CS34: Non-Local Operators in Probability: Anomalous Transport, Stochastic Resettings and Diffusions with Memory

Organizer: Lorenzo Cristofaro & Giacomo Ascione (University of Luxembourg & Scuola Superiore Meridionale)

Non-Local Boudary Value Problems and Stochastic Resettings

Fausto Colantoni

We investigate the connection between Non-local Boundary Value Problems (NLBVPs) for the heat equation on the positive half-line and Brownian motion with Poissonian resetting. By NLBVPs, we refer to problems involving the heat equation in the domain with non-local operators at the boundary, such as Marchaud-type derivatives. In this case, Brownian motion tends to escape from the boundary via jumps, with each jump corresponding to the last jump of the subordinator associated with the non-local operator. In contrast, when Brownian motion is subject to Poissonian resetting, it is reset to the origin at exponentially distributed random times, leading to a concentration near the boundary. We show that these two dynamics are related when viewed through time reversal.

Bibliography

\([1]\) F. Colantoni. "Non-local skew and non-local skew sticky Brownian motions." Journal of Evolution Equations, vol. 25, Art. 39, 2025.

\([2]\) S. Bonaccorsi, F. Colantoni, M. D’Ovidio, and G. Pagnini. "Non-local Boundary Value Problems, stochastic resetting and Brownian motions on graphs." Submitted, arXiv:2209.14135, 2024.

\([3]\) F. Colantoni, M. D’Ovidio, and G. Pagnini. "Time reversal of Brownian motion with Poissonian resetting." Submitted, arXiv:2505.15639, 2025.