Strong propagation of chaos for systems of interacting particles with nearly stable jumps

TL;DR

This paper studies a system of interacting particles with jumps influenced by heavy-tailed distributions, demonstrating strong propagation of chaos and convergence to a non-linear SDE driven by an alpha-stable process, relevant for neural network modeling.

Contribution

It establishes strong propagation of chaos for particle systems with nearly stable jumps and proves convergence to a non-linear alpha-stable driven SDE, including error bounds.

Findings

The empirical measure converges to a conditional distribution of a typical particle.

The limit system has a unique strong solution driven by an alpha-stable process.

Finite systems exhibit strong convergence with explicit error estimates.

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 $\alpha-$ stable law and scaled by $N^{-1/\alpha},$ where $0 < \alpha <2 .$ We call these jumps collateral jumps. Moreover, in case $ 0 < \alpha < 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 $ \alpha-$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.