arXiv subjects

Eliana Fausti

Publications and source records attributed to Eliana Fausti.

5 recordsLinked to original sources

arXiv2026

An interacting particle system for the front of an epidemic advancing through a susceptible population

We introduce an interacting particle system that models the spread of an epidemic in terms of heterogeneous diffusive dynamics, rather than exogenous contact and transmission rates at the population level as in classical compartmental models. Each individual has a one-dimensional level of shielding that evolves according to a stochastic differential equation reflected at the advancing front of the epidemic. The front is driven by cumulative infections, and collisions with it represent at-risk situations which may lead to infection depending on a non-Markovian mechanism that involves the local time, the intrinsic transmissibility, and the current contagiousness within the population. We give a rigorous construction of the system and develop two key technical tools: a compensated martingale property for the infected proportion and a general result on how local time transforms under a random time-dependent bijection of the state space. The former yields a decomposition of the expected number of new infections that parallels a corresponding decomposition in the SIR model. The latter allows us to represent the law of each particle, after suitable conditioning, as a generalised elastic Brownian motion with drift.

math.PR↗

arXiv2026

A free boundary problem for the mean-field limit of diffusing particles with nonlinear boundary reactivity

Consider a finite system of diffusing particles coupled through a reactive boundary. Each particle is reflected, but may react with the boundary according to a killing mechanism which depends on the current reactivity of the boundary and the particle's local time along it. With every such reaction, the boundary moves and its reactivity adjusts. We show that this system admits a unique mean-field limit, described by a free boundary problem with nonlinear and nonlocal reactivity. The latter generalises the classical Robin condition for the case of a fixed boundary with constant reactivity. Via Skorokhod's M1 topology and a characterisation of the particles' behaviour near the boundary, we first identify the weak limit points of the empirical measure flows with killing. Then, we combine a probabilistic decoupling technique and energy estimates to prove uniqueness and deduce convergence. Our analysis gives a rigorous mean-field description of a model of epidemic spreading. Moreover, it contributes to the literature on inert drift systems and yields a novel mean-field perspective on the recent encounter-based framework for diffusion-mediated surface reaction from [Phys. Rev. Lett. 125 (2020) 078102].

math.PR↗

arXiv2025

A localized particle filter for geophysical data assimilation

Particle filters are computational techniques for estimating the state of dynamical systems by integrating observational data with model predictions. This work introduces a class of Localized Particle Filters (LPFs) that exploit spatial localization to reduce computational costs and mitigate particle degeneracy in high-dimensional systems. By partitioning the state space into smaller regions and performing particle weight updates and resampling separately within each region, these filters leverage assumptions of limited spatial correlation to achieve substantial computational gains. This approach proves particularly valuable for geophysical data assimilation applications, including weather forecasting and ocean modeling, where system dimensions are vast, and complex interactions and nonlinearities demand efficient yet accurate state estimation methods. We demonstrate the methodology on a partially observed rotating shallow water system, achieving favourable performance in terms of algorithm stability and error estimates.

stat.AP↗

arXiv2024

Hyperbolic contractivity and the Hilbert metric on probability measures

This paper gives a self-contained introduction to the Hilbert projective metric $\mathcal{H}$ and its fundamental properties, with a particular focus on the space of probability measures. We start by defining the Hilbert pseudo-metric on convex cones, focusing mainly on dual formulations of $\mathcal{H}$ . We show that linear operators on convex cones contract in the distance given by the hyperbolic tangent of $\mathcal{H}$, which in particular implies Birkhoff's classical contraction result for $\mathcal{H}$. Turning to spaces of probability measures, where $\mathcal{H}$ is a metric, we analyse the dual formulation of $\mathcal{H}$ in the general setting, and explore the geometry of the probability simplex under $\mathcal{H}$ in the special case of discrete probability measures. Throughout, we compare $\mathcal{H}$ with other distances between probability measures. In particular, we show how convergence in $\mathcal{H}$ implies convergence in total variation, $p$-Wasserstein distance, and any $f$-divergence. Furthermore, we derive a novel sharp bound for the total variation between two probability measures in terms of their Hilbert distance.

math.PR↗

arXiv2023

Exponential contractions and robustness for approximate Wonham filters

We consider the problem of estimating the state of a continuous-time Markov chain from noisy observations. We show that the corresponding optimal filter is strictly contracting pathwise, when considered in the Hilbert projective space, and give explicit deterministic and pathwise rates of convergence. Using this, we provide alternative proofs of the robustness of optimal filters, improving on known error estimates, and derive rigorous and computable error bounds for approximate filters.

math.ST↗