McKean-Vlasov limit for interacting systems with simultaneous jumps

Motivated by several applications, including neuronal models, we consider the McKean-Vlasov limit for a general class of mean-field systems of interacting diffusions characterized by an interaction via simultaneous jumps. We focus our interest on systems where the rate of the jumps is unbounded, which are rarely treated in the mean-field literature, and we prove well-posedness of the McKean-Vlasov limit together with propagation of chaos via a coupling technique. To highlight the role of simultaneous jumps, we introduce an intermediate process which is close to the original particle system but does not display simultaneous jumps. This shows in particular that the simultaneous jumps contribute to the overall rate of convergence of the $N$-particle empirical measures by a term of order $1/\sqrt{N}$.