Description | The workshop aims to explore recent advances in the study of superdiffusive dynamics, with a particular focus on random motion in random Lévy media. We also aim to bring together researchers from various branches of mathematics at the intersection of probability and mathematical physics, fostering discussions on the most significant open problems in this field. We especially welcome the participation of young researchers and PhD students. |
Planning | | 10:00 ‑ 10:30 | Welcoming and coffee break | | 10:30 ‑ 11:20 | Marco Lenci ( Università degli studi di Bologna)
Title: An overview of semi-recent results on RWs in 1D Lévy random media
Abtract: I will speak of several versions of a continuous-time random walk which travels with constant speed between the points of a Lévy random medium, i.e., a random point process on the real line such that the distances between neighboring points are i.i.d. and heavy-tailed, The laws of the walk and of the medium are independent. These models generalize the so-called Lévy-Lorentz gas introduced by Barkai, Fleurer and Klafter in 2000, as a toy model for phenomena of anomalous diffusion in physics. I will give an overview of recent and less recent limit theorems, focusing a bit more on those studied by Bianchi, Cristadoro, Ligabò, Pène and myself. | | 11:30 ‑ 12:20 | Marco Zamparo (Università degli Studi del Piemonte Orientale "Amedeo Avogadro")
Title: Transport Properties of the Lévy-Lorentz gas
Abtract:The Lévy-Lorentz gas describes the motion of a particle on the real line in the presence of a random array of scattering points, whose distances between neighboring points are heavy-tailed i.i.d. random variables with finite mean. The motion is a continuous-time, constant-speed interpolation of the simple symmetric random walk on the marked points. In this seminar I review the model and discuss its transport properties, both annealed and quenched, under the assumption that the tail distribution of the spacing between scatterers is regularly varying at infinity. In particular, by presenting large deviation estimates and the asymptotics of moments for the particle displacement, I show that the motion is superdiffusive in the annealed framework, whereas a normal diffusive behaviorcharacterizes the motion conditional on a typical realization of the scatterers. Finally, supposing to average over identical copies of the system, I propose some preliminary results on the quenched to annealed transition as the number of copies increases.
[M. Zamparo, Large fluctuations and transport properties of the Lévy-Lorentz gas, Ann. Inst. H. Poincaré Probab. Statist. 59 (2023), 621-661] | | 12:30 ‑ 14:30 | Lunch break | | 14:30 ‑ 15:20 | Gaia Pozzoli (CY Cergy Paris Université)
Title: Ladder costs for random walks in Lévy random media
Abtract: We consider a random walk moving on a Lévy random medium, that is a one-dimensional renewal point process with i.i.d. inter-distances in the domain of attraction of a stable law, focusing on the characterization of the law of its first-ladder height and length. The study relies on the construction of a broader class of processes, denoted as Random Walks in Random Scenery on Bonds (RWRSB), consisting of a scenery which associates two random variables with each bond of the integer lattice Z (corresponding to the two possible crossing directions of that bond) and a random walk S on Z that collects the scenery values of the bond it traverses. Under suitable assumptions, we characterize the tail distribution of the sum of the scenery values collected up to the first-passage time in the positive semi-axis. This setting will be applied to obtain results for the first-ladder quantities of random walks in Lévy media, with an additional application in the context of the (generalized) Lévy-Lorentz gas. The main tools of investigation are a generalized Spitzer-Baxter identity and a suitable representation of the RWRSB in terms of local times of the random walk S.
[Joint work with A. Bianchi and G. Cristadoro]. | | 15:30 ‑ 16:20 | Giampaolo Cristadoro (Università degli studi di Milano-Bicocca)
Title: Precise large deviations through a uniform Tauberian theorem
Abtract: I will derive a large deviation principle for families of random variables in the basin of attraction of spectrally positive stable distributions by proving a uniform version of the Tauberian theorem for Laplace-Stieltjes transforms. The main advantage of this method is that it can be easily applied to cases that are beyond the reach of the techniques currently used in the literature, such as random walks with long-ranged memory kernels or randomly stopped sums where the random time is not concentrated around its expectation. The method reveals the role of the characteristic function when Cramér's condition is violated and provides a unified approach within regular variation.
[Joint work with G. Pozzoli]. | |
Organizers | David Barbato (Dipartimento di Matematica, Università di Padova, Italy)
Alessandra Bianchi (Dipartimento di Matematica, Università di Padova, Italy)
Bernardo D'Auria (Dipartimento di Matematica, Università di Padova, Italy)
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Contacts | For information please mail the organizers. |
Financial
Support | The organization of the workshop has been supported by the Department of Mathematics of the Università of Padova
through the BIRD project 239937 - "Stochastic dynamics on graphs and random structures". |
