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Multivariate Hawkes processes on inhomogeneous random graphs

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  • Agathe-Nerine, Zoé

Abstract

We consider a population of N interacting neurons, represented by a multivariate Hawkes process: The firing rate of each neuron depends on the history of the connected neurons. Contrary to the mean-field framework where the interaction occurs on the complete graph, the connectivity between particles is given by a random possibly diluted and inhomogeneous graph where the probability of presence of each edge depends on the spatial position of its vertices. We address the well-posedness of this system and Law of Large Numbers results as N→∞. A crucial issue will be to understand how spatial inhomogeneity influences the large time behavior of the system.

Suggested Citation

  • Agathe-Nerine, Zoé, 2022. "Multivariate Hawkes processes on inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 86-148.
    • Handle: RePEc:eee:spapps:v:152:y:2022:i:c:p:86-148
    DOI: 10.1016/j.spa.2022.06.019
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    References listed on IDEAS

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    1. Ditlevsen, Susanne & Löcherbach, Eva, 2017. "Multi-class oscillating systems of interacting neurons," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1840-1869.
    2. Chevallier, J. & Duarte, A. & Löcherbach, E. & Ost, G., 2019. "Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 1-27.
    3. Cormier, Quentin & Tanré, Etienne & Veltz, Romain, 2020. "Long time behavior of a mean-field model of interacting neurons," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2553-2595.
    4. Budhiraja, Amarjit & Wu, Ruoyu, 2016. "Some fluctuation results for weakly interacting multi-type particle systems," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2253-2296.
    5. Chevallier, Julien, 2017. "Mean-field limit of generalized Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3870-3912.
    6. Heesen, Sophie & Stannat, Wilhelm, 2021. "Fluctuation limits for mean-field interacting nonlinear Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 139(C), pages 280-297.
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