Giacomo Ascione [pdf]
Titolo: Rollnik classes and Lieb-Thirring inequalities for pseudo-relativistic Schrodinger equations
Abstract: In this talk, we consider some extensions of the classical Rollnik class to the case of the pseudo-relativistic Schrodinger equation. The fractional Rollnik classes here introduced share a non-empty intersection with the Kato class and include Coulomb-type potentials with singularity up to the critical one of the Hardy potential. In particular, by means of the Birman-Swinger principle, we are able to show that pseudo-relativistic Schrodinger operators with fractional Rollnik potentials are still self-adjoint. In particular, during the talk, we will focus on a 0-exponent Lieb-Thirring inequality (i.e., a bound on the number of negative eigenvalues) and on the non-existence of negative or zero eigenvalues. This is a joint work with Atsuhide Ishida from Tokyo University of Science and Joszef Lorinczi from Renyi Institute of Mathematics.
Fausto Colantoni [pdf]
Titolo: Elastic Brownian motion with random jumps from the boundary
Abstract: We study an elastic Brownian motion on smooth domains, where the particle, instead of being killed at the boundary, restarts from a random position inside the domain. We characterize the process through its SDE and generator, and describe its invariant measure. Through time reversal, we explore connections with statistical mechanics and with Brownian motion under Poissonian resetting.
Simone Creo [pdf]
Titolo: On (fractional) inverse problems in irregular domains
Abstract: In this talk we consider parabolic inverse problems in irregular domains and in suitable smoother approximating structures. After proving well-posedness results, we prove that the solutions of the approximating problems converge in a suitable sense to the solution of the problem in the irregular
domain via Mosco convergence. We also present some applications. We then conclude by mentioning some recent results in the time-fractional case, i.e. we prove well-posedness results for inverse problems with Caputo-type fractional derivatives. These results are obtained in collaboration with M. R. Lancia, A. Mola, G. Mola and S. Romanelli.
Alessandra Meoli [pdf]
Titolo: Tempered stable time changes of bivariate Poisson counts with applications to shock models
Abstract: In this talk I introduce the bivariate tempered space-fractional Poisson process (BTSFPP), obtained by running two independent Poisson counters on a common tempered α-stable subordinator. I will focus on the main distributional results: probability generating function, evolution equation, and an explicit discrete Lévy measure. Then, as a main application, I will consider competing-risk shock models driven by a BTSFPP, where a system fails when the cumulative number of shocks reaches a random threshold.
Verdiana Mustaro [pdf]
Titolo: On some spatial transformations of multidimensional diffusion processes and related absorption problems
Abstract: The aim of this talk is to present a general framework for the construction of new multidimensional diffusion processes through a spatial transformation. The idea is to express the probability density function (p.d.f.) of the new diffusion in terms of the former p.d.f. This is done by means of a product-form relation involving a weight function w, which is obtained as the solution of a partial differential equation and only affects the drift vector of the diffusion. This approach makes it possible to extend the issue of transformed diffusions, typically addressed in one dimension, to the multidimensional setting.
Moreover, it allows to obtain closed-form relations between the original and transformed processes, with applications to stochastic ordering, Poissonian resetting, and diffusion in potential fields.
Particular attention is devoted to the analysis of two-dimensional processes. Indeed, in the case of two-dimensional diffusions satisfying specific symmetry properties, explicit results are derived for the transition probability in the presence of an absorbing boundary. Two case studies, concerning the Wiener and Ornstein–Uhlenbeck processes, are also extensively treated. For such processes, explicit expressions of the weight function, transition densities, and potential functions are derived. A remarkable result is that the Wiener case admits a mixture representation which reveals the emergence of multimodal behaviors. Finally, specific analytical results are presented for the two aforementioned processes in the two-dimensional case
Luca Schiavone [pdf]
Titolo: Stochastic Hamiltonian Field Theories on globally hyperbolic spacetimes: a multisymplectic approach
Abstract: We develop a novel, fully geometric framework for first-order Hamiltonian field theories subject to stochastic influences.
Our construction combines the multisymplectic approach to classical field theories of F. Gay-Balmaz, J. C. Marrero, and N. Martínez [BMM2022], which employs a canonical affine bracket on a finite-dimensional phase space, with the formulation of stochastic Hamiltonian mechanics on manifolds due to J. A. Lazaro-Camì and J. P. Ortega [LCO2008]. By modeling the dynamics of a classical field as an evolution along stochastic worldlines on a globally hyperbolic spacetime, we derive a stochastic generalization of the de Donder-Weyl equations from a stochastic Heisenberg-like equation for currents. A stochastic version of Noether's Theorem for field theories is given, which provides a criterion to determine whether classical conservation laws are preserved in the presence of noise.
[BMM2022] F. Gay-Balmaz, J. C. Marrero, and N. Martínez. A new canonical affine bracket formulation of Hamiltonian Classical Field theories of first order. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 118: 1–60, 2024.
[LCO2008] J. A. Lazaro-Camì and J. P. Ortega. Stochastic Hamiltonian dynamical systems. Reports on Mathematical Physics, 61: 65–122, 2008.
Serena Spina [pdf]
Titolo: A generalized Ehrenfest model with catastrophes and the related Ornstein-Uhlenbeck process
Abstract: We study a multi-type Ehrenfest process modeled as a finite quasi-birth-death (QBD) process. The QBD process is defined by two coordinates: the level, which tracks the count of presences within the system at time t, and the phase, which represents the system's internal state or the state of its governing environment at time t within that level. In the present model, we assume that the transitions are allowed only to the two adjacent levels of the same phase origin and are characterized by linear rates. The crucial element lies in the phase switching mechanism at the origin, which is governed by an irreducible stochastic matrix. The process evolution is interrupted by catastrophic events, whose occurrences are controlled by a Poisson process. Each catastrophe resets the system's state to zero, initiating a new cycle of evolution until the next resetting event. We conduct a comprehensive analysis, addressing both the transient and long-term behavior of this process. Furthermore, we derive diffusive approximation for the process, by proving its convergence to a reflected Ornstein–Uhlenbeck jump diffusion process.
Maria Chiara Bovier, Space-dependent Continuous Time Random Walk for Soil Moisture Modeling, [pdf]
Mirko D'Ovidio, On the non-local dynamic problems and non-Markov processes, [pdf]
Daniel Eduardo Cedeño Giron, Sampling Killed Anomalous Subdiffusions, [pdf]
Francesco Virgili, On the infinitesimal behavior of semi-Markovian CTRWs, [pdf]