Stability and mean-field limits of age dependent Hawkes processes. (English. French summary) Zbl 1466.60102
This paper introduces a multivariate Hawkes process with a special type of age dependent rate function. “It incorporates individual post-jump behavior into the framework of the usual Hawkes model. This allows to model recovery properties such as refractory periods, where the effects of the network are momentarily being suppressed or altered. We show how classical stability results for Hawkes processes can be improved by introducing age into the system. We neither need to a priori bound the intensities nor to impose any conditions on the Lipschitz constants. When the interactions between neurons are of mean-field type, we study large network limits and establish the propagation of chaos property of the system. The results we obtain are similar to what has been shown for ordinary nonlinear Hawkes processes in [P. Brémaud and L. Massoulié, Ann. Probab. 24, No. 3, 1563–1588 (1996; Zbl 0870.60043)], and [M. Costa et al., “ Renewal in Hawkes processes with self-excitation and inhibition”, Preprint, arXiv:1801.04645; https://hal.archives-ouvertes.fr/hal-01683954]”.
Reviewer: H. N. Nagaraja (Columbus)
MSC:
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 60G57 | Random measures |
| 60K05 | Renewal theory |
Keywords:
multivariate nonlinear Hawkes processes; mean-field approximations; age dependency; stability; coupling; piecewise deterministic Markov processesCitations:
Zbl 0870.60043References:
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