CS34: Non-local operators in probability: anomalous transport, stochastic resettings and diffusions with memory

date: 7/17/2025, time: 14:00-15:30, room: IM B

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

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

Non-Local Boudary Value Problems and Stochastic Resettings

Fausto Colantoni (Sapienza University of Rome)

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.

Time-Changed spherical Brownian motions with drift and their anomalous behaviour

Giacomo Ascione (Scuola Superiore Meridionale)

Anomalous diffusions in \(\mathbb{R}^d\) are characterized by a nonlinear relation between the mean-square displacement of the diffusing particles and the time-variable. It is clear, however, that this definition cannot be applied to diffusing particles in a compact Riemannian manifold. Even for the Brownian motion, while the linear character of the mean-square displacement is preserved for small times, the geometric properties of the considered domain clearly imposes an upper bound. As a consequence, one cannot recognize anomalous diffusions from standard ones by the observation of the asymptotic behaviour of the mean-square displacement. Nevertheless, even in \(\mathbb{R}^d\), one can recognize anomalous diffusions that exhibit a linear behaviour for small times, while deviating from this as time increases: hence, even the small-time asymptotic of the mean-square displacement could be not sufficient.

In this talk, we introduce a prototype of anomalous diffusion on the sphere. Precisely, we will consider a spherical Brownian motion (possibly with a special drift described by a suitable vector field) time-changed by means of an independent inverse subordinator. These models were first considered (without drift) in \([1,2,3]\), while the drifted model has been described in \([4]\). We first study the inter-connection between such processes and some time-nonlocal Kolmogorov equations on the sphere. This is done by providing an explicit representation, in terms of spherical harmonics, of strong solutions of such equations when the initial data belongs to \(H^s(\mathbb{S}^2)\) for some \(s>1\), where \(\{H^s(\mathbb{S}^2)\}_{s \in \mathbb{R}}\) is the Sobolev scale induced by the Laplace-Beltrami operator on the sphere. Thanks to this representation, we are able to show that such models still exhibit the same stationary/limit measure as the spherical Brownian motion (i.e., the uniform distribution on the sphere), but they converge to such a state in a non-exponential way. The latter phenomenon is called anomalous relaxation: we propose this characteristic to actually recognize anomalous diffusive behaviours on compact manifolds.

Bibliography

\([1]\) Mirko D’Ovidio and Erkan Nane. "Time dependent random fields on spherical non-homogeneous surfaces." Stochastic Processes and their Applications vol. 124, no. 6, 2014, pp. 2098-2131.

\([2]\) Mirko D’Ovidio and Erkan Nane. "Fractional Cauchy problems on compact manifolds." Stochastic Analysis and Applications vol. 34, no. 2, 2016, pp. 232-257.

\([3]\) Mirko D’Ovidio, Enzo Orsingher,and Lyudmyla Sakhno. "Models of space-time random fields on the sphere." Modern Stochastics: Theory and Applications vol. 9, no. 2, 2022, pp. 139-156.

\([4]\) Giacomo Ascione and Anna Vidotto. "Time changed spherical Brownian motions with longitudinal drifts." Stochastic Processes and their Applications vol. 181m, 2025, pp. 104547.

Anomalous Random Flights and Time-Changed Random Evolutions

Francesco Iafrate (University of Hamburg)

Random flights represent finite velocity random motions changing direction at any Poissonian time. We present an extension of such models to time-fractional processes arising from a non-local generalization of the kinetic equations. The probabilistic interpretation of the solutions of the time-fractional equations leads to a time-changed version of the original transport processes. The obtained results provide a clear picture of the role played by the time-fractional derivatives in this kind of random motions. The anomalous behavior of these random scattering models is useful to describe several complex systems arising in statistical physics and biology. In the case of the one-dimensional random flight, called telegraph process, we study the time-fractional version of the classical telegraph equation. Furthermore, by exploiting the Kac’s approach we give a suitable stochastic interpretation of the solution of the fractional telegraph equation.

We further pursue an extension of our framework to random evolutions, which describe Markovian (or semi‑Markovian) systems undergoing stochastic regime switches. The aforementioned telegraph process serves as a prototypical example of such dynamics. We proceed to construct their time‑changed counterparts and tackle the associated non‑local evolution equations in a general abstract setting.

Bibliography

\([1]\) L. Angelani, A. De Gregorio, R. Garra and F. Iafrate, Anomalous random flights and time-fractional run-and-tumble equations . J Stat Phys, 191, 129 (2024)

\([2]\) A. De Gregorio and F. Iafrate, Time-changed random evolutions and related non-local equations . Working paper (2025)