 29-59.

[2] M. D’Ovidio. Fractional Boundary Value Problems and Elastic Sticky Brownian Motions, I: The half line. Submitted, arXiv:2402.12982

[3] M. D’Ovidio. Fractional Boundary Value Problems and Elastic Sticky Brownian Motions, II: The bounded domain. Submitted, arXiv:2205.04162

[4] S. Bonaccorsi, M. D'Ovidio. Sticky Brownian motions on star graphs. Submitted, arXiv:2311.07521

[5] S. Bonaccorsi, F. Colantoni, M. D'Ovidio. Non-local Boundary Value Problems, Stochastic resetting and Brownian motions on Graphs. To be submitted.