Strong propagation of chaos for systems of interacting particles with nearly stable jumps
Eva Löcherbach, Dasha Loukianova, Elisa Marini
TL;DR
The paper studies the large-N limit of interacting particles with collateral jumps driven by heavy-tailed, α-stable–attracted noise, revealing a conditional propagation of chaos where the limit empirical measure equals the law of a typical particle given the common α-stable noise. It introduces a novel coupling that represents the finite-system interaction as a stochastic integral against an α-stable process, enabling strong convergence to a limit nonlinear SDE driven by an α-stable process. Strong existence and uniqueness are established for both the finite system and the limit system, with a detailed analysis distinguishing the α>1 and α<1 regimes and handling main jumps accordingly. The main contributions include a rigorous representation of finite-particle interactions, a constructive coupling to the limit, and explicit finite-time error bounds and rates that quantify how fast the finite system approximates the conditional limit dynamics, with relevance to large neural-network models. The results extend diffusion-based mean-field limits to heavy-tailed Lévy noise, highlighting conditional chaos and the role of common noise in high-dimensional interacting systems.
Abstract
We consider a system of $N$ interacting particles, described by SDEs driven by Poisson random measures, where the coefficients depend on the empirical measure of the system. Every particle jumps with a jump rate depending on its position. When this happens, all the other particles of the system receive a small random kick which is distributed according to a heavy tailed random variable belonging to the domain of attraction of an $α-$ stable law and scaled by $N^{-1/α},$ where $0 < α<2 .$ We call these jumps collateral jumps. Moreover, in case $ 0 < α< 1, $ the jumping particle itself undergoes a macroscopic, main jump. Such systems appear in the modeling of large neural networks, such as the human brain. The particular scaling of the collateral jumps implies that the limit of the empirical measures of the system is random and equals the conditional distribution of one typical particle in the limit system, given the source of common noise. Thus the system exhibits the conditional propagation of chaos property. The limit system turns out to be solution of a non-linear SDE, driven by an $ α-$stable process. We prove strong unique existence of the limit system and introduce a suitable coupling to obtain the strong convergence of the finite to the limit system, together with precise error bounds for finite time marginals.