CS28: Propagation of Chaos in Life Science Models

Organizer: Dasha Loukianova (Université d’Évry-Paris-Saclay)

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

Elisa Marini

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.