Local stability and rates of convergence to equilibrium for the nonlinear renewal equation; applications to Hawkes processes

We study the asymptotic properties of the solutions of a nonlinear renewal equation. The main contribution of the present article is to provide stability and convergence results around equilibrium solutions, under some local subcritical condition. Quantitative rates of convergence to equilibrium are established. Instability results are given in both critical and supercritical cases. As an implication of these results, we establish a Central Limit Theorem for Hawkes processes in a mean-field interaction.