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 rate depending on its position. When this happens, all the other particles of the system receive a same random kick distributed according to a heavy-tailed random variable belonging to the domain of attraction of an $\alpha$-stable law and scaled by $N^{-1/\alpha}$, $\alpha \in (0,2)\setminus \{1\}$. We call these jumps collateral jumps. Moreover, in case $0<\alpha<1$, the jumping particle itself undergoes a macroscopic, main jump. Similar systems are employed to model families of interacting neurons and, in that context, main and collateral jumps represent respectively the hyperpolarization of a neuron after a spike and the synaptic inputs received by post-synaptic neurons from pre-synaptic ones.  

We prove that our system has the conditional propagation of chaos property: as $N\to +\infty$, the finite particle system converges to an infinite exchangeable system which obeys a McKean-Vlasov SDE driven by an $\alpha$-stable process, and particles in the limit system are independent, conditionally on the driving $\alpha$-stable process.

This talk is based on joint work with Eva Locherbach and Dasha Loukianova.