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We consider a multivariate non-linear Hawkes process in a multi-class setup where particles are organised within two populations of possibly different sizes, such that one of the populations acts excitatory on the system while the other population acts inhibitory on the system. The goal of this note is to present a class of Hawkes Processes with stable dynamics without assumptions on the spectral radius of the associated weight function matrix. This illustrates how inhibition in a Hawkes system significantly affects the stability properties of the system.

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In the last decade Hawkes processes have received much attention as models for functional connectivity in neural spiking networks and other dynamical systems with a cascade behavior. In this paper we establish a renewal approach for analyzing this process. We consider the ordinary nonlinear Hawkes process as well as the more recently described age dependent Hawkes process. We construct renewal-times and establish moment results for these. This gives rise to study the Hawkes process as a Markov chain. As an application, we prove asymptotic results such as a functional CLT and a time-average CLT.