Mean-field limits for nearly unstable Hawkes processes

In this paper, we establish general scaling limits for nearly unstable Hawkes processes in a mean-field regime by extending the method introduced by Jaisson and Rosenbaum [41]. Under a mild asymptotic criticality condition on the self-exciting kernels {ϕn}, specifically ‖ϕn‖L1→1, we first show that the scaling limits of these Hawkes processes are necessarily stochastic Volterra diffusions of affine type. Moreover, we establish a propagation of chaos result for Hawkes systems with mean-field interactions, highlighting three distinct regimes for the limiting processes, which depend on the asymptotics of n(1−‖ϕn‖L1)2. These results provide a significant generalization of the findings by Delattre et al. [22].