Skip to main content
SpringerLink
Log in

Empirical Measure and Small Noise Asymptotics Under Large Deviation Scaling for Interacting Diffusions

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and also through the drift and diffusion coefficients that depend on the state empirical measure. We study large deviation behavior of the empirical measure process which is governed by two types of scaling, one corresponding to mean field asymptotics and the other to the Freidlin–Wentzell small noise asymptotics. Different levels of intensity of the small common noise lead to different types of large deviation behavior, and we provide a precise characterization of the various regimes. The rate functions can be interpreted as the value functions of certain stochastic control problems in which there are two types of controls; one of the controls is random and nonanticipative and arises from the aggregated contributions of the individual Brownian noises, whereas the second control is nonrandom and corresponds to the small common Brownian noise that impacts all particles. We also study large deviation behavior of interacting particle systems approximating various types of Feynman–Kac functionals. Proofs are based on stochastic control representations for exponential functionals of Brownian motions and on uniqueness results for weak solutions of stochastic differential equations associated with controlled nonlinear Markov processes

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Antunes, N., Fricker, C., Robert, P., Tibi, D.: Stochastic networks with multiple stable points. Ann. Probab. 36, 255–278 (2008)

    Article  MathSciNet  Google Scholar 

  2. Baladron, J., Fasoli, D., Faugeras, O., Touboul, J.: Mean-field description and propagation of chaos in networks of Hodgkin–Huxley and FitzHugh–Nagumo neurons. J. Math. Neurosci. 2(1), 10 (2012)

    Article  MathSciNet  Google Scholar 

  3. Bordenave, C., Macdonald, D., Proutiere, A.: Performance of random medium access control, an asymptotic approach. In: Proceedings of ACM Sigmetrics, pp. 1–12 (2008)

  4. Bordenave, C., McDonald, D., Proutiere, A.: Random multi-access algorithms in networks with partial interaction: a mean field analysis. Netw. Heterog. Media 5(1), 31–62 (2010)

    Article  MathSciNet  Google Scholar 

  5. Boué, M., Dupuis, P.: A variational representation for certain functionals of Brownian motion. Ann. Probab. 26, 1641–1659 (1998)

    Article  MathSciNet  Google Scholar 

  6. Braun, W., Hepp, K.: The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Commun. Math. Phys. 56(2), 101–113 (1977)

    Article  MathSciNet  Google Scholar 

  7. Budhiraja, A., Dupuis, P.: A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Stat. 20, 39–61 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Budhiraja, A., Dupuis, P., Fischer, M.: Large deviation properties of weakly interacting processes via weak convergence methods. Ann. Probab. 40, 74–102 (2012)

    Article  MathSciNet  Google Scholar 

  9. Dawson, D.A.: Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Stat. Phys. 31(1), 29–85 (1983)

    Article  MathSciNet  Google Scholar 

  10. Dawson, D.A., Gärtner, J.: Large deviations from the McKean–Vlasov limit for weakly interacting diffusions. Stochastics 20(4), 247–308 (1987)

    Article  MathSciNet  Google Scholar 

  11. Del Moral, P.: Feynman–Kac formulae. Probability and its Applications (New York). Springer, New York (2004). Genealogical and interacting particle systems with applications

  12. Dupuis, P., Ellis, R.S.: A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York (1997)

    Book  Google Scholar 

  13. Dupuis, P., Spiliopoulos, K.: Large deviations for multiscale problems via weak convergence methods. Stoch. Proc. Appl. 122, 1947–1987 (2012)

    Article  Google Scholar 

  14. Garnier, J., Papanicolaou, G., Yang, T.-W.: Large deviations for a mean field model of systemic risk. SIAM J. Financ. Math. 4(1), 151–184 (2013)

    Article  MathSciNet  Google Scholar 

  15. Giesecke, K., Spiliopoulos, K., Sowers, R.B.: Default clustering in large portfolios: typical events. Ann. Appl. Probab. 23(1), 348–385 (2013)

    Article  MathSciNet  Google Scholar 

  16. Goméz-Serrano, J., Graham, C., Boudec, J.-Y.L.: The bounded confidence model of opinion dynamics. Math. Models Methods Appl. Sci 22(02), 1150007 (2012)

    Article  MathSciNet  Google Scholar 

  17. Graham, C., Méléard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25(1), 115–132 (1997)

    Article  MathSciNet  Google Scholar 

  18. Graham, C., Robert, P.: A multi-class mean-field model with graph structure for TCP flows. In: Progress in Industrial Mathematics at ECMI 2008, pp. 125–131. Springer (2010)

  19. Hu, W., Salins, M., Spiliopoulos, K.: Large deviations and averaging for systems of slow-fast stochastic reaction–diffusion equations. Stoch. PDE Anal. Comp. 7, 808–874 (2019)

    Article  MathSciNet  Google Scholar 

  20. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, Volume 24 of North-Holland Mathematical Library, 2nd edn. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo (1989)

  21. Kurtz, T.G., Xiong, J.: Particle representations for a class of nonlinear SPDEs. Stoch. Process. Appl. 83(1), 103–126 (1999)

    Article  MathSciNet  Google Scholar 

  22. McKean, H.P.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56(6), 1907 (1966)

    Article  MathSciNet  Google Scholar 

  23. Méléard, S.: Asymptotic behaviour of some interacting particle systems; McKean–Vlasov and Boltzmann models. In: Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), Volume 1627 of Lecture Notes in Mathematics, pp. 42–95. Springer, Berlin (1996)

  24. Oelschläger, K.: A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12(2), 458–479 (1984)

    Article  MathSciNet  Google Scholar 

  25. Orrieri, Carlo.: Large deviations for interacting particle systems: joint mean-field and small-noise limit (2018). arXiv preprint arXiv:1810.12636

  26. Shiga, T., Tanaka, H.: Central limit theorem for a system of Markovian particles with mean field interactions. Probab. Theory Relat. Fields 69(3), 439–459 (1985)

    MathSciNet  MATH  Google Scholar 

  27. Sun, X., Wang, R., Xu, L., Yang, X.: Large deviations for two-time-scale stochastic Burgers equation (2020). arXiv:1811.00290

  28. Sznitman, A.-S.: Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56(3), 311–336 (1984)

    Article  MathSciNet  Google Scholar 

  29. Sznitman, A.-S.: Topics in Propagation of Chaos, vol. 1464, pp. 167–251. Springer, Berlin (1991)

    MATH  Google Scholar 

  30. Tanaka, H.: Limit theorems for certain diffusion processes with interaction. Stochastic analysis (Katata/Kyoto, 1982), Volume 32 of North-Holland Math. Library, pp. 469–488. North-Holland, Amsterdam (1984)

Download references

Acknowledgements

This research was supported in part by the NSF (DMS-1814894, DMS-1853968). We thank an anonymous reviewer for directing us to the related works [19, 27].

Author information

Authors and Affiliations

  1. Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC, 27599, USA

    Amarjit Budhiraja & Michael Conroy

Authors
  1. Amarjit Budhiraja
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Conroy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Conroy.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Budhiraja, A., Conroy, M. Empirical Measure and Small Noise Asymptotics Under Large Deviation Scaling for Interacting Diffusions. J Theor Probab 35, 295–349 (2022). https://doi.org/10.1007/s10959-020-01071-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-020-01071-4

Keywords

Mathematics Subject Classification (2020)

Search

Navigation