  We consider a sequence of systems of Hawkes processes having mean field
interactions in a diffusive regime. The stochastic intensity of each process is
a solution of a stochastic differential equation driven by N independent
Poisson random measures. We show that, as the number of interacting components
N tends to infinity, this intensity converges in distribution in Skorohod space
to a CIR-type diffusion. Moreover, we prove the convergence in distribution of
the Hawkes processes to the limit point process having the limit diffusion as
intensity. To prove the convergence results, we use analytical technics based
on the convergence of the associated infinitesimal generators and Markovian
semigroups.
