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Journal of Theoretical Probability

, Volume 32, Issue 1, pp 131–162 | Cite as

Large Deviations for Cascades of Diffusions Arising in Oscillating Systems of Interacting Hawkes Processes

  • E. LöcherbachEmail author
Article

Abstract

We consider oscillatory systems of interacting Hawkes processes introduced in Ditlevsen and Löcherbach (Stoch Process Appl 2017, http://arxiv.org/abs/1512.00265) to model multi-class systems of interacting neurons together with the diffusion approximations of their intensity processes. This diffusion, which incorporates the memory terms defining the dynamics of the Hawkes process, is hypo-elliptic. It is given by a high-dimensional chain of differential equations driven by 2-dimensional Brownian motion. We study the large population, i.e., small noise limit of its invariant measure for which we establish a large deviation result in the spirit of Freidlin and Wentzell.

Keywords

Hawkes processes Piecewise deterministic Markov processes Diffusion approximation Sample path large deviations for degenerate diffusions Control theory for degenerate diffusions 

Mathematics Subject Classification (2010)

60G17 60G55 60J60 

Notes

Acknowledgements

I would like to thank an anonymous reviewer for his valuable comments and suggestions which helped me to improve the paper. This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01) and as part of the activities of FAPESP Research, Dissemination and Innovation Center for Neuromathematics (Grant 2013/07699-0, S. Paulo Research Foundation).

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.CNRS UMR 8088, Département de MathématiquesUniversité de Cergy-PontoiseCergy-Pontoise CedexFrance

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