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We consider a system of $N$ particles, described by SDEs driven by Poissonrandom measure, where the coefficients depend on the empirical measure of thesystem. Every particle jumps with a jump rate depending on its position. Whenjumping, a macroscopic quantity is added to its own position. Moreover,simultaneously, all other particles of the system receive a small random kickwhich is distributed according to an $\alpha-$ stable law and scaled in$N^{-1/\alpha},$ where $0 < \alpha < 1.$ This particular scaling implies thatthe limit of the empirical measures of the system is random and equals theconditional distribution of one typical particle in the limit system, given thesource of common noise. Thus the system exhibits the {\it conditionalpropagation of chaos property}. The limit system turns out to be solution of anon-linear SDE, driven by an $ \alpha-$stable subordinator. We prove strongunique existence of the limit system and introduce a suitable coupling toobtain the strong convergence of the finite to the limit system, together withprecise error bounds for finite time marginales.

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