Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Large deviations of the empirical flow for continuous time Markov chains

Lorenzo Bertini, Alessandra Faggionato, and Davide Gabrielli

Full-text: Open access


We consider a continuous time Markov chain on a countable state space and prove a joint large deviation principle for the empirical measure and the empirical flow, which accounts for the total number of jumps between pairs of states. We give a direct proof using tilting and an indirect one by contraction from the empirical process.


On considère une chaîne de Markov en temps continu à espace d’états denombrable, et on prouve un principe de grandes déviations commun pour la mesure empirique et le courant empirique, qui représente le nombre total de sauts entre les paires d’états. On donne une preuve directe à l’aide d’un tilting, et une preuve indirecte par contraction, à partir du processus empirique.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 3 (2015), 867-900.

Received: 1 May 2013
Revised: 6 January 2014
Accepted: 20 January 2014
First available in Project Euclid: 1 July 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 82C05: Classical dynamic and nonequilibrium statistical mechanics (general)

Markov chain Large deviations principle Entropy Empirical flow


Bertini, Lorenzo; Faggionato, Alessandra; Gabrielli, Davide. Large deviations of the empirical flow for continuous time Markov chains. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 3, 867--900. doi:10.1214/14-AIHP601.

Export citation


  • [1] M. Baiesi, C. Maes and K. Netočný. Computation of current cumulants for small nonequilibrium systems. J. Stat. Phys. 135 (1) (2009) 57–75.
  • [2] P. Baldi and M. Piccioni. A representation formula for the large deviation rate function for the empirical law of a continuous time Markov chain. Statist. Probab. Lett. 41 (2) (1999) 107–115.
  • [3] G. Basile and L. Bertini. Donsker–Varadhan asymptotics for degenerate jump Markov processes. Preprint, 2013. Available at arXiv:1310.5829.
  • [4] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim. Non equilibrium current fluctuations in stochastic lattice gases. J. Stat. Phys. 123 (2006) 237–276.
  • [5] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim. Large deviations of the empirical current in interacting particle systems. Theory Probab. Appl. 51 (2007) 2–27.
  • [6] L. Bertini, A. Faggionato and D. Gabrielli. From level 2.5 to level 2 large deviations for continuous time Markov chains. Markov Process. Related Fields 20 (2014) 545–562.
  • [7] L. Bertini, A. Faggionato and D. Gabrielli. Flows, currents and symmetries for continuous time Markov chains: A large deviation approach. Preprint.
  • [8] T. Bodineau and B. Derrida. Current fluctuations in non-equilibrium diffusive systems: An additivity principle. Phys. Rev. Lett. 92 (2004) 180601.
  • [9] T. Bodineau and B. Derrida. Current large deviations for asymmetric exclusion processes with open boundaries. J. Stat. Phys. 123 (2) (2006) 277–300.
  • [10] T. Bodineau, B. Derrida and J. L. Lebowitz. Vortices in the two-dimensional simple exclusion process. J. Stat. Phys. 131 (2008) 821–841.
  • [11] T. Bodineau, V. Lecomte