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


We consider oscillatory systems of interacting Hawkes processes introduced in Ditlevsen and Löcherbach (Stoch Process Appl 2017, 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.


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 



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).


  1. 1.
    Benaïm, M., Hirsch, M.W.: Mixed equilibria and dynamical systems arising from fictitious play in perturbed games. Games Econ. Behav. 29, 36–72 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bianchini, R.M., Stefani, G.: Normal local controllability of order one. Int. J. Control 39, 701–704 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brémaud, P., Massoulié, L.: Stability of nonlinear Hawkes processes. Ann. Probab. 24(3), 1563–1588 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chevallier, J.: Mean-field limit of generalized Hawkes processes. Stoch. Process. Appl. (2015). doi: 10.1016/ zbMATHGoogle Scholar
  5. 5.
    Coron, J.-M.: Control and Nonlinearity. American Mathematical Society (AMS), Providence, RI (2007)Google Scholar
  6. 6.
    Delarue, F., Menozzi, S.: Density estimates for a random noise propagating through a chain of differential equations. J. Funct. Anal. 259, 1577–1630 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Delattre, S., Robert, C.Y., Rosenbaum, M.: Estimating the efficient price from the order flow: a Brownian Cox process approach. Stoch. Process. Appl. 123(7), 2603–2619 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dembo, A., Zeitouni, O.: Large deviation techniques and applications. In: Stochastic Modelling and Applied Probability, vol. 38. Springer Berlin Heidelberg (2010)Google Scholar
  9. 9.
    Ditlevsen, S., Löcherbach, E.: Multi-class oscillating systems of interacting neurons. Stoch. Process. Appl. 127, 1840–1869 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Transl. from the Russian by Joseph Szuecs, 2nd ed. Springer, New York, NY (1998)Google Scholar
  11. 11.
    Hansen, N., Reynaud-Bouret, P., Rivoirard, V.: Lasso and probabilistic inequalities for multivariate point processes. Bernoulli 21(1), 83–143 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hawkes, A.G.: Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hawkes, A.G., Oakes, D.: A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 93–503 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Höpfner, R., Löcherbach, E., Thieullen, M.: Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Application to a stochastic Hodgkin–Huxley model. ESAIM P & S 20, 527–554 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lee, E.B., Markus, L.: Foundations of Optimal Control Theory. The SIAM Series in Applied Mathematics. Wiley, New York (1967)Google Scholar
  16. 16.
    Lewis, A.D.: A brief on controllability of nonlinear systems.
  17. 17.
    Mallet-Paret, J., Smith, H.L.: The Poincaré–Bendixson theorem for monotone cyclic feedback systems. J. Dyn. Differ. Equ. 2(4), 367–421 (1990)CrossRefzbMATHGoogle Scholar
  18. 18.
    Millet, A., Sanz-Solé, M.: A simple proof of the support theorem for diffusion processes. Semin Probab (Strasbourg) 28, 26–48 (1994)zbMATHGoogle Scholar
  19. 19.
    Pigato, P.: Tube estimates for diffusion processes under a weak Hörmander condition (2014).
  20. 20.
    Rey-Bellet, L., Thomas, L.E.: Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators. Commun. Math. Phys. 215, 1–24 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Stroock, D., Varadhan, S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, vol. III, pp. 333–359 (1972)Google Scholar
  22. 22.
    Sussmann, H.J.: A sufficient condition for local controllability. SIAM J. Control Optim. 16, 790–802 (1978)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

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

Personalised recommendations