2017
DOI: 10.1016/j.spa.2017.02.012
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Mean-field limit of generalized Hawkes processes

Abstract: We generalize multivariate Hawkes processes mainly by including a dependence with respect to the age of the process, i.e. the delay since the last point.Within this class, we investigate the limit behaviour, when n goes to infinity, of a system of n mean-field interacting age-dependent Hawkes processes. We prove that such a system can be approximated by independent and identically distributed age dependent point processes interacting with their own mean intensity. This result generalizes the study performed by… Show more

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Cited by 77 publications
(108 citation statements)
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“…Various other types of kinetic models have been derived during the past decades depending on the hypotheses assumed for the dynamics of the emission of an action potential. They include for example integrateand-fire neural networks [8,9,12] and time-elapsed neuronal models [13][14][15]36].…”
Section: Introductionmentioning
confidence: 99%
“…Various other types of kinetic models have been derived during the past decades depending on the hypotheses assumed for the dynamics of the emission of an action potential. They include for example integrateand-fire neural networks [8,9,12] and time-elapsed neuronal models [13][14][15]36].…”
Section: Introductionmentioning
confidence: 99%
“…They derived a Volterra integral equation and used it to obtain the stability criteria. More recently, [24,25,23] have re-explored these models for neuroscience applications (see [8,7] for a rigorous derivation of some of these PDEs using Hawkes processes). PDE (3) differs from theirs in the sense that we have a non-linear transport term (theirs is constant and equal to one) and our boundary condition is more complex.…”
Section: Introductionmentioning
confidence: 99%
“…This typically involves stochastic processes, such as (probabilistic) cellular automata [8][9][10], contact processes [10,11], or interacting Hawkes processes [12]. In particular, infectious diseases have been modeled by so-called susceptible-infectious models or generalizations thereof [13], whereas spikepropagation in neural networks has been modeled by socalled branching networks [14][15][16][17][18][19][20][21][22], Hawkes processes [23][24][25], or probabilistic integrate-and-fire networks [26,27]. These models can be either constructed as independentinteraction models (static interactions), or as threshold models with interactions depending on the states of the interacting partners [28].…”
Section: Introductionmentioning
confidence: 99%