Weak conditional propagation of chaos for systems of interacting particles with nearly stable jumps

TL;DR

This work analyzes the weak convergence of a system of $N$ interacting particles with collateral jumps driven by Poisson measures to an infinite-exchangeable limit governed by a common $\alpha$-stable noise. The authors introduce a time-change representation for the collateral-jump sum, enabling a stable-limit description and a conditional propagation of chaos: as $N\to\infty$, the finite-particle dynamics converge to an infinite-system where particles are conditionally independent given the driving stable process $S^\alpha$. The main contributions are (i) a weak-pathwise convergence result for the empirical measure to a directing measure $\bar{\mu}$, (ii) a detailed semimartingale analysis identifying the limit via characteristics, and (iii) a demonstration that the limit system is the unique strong solution to the McKean–Vlasov-type equation driven by the common stable noise, with the directing measure equal to the conditional law given $S^\alpha$. These results provide a novel weak-approximation framework for systems where common heavy-tailed jumps appear only in the limit, with potential applications to large neural networks and related stochastic networks.

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. Using a representation of the collateral jump sum as a time-changed random walk, we prove the convergence in law, in Skorokhod space, of this system to a limit infinite-exchangeable system of SDEs driven by a common stable process. This stable process arises due to the stable central limit theorem, and the particles in the limit system are independent and identically distributed, conditionally on that. That is, the $N$-particle system exhibits the conditional propagation of chaos property.

Theorems & Definitions (33)

• Definition 3.1: Domain of attraction
• Example 1
• Theorem 3.2
• Corollary 3.3
• Proposition 3.4
• Theorem 3.5
• Theorem 3.6
• Proposition 4.1
• proof
• proof : Proof of Proposition \ref{['thm:tightness']}