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A sufficient condition for the quasipotential to be the rate function of the invariant measure of countable-state mean-field interacting particle systems

Published online by Cambridge University Press:  21 March 2024

Sarath Yasodharan*
Brown University
Rajesh Sundaresan*
Indian Institute of Science
*Postal address: Division of Applied Mathematics, 182 George Street, Providence, RI 02912, USA. Email address:
**Postal address: Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru 560012, India. Email address:


This paper considers the family of invariant measures of Markovian mean-field interacting particle systems on a countably infinite state space and studies its large deviation asymptotics. The Freidlin–Wentzell quasipotential is the usual candidate rate function for the sequence of invariant measures indexed by the number of particles. The paper provides two counterexamples where the quasipotential is not the rate function. The quasipotential arises from finite-horizon considerations. However, there are certain barriers that cannot be surmounted easily in any finite time horizon, but these barriers can be crossed in the stationary regime. Consequently, the quasipotential is infinite at some points where the rate function is finite. After highlighting this phenomenon, the paper studies some sufficient conditions on a class of interacting particle systems under which one can continue to assert that the Freidlin–Wentzell quasipotential is indeed the rate function.

Original Article
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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Berge, C. (1997). Topological Spaces: Including a Treatment of Multi-valued Functions, Vector Spaces, and Convexity. Dover, Mineola, NY.Google Scholar
Bertini, L. et al. (2002). Macroscopic fluctuation theory for stationary non-equilibrium states. J. Statist. Phys. 107, 635675.CrossRefGoogle Scholar
Bertini, L. et al. (2003). Large deviations for the boundary driven symmetric simple exclusion process. Math. Phys. Anal. Geom. 6, 231267.CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Bodineau, T. and Giacomin, G. (2004). From dynamic to static large deviations in boundary driven exclusion particle systems. Stoch. Process. Appl. 110, 6781.CrossRefGoogle Scholar
Bordenave, C., McDonald, D. and Proutiere, A. (2010). A particle system in interaction with a rapidly varying environment: mean field limits and applications. Networks Heterog. Media 5, 3162.CrossRefGoogle Scholar
Borkar, V. S. and Sundaresan, R. (2012). Asymptotics of the invariant measure in mean field models with jumps. Stoch. Systems 2, 322380.CrossRefGoogle Scholar
Budhiraja, A. and Dupuis, P. (2019). Analysis and Approximation of Rare Events. Springer, New York.CrossRefGoogle Scholar
Cerrai, S. and Paskal, N. (2022). Large deviations principle for the invariant measures of the 2D stochastic Navier–Stokes equations with vanishing noise correlation. Stoch. Partial Differential Equat. 10, 16511681.Google Scholar
Cerrai, S. and Röckner, M. (2004). Large deviations for stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann. Prob. 32, 11001139.CrossRefGoogle Scholar
Cerrai, S. and Röckner, M. (2005). Large deviations for invariant measures of stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann. Inst. H. Poincaré Prob. Statist. 41, 69105.CrossRefGoogle Scholar
Dawson, D. A. and Gärtner, J. (1987). Large deviations from the McKean–Vlasov limit for weakly interacting diffusions. Stochastics 20, 247308.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications, 2nd edn. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Donsker, M. D. and Varadhan, S. R. S. (1975). Asymptotic evaluation of certain Markov process expectations for large time, I. Commun. Pure Appl. Math. 28, 147.CrossRefGoogle Scholar
Durrett, R. (2019). Probability: Theory and Examples, 5th edn. Cambridge University Press.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
Farfán, J., Landim, C. and Tsunoda, K. (2019). Static large deviations for a reaction–diffusion model. Prob. Theory Relat. Fields 174, 49101.CrossRefGoogle Scholar
Freidlin, M. I. and Wentzell, A. D. (2012). Random Perturbations of Dynamical Systems, 3rd edn. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Khasminskii, R. (2012). Stochastic Stability of Differential Equations. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Kumar, A., Altman, E., Miorandi, D. and Goyal, M. (2006). New insights from a fixed point analysis of single cell IEEE 802.11 WLANs. In Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies, Institute of Electrical and Electronics Engineers, Piscataway, NJ, pp. 1550–1561.Google Scholar
Léonard, C. (1995). Large deviations for long range interacting particle systems with jumps. Ann. Inst. H. Poincaré Prob. Statist. 31, 289323.Google Scholar
Léonard, C. (1995). On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations. Prob. Theory Relat. Fields 101, 144.CrossRefGoogle Scholar
Liptser, R. (1996). Large deviations for two scaled diffusions. Prob. Theory Relat. Fields 106, 71104.CrossRefGoogle Scholar
Martirosyan, D. (2017). Large deviations for stationary measures of stochastic nonlinear wave equations with smooth white noise. Commun. Pure Appl. Math. 70, 17541797.CrossRefGoogle Scholar
McKean, H. P. (1967). Propagation of chaos for a class of non-linear parabolic equations. In Stochastic Differential Equations (Lecture Series in Differential Equations 7), Catholic University, Washington, DC, pp. 41–57.Google Scholar
Meyn, S. P. et al. (2015). Ancillary service to the grid using intelligent deferrable loads. IEEE Trans. Automatic Control 60, 28472862.CrossRefGoogle Scholar
Mufa, C. (1994). Optimal Markovian couplings and applications. Acta Math. Sinica 10, 260275.CrossRefGoogle Scholar
Puhalskii, A. (2019). Large deviations of the long term distribution of a non Markov process. Electron. Commun. Prob. 24, article no. 35.CrossRefGoogle Scholar
Puhalskii, A. A. (2016). On large deviations of coupled diffusions with time scale separation. Ann. Prob. 44, 31113186.CrossRefGoogle Scholar
Puhalskii, A. A. (2020). Large deviation limits of invariant measures. Preprint. Available at Scholar
Salins, M., Budhiraja, A. and Dupuis, P. (2019). Uniform large deviation principles for Banach space valued stochastic differential equations. Trans. Amer. Math. Soc. 372, 83638421.CrossRefGoogle Scholar
Salins, M. and Spiliopoulos, K. (2021). Metastability and exit problems for systems of stochastic reaction–diffusion equations. Ann. Prob. 49, 23172370.CrossRefGoogle Scholar
Sowers, R. (1992). Large deviations for the invariant measure of a reaction–diffusion equation with non-Gaussian perturbations. Prob. Theory Relat. Fields 92, 393421.CrossRefGoogle Scholar
Sowers, R. B. (1992). Large deviations for a reaction–diffusion equation with non-Gaussian perturbations. Ann. Prob. 20, 504537.CrossRefGoogle Scholar
Veretennikov, A. Y. (2000). On large deviations for SDEs with small diffusion and averaging. Stoch. Process. Appl. 89, 6979.CrossRefGoogle Scholar
Yasodharan, S. and Sundaresan, R. (2023). Large time behaviour and the second eigenvalue problem for finite state mean-field interacting particle systems. Adv. Appl. Prob. 55, 85125.CrossRefGoogle Scholar