Electronic Journal of Probability

Path large deviations for interacting diffusions with local mean-field interactions in random environment

Patrick E. Müller

Full-text: Open access

Abstract

We consider a system of $N^{d}$ spins in random environment with a random local mean-field type interaction. Each spin has a fixed spatial position on the torus $\mathbb{T} ^{d}$, an attached random environment and a spin value in $\mathbb{R} $ that evolves according to a space and environment dependent Langevin dynamic. The interaction between two spins depends on the spin values, the spatial distance and the random environment of both spins. We prove the path large deviation principle from the hydrodynamic (or local mean-field McKean-Vlasov) limit and derive different expressions of the rate function for the empirical process and for the empirical measure of the paths. To this end we generalize an approach of Dawson and Gärtner.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 76, 56 pp.

Dates
Received: 29 January 2017
Accepted: 14 August 2017
First available in Project Euclid: 16 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1505527232

Digital Object Identifier
doi:10.1214/17-EJP94

Mathematical Reviews number (MathSciNet)
MR3710796

Zentralblatt MATH identifier
06797886

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F10: Large deviations 82C22: Interacting particle systems [See also 60K35]

Keywords
large deviations interacting diffusion interacting particle systems local mean-field McKean-Vlasov equation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Müller, Patrick E. Path large deviations for interacting diffusions with local mean-field interactions in random environment. Electron. J. Probab. 22 (2017), paper no. 76, 56 pp. doi:10.1214/17-EJP94. https://projecteuclid.org/euclid.ejp/1505527232


Export citation

References

  • [1] Robert B. Ash. Real analysis and probability. Academic Press, New York, 1972. Probability and Mathematical Statistics, No. 11.
  • [2] Hedy Attouch, Giuseppe Buttazzo, and Gérard Michaille. Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006.
  • [3] Javier Baladron, Diego Fasoli, Olivier Faugeras, and Jonathan Touboul. Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons. J. Math. Neurosci., 2:Art. 10, 50, 2012.
  • [4] Gérard Ben Arous and Alice Guionnet. Large deviations for Langevin spin glass dynamics. Probab. Theory Related Fields, 102(4):455–509, 1995.
  • [5] Patrick Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition, 1999.
  • [6] Marc Brunaud. Finite Kullback information diffusion laws with fixed marginals and associated large deviations functionals. Stochastic Process. Appl., 44(2):329–345, 1993.
  • [7] Tanguy Cabana and Jonathan Touboul. Large deviations for randomly connected neural networks: I. spatially extended systems. Preprint arXiv:1510.06957, 2017.
  • [8] Tanguy Cabana and Jonathan Touboul. Large deviations for randomly connected neural networks: Ii. state-dependent interactions. Preprint arXiv:1601.00985, 2017.
  • [9] Patrick Cattiaux and Christian Léonard. Large deviations and Nelson processes. Forum Math., 7(1):95–115, 1995.
  • [10] Francis Comets. Nucleation for a long range magnetic model. Ann. Inst. H. Poincaré Probab. Statist., 23(2):135–178, 1987.
  • [11] Paolo Dai Pra and Frank den Hollander. McKean-Vlasov limit for interacting random processes in random media. J. Statist. Phys., 84(3-4):735–772, 1996.
  • [12] D. A. Dawson and J. Gärtner. Large deviations, free energy functional and quasi-potential for a mean field model of interacting diffusions. Mem. Amer. Math. Soc., 78(398):iv+94, 1989.
  • [13] Donald A. Dawson and Jürgen Gärtner. Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics, 20(4):247–308, 1987.
  • [14] Amir Dembo and Ofer Zeitouni. Large deviations techniques and applications, volume 38 of Applications of Mathematics (New York). Springer-Verlag, New York, second edition, 1998.
  • [15] Paul Dupuis and Richard S. Ellis. A weak convergence approach to the theory of large deviations. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, 1997.
  • [16] Jin Feng and Thomas G. Kurtz. Large deviations for stochastic processes, volume 131 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2006.
  • [17] Hans Föllmer. Random fields and diffusion processes. In École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, volume 1362 of Lecture Notes in Math., pages 101–203. Springer, Berlin, 1988.
  • [18] M. I. Freidlin and A. D. Wentzell. Random perturbations of dynamical systems, volume 260 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, second edition, 1998. Translated from the 1979 Russian original by Joseph Szücs.
  • [19] Josselin Garnier, George Papanicolaou, and Tzu-Wei Yang. Large deviations for a mean field model of systemic risk. SIAM J. Financial Math., 4(1):151–184, 2013.
  • [20] Jürgen Gärtner. On the McKean-Vlasov limit for interacting diffusions. Math. Nachr., 137:197–248, 1988.
  • [21] A. Guionnet. Averaged and quenched propagation of chaos for spin glass dynamics. Probab. Theory Related Fields, 109(2):183–215, 1997.
  • [22] Alice Guionnet. Large deviations and stochastic calculus for large random matrices. Probab. Surv., 1:72–172, 2004.
  • [23] Shamik Gupta, Max Potters, and Stefano Ruffo. One-dimensional lattice of oscillators coupled through power-law interactions: Continuum limit and dynamics of spatial Fourier modes. Phys. Rev. E, 85(6), 2012.
  • [24] Claude Kipnis and Stefano Olla. Large deviations from the hydrodynamical limit for a system of independent Brownian particles. Stochastics Stochastics Rep., 33(1-2):17–25, 1990.
  • [25] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural’ceva. Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, volume 23. American Mathematical Society, Providence, R.I., 1968.
  • [26] Robert S. Liptser and Albert N. Shiryaev. Statistics of random processes. I, volume 5 of Applications of Mathematics (New York). Springer-Verlag, Berlin, expanded edition, 2001. General theory, Translated from the 1974 Russian original by A. B. Aries, Stochastic Modelling and Applied Probability.
  • [27] Eric Luçon and Wilhelm Stannat. Mean field limit for disordered diffusions with singular interactions. Ann. Appl. Probab., 24(5):1946–1993, 2014.
  • [28] Máté Maródi, Francesco d’Ovidio, and Tamás Vicsek. Synchronization of oscillators with long range interaction: Phase transition and anomalous finite size effects. Phys. Rev. E, 66(1), 2002.
  • [29] Julien Michel and Raoul Robert. Large deviations for Young measures and statistical mechanics of infinite-dimensional dynamical systems with conservation law. Comm. Math. Phys., 159(1):195–215, 1994.
  • [30] S. Mischler, C. Quiñinao, and J. Touboul. On a kinetic Fitzhugh-Nagumo model of neuronal network. Comm. Math. Phys., 342(3):1001–1042, 2016.
  • [31] Patrick E Müller. Limiting Properties of a Continuous Local Mean-Field Interacting Spin System: Hydrodynamic Limit, Propagation of Chaos, Energy Landscape and Large Deviations. PhD thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, 2016.
  • [32] Jeffrey L Rogers and Luc T Wille. Phase transitions in nonlinear oscillator chains. Phys. Rev. E, 54(3):R2193, 1996.
  • [33] Michael Scheutzow. Periodic behavior of the stochastic Brusselator in the mean-field limit. Probab. Theory Related Fields, 72(3):425–462, 1986.
  • [34] Daniel W. Stroock and S. R. Srinivasa Varadhan. Multidimensional diffusion processes, volume 233 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1979.
  • [35] Hiroshi Tanaka. Limit theorems for certain diffusion processes with interaction. In Stochastic analysis (Katata/Kyoto, 1982), volume 32 of North-Holland Math. Library, pages 469–488. North-Holland, Amsterdam, 1984.