IS06: Levy-type processes

date: 7/17/2025, time: 16:00-17:30, room: IM EM

Organizer: Victoria Knopova (Kyiv Taras Shevchenko National University)

Chair: Victoria Knopova (Kyiv Taras Shevchenko National University)

Semigroups under resetting: construction and convergence

Karol Szczypkowski (Wrocław University of Science and Technology)

Using the language of integral kernels, we will discuss the mechanism of resetting at Poissonian times through successive terms of a sequence whose coordinates are realizations of independent random variables taking values in \([0,1]\).
Next, applications to multidimensional strictly stable Lévy processes under resetting will be presented. In particular, we are interested in the properties of the density \(p(t; x, y)\) of these processes. One such property is its behavior relative to the density of the ergodic measure \(\rho(y)\), as \(t \to +\infty\) and \(y = y(t)\), depending on the region \([0, \infty) \times \mathbb{R}^d\) to which \((t, y)\) belongs. For example, for Brownian motion under resetting by a constant sequence \((c, c, c, \ldots)\), where \(c \in (0,1)\), we observe a change in behavior (a phase transition) relative to regions separated by the hyperplane \(|y| = 2t\).
The results were obtained jointly with T. Grzywna, Z. Palmowski, and B. Trojan.

Bibliography

\([1]\) Grzywny Tomasz, Palmowski Zbigniew, Szczypkowski Karol, Trojan Bartosz Stationary states for stable processes with partial resetting, arXiv:2412.15626

\([2]\) Costantino Di Bello, Aleksei Chechkin, Grzywny Tomasz, Palmowski Zbigniew, Szczypkowski Karol, Trojan Bartosz Partial versus total resetting for Levy flights in d dimensions: Similarities and discrepancies, Chaos 35, 043129 (2025).

Super-diffusive asymptotic behaviour of an interface kinetic model

Lorenzo Marino (Scuola Normale Superiore di Pisa)

We study the scaling limit of solutions to a linearised Boltzmann phonon equation in one spatial dimension, subject to a random mechanism of transmission, reflection or absorption at an interface. Assuming a fast enough degeneracy of the scattering kernel and a slow decay for the probability of absorption at low frequencies, we show that the solutions to the interface model exhibit a super-diffusive behaviour in the long time limit, with the scaling parameter depending only on the interplay between the decay velocities of the scattering kernel and the drift. We also characterise such a limit as the unique weak solution to a non-local in space evolution equation, subject to boundary conditions at the interface. Our proof relies on the probabilistic formulation of the above problem, establishing a new invariance principle for the associated stochastic processes.

This talk is based on a work in collaboration with Tomasz Komorowski and Krzysztof Bogdan. The activity was carried out within the project "Noise in Fluids", Grant Agreement 101053472, CUP E53C22001720006.

Liouville theorems for Fourier Multipliers

David Berger (TU Dresden)

In this talk we will discuss Liouville theorems for a class of Fourier multipliers, which contains especially the generators of Lévy operators. We are looking at the bounded case and show that with some adjustments the proof goes through for functions satisfying certain growth conditions. In particular, we present new results for Fourier Mulipliers in dimension \(1\). At the end we present new results for the unique continuation properties of non-local operators.

Bibliography

\([1]\) D. Berger, R. Schilling. The (strong) Liouville property for a class of non-local operators, Math. Scand. (2022), 128(2).

\([2]\) D. Berger, R. Schilling, E. Shargordosky The Liouville Theorem for a class of Fourier Multipliers and its connection to coupling. B. Lond. Math. Soc., appeared online first, DOI: 10.1112/blms.1306 (2024).

\([3]\) D. Berger, R. Schilling, E. Shargordosky, T. Sharia. An extension of the Liouville theorem for Fourier multipliers to sub-exponentially growing solutions. J. Spectr. Theory (2024), 14(2).

\([4]\) D. Berger, R. Schilling. On the unique continuation property of non-local operators. Working paper.