STABILITY OF NONLINEAR HAWKES PROCESSES

We address the problem of the convergence to equilibrium of a general class of point processes, containing, in particular, the nonlinear mut ually exciting point processes, an extension of the linear Hawkes proc esses, and give general conditions guaranteeing the existence of a sta tionary version and the convergence to equilibrium of a nonstationary version, both in distribution and in variation. We also give a new pro of of a result of Kerstan concerning point processes with bounded inte nsity and general nonlinear dynamics satisfying a Lipschitz condition.