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

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. 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.