The Annals of Applied Probability

Equilibrium large deviations for mean-field systems with translation invariance

Julien Reygner

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider particle systems with mean-field interactions whose distribution is invariant by translations. Under the assumption that the system seen from its centre of mass be reversible with respect to a Gibbs measure, we establish large deviation principles for its empirical measure at equilibrium. Our study covers the cases of McKean–Vlasov particle systems without external potential, and systems of rank-based interacting diffusions. Depending on the strength of the interaction, the large deviation principles are stated in the space of centered probability measures endowed with the Wasserstein topology of appropriate order, or in the orbit space of the action of translations on probability measures. An application to the study of atypical capital distribution is detailed.

Article information

Ann. Appl. Probab., Volume 28, Number 5 (2018), 2922-2965.

Received: July 2017
Revised: December 2017
First available in Project Euclid: 28 August 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations 60J60: Diffusion processes [See also 58J65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Large deviations mean-field systems McKean–Vlasov particle systems rank-based interacting diffusions free energy


Reygner, Julien. Equilibrium large deviations for mean-field systems with translation invariance. Ann. Appl. Probab. 28 (2018), no. 5, 2922--2965. doi:10.1214/17-AAP1379.

Export citation


  • [1] Ambrosio, L., Gigli, N. and Savaré, G. (2008). Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed. Birkhäuser, Basel.
  • [2] Banner, A. D., Fernholz, R. and Karatzas, I. (2005). Atlas models of equity markets. Ann. Appl. Probab. 15 2296–2330.
  • [3] Benachour, S., Roynette, B., Talay, D. and Vallois, P. (1998). Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos. Stochastic Process. Appl. 75 173–201.
  • [4] Benachour, S., Roynette, B. and Vallois, P. (1998). Nonlinear self-stabilizing processes. II. Convergence to invariant probability. Stochastic Process. Appl. 75 203–224.
  • [5] Benedetto, D., Caglioti, E., Carrillo, J. A. and Pulvirenti, M. (1998). A non-Maxwellian steady distribution for one-dimensional granular media. J. Stat. Phys. 91 979–990.
  • [6] Benedetto, D., Caglioti, E. and Pulvirenti, M. (1997). A kinetic equation for granular media. RAIRO Modél. Math. Anal. Numér. 31 615–641.
  • [7] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [8] Bobkov, S. and Ledoux, M. (2016). One-dimensional empirical measures, order statistics and Kantorovich transport distances. Mem. Amer. Math. Soc. To appear.
  • [9] Bossy, M. and Talay, D. (1996). Convergence rate for the approximation of the limit law of weakly interacting particles: Application to the Burgers equation. Ann. Appl. Probab. 6 818–861.
  • [10] Bossy, M. and Talay, D. (1997). A stochastic particle method for the McKean–Vlasov and the Burgers equation. Math. Comp. 66 157–192.
  • [11] Bruggeman, C. (2016). Dynamics of Large Rank-Based Systems of Interacting Diffusions. Ph.D. thesis, Columbia Univ., New York, NY.
  • [12] Carrillo, J. A., McCann, R. J. and Villani, C. (2003). Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates. Rev. Mat. Iberoam. 19 971–1018.
  • [13] Carrillo, J. A., McCann, R. J. and Villani, C. (2006). Contractions in the $2$-Wasserstein length space and thermalization of granular media. Arch. Ration. Mech. Anal. 179 217–263.
  • [14] Cattiaux, P., Guillin, A. and Malrieu, F. (2008). Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Related Fields 140 19–40.
  • [15] Cattiaux, P. and Pédèches, L. (2016). The 2-D stochastic Keller–Segel particle model: Existence and uniqueness. ALEA Lat. Am. J. Probab. Math. Stat. 13 447–463.
  • [16] Chafaï, D., Gozlan, N. and Zitt, P.-A. (2014). First-order global asymptotics for confined particles with singular pair repulsion. Ann. Appl. Probab. 24 2371–2413.
  • [17] Chatterjee, S. and Pal, S. (2010). A phase transition behavior for Brownian motions interacting through their ranks. Probab. Theory Related Fields 147 123–159.
