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Stability of nonlinear Hawkes processes. (English) Zbl 0870.60043
The present work is concerned with a family \(N=(N_i)_{1\leq i\leq k}\) of simple point processes without common points admitting respective \({\mathcal F}^N_t\)-intensities \(\lambda_i(t)=\varphi_i\left(\sum^k_{j=1} \int_{(-\infty, t)} h_{ji}(t- s)N_j(ds)\right)\), where \({\mathcal F}^N_t= \bigvee^k_{i=1}{\mathcal F}^{N_i}_t\) denotes the internal history of \(N\), \(\varphi_i:\mathbb{R}\to\mathbb{R}^+\) and \(h_{ji}:\mathbb{R}^+\to\mathbb{R}\). This class of processes contains univariate and multivariate linear Hawkes processes as well as nonlinear mutually exciting point processes, which are used in modelling neuronal activity. The goal of this work is to find conditions bearing on the functions \(\varphi_i\) and \(h_{ji}\), guaranteeing the existence and uniqueness of a stationary version of \(N\) and the convergence to equilibrium of a nonstationary version, both in distribution and in variation.
Reviewer: P.Weiß (Linz)

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60H20 Stochastic integral equations
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