  • [18] Dawson, D. A. and Gärtner, J. (1986). Large deviations and tunnelling for particle systems with mean field interaction. C. R. Math. Rep. Acad. Sci. Can. 8 387–392.
  • [19] Dawson, D. A. and Gärtner, J. (1987). Large deviations from the McKean–Vlasov limit for weakly interacting diffusions. Stochastics 20 247–308.
  • [20] Dawson, D. A. and Gärtner, J. (1989). Large deviations, free energy functional and quasi-potential for a mean field model of interacting diffusions. Mem. Amer. Math. Soc. 78 iv $+$ 94.
  • [21] Dembo, A., Shkolnikov, M., Varadhan, S. R. S. and Zeitouni, O. (2016). Large deviations for diffusions interacting through their ranks. Comm. Pure Appl. Math. 69 1259–1313.
  • [22] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin. Corrected reprint of the second (1998) edition.
  • [23] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York.
  • [24] Dupuis, P., Laschos, V. and Ramanan, K. (2015). Large deviations for empirical measures generated by Gibbs measures with singular energy functionals. Preprint. Available at arXiv:1511.06928.
  • [25] Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 271. Springer, New York.
  • [26] Fernholz, E. R. (2002). Stochastic Portfolio Theory. Applications of Mathematics (New York) 48. Springer, New York.
  • [27] Fournier, N. and Jourdain, B. (2017). Stochastic particle approximation of the Keller–Segel equation and two-dimensional generalization of Bessel processes. Ann. Appl. Probab. 27 2807–2861.
  • [28] Jordan, R., Kinderlehrer, D. and Otto, F. (1998). The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29 1–17.
  • [29] Jourdain, B. (2000). Diffusion processes associated with nonlinear evolution equations for signed measures. Methodol. Comput. Appl. Probab. 2 69–91.
  • [30] Jourdain, B. and Malrieu, F. (2008). Propagation of chaos and Poincaré inequalities for a system of particles interacting through their CDF. Ann. Appl. Probab. 18 1706–1736.
  • [31] Jourdain, B. and Reygner, J. (2013). Propogation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation. Stoch. Partial Differ. Equ., Anal. Computat. 1 455–506.
  • [32] Jourdain, B. and Reygner, J. (2015). Capital distribution and portfolio performance in the mean-field Atlas model. Ann. Finance 11 151–198.
  • [33] Kolli, P. and Shkolnikov, M. (2018). SPDE limit of the global fluctuations in rank-based models. Ann. Probab. 46 1042–1069.
  • [34] Leblé, T. and Serfaty, S. (2017). Large deviation principle for empirical fields of log and Riesz gases. Invent. Math. 210 645–757.
  • [35] Léonard, C. (1987). Large deviations and law of large numbers for a mean field type interacting particle system. Stochastic Process. Appl. 25 215–235.
  • [36] Malrieu, F. (2003). Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13 540–560.
  • [37] Mukherjee, C. and Varadhan, S. R. S. (2016). Brownian occupation measures, compactness and large deviations. Ann. Probab. 44 3934–3964.
  • [38] Otto, F. (2001). The geometry of dissipative evolution equations: The porous medium equation. Comm. Partial Differential Equations 26 101–174.
  • [39] Pal, S. and Pitman, J. (2008). One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Probab. 18 2179–2207.
  • [40] Rey-Bellet, L. and Spiliopoulos, K. (2015). Irreversible Langevin samplers and variance reduction: A large deviations approach. Nonlinearity 28 2081–2103.
  • [41] Reygner, J. (2015). Chaoticity of the stationary distribution of rank-based interacting diffusions. Electron. Commun. Probab. 20 no. 60, 1–20.
  • [42] Reygner, J. (2017). Long time behaviour and mean-field limit of Atlas models. ESAIM Proc. Surv. 60 132–143.
  • [43] Shkolnikov, M. (2012). Large systems of diffusions interacting through their ranks. Stochastic Process. Appl. 122 1730–1747.
  • [44] Sun, L.-H. (2013). Systemic risk illustrated. In Handbook on Systemic Risk 444–452. Cambridge Univ. Press, Cambridge.
  • [45] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
  • [46] Wang, R., Wang, X. and Wu, L. (2010). Sanov’s theorem in the Wasserstein distance: A necessary and sufficient condition. Statist. Probab. Lett. 80 505–512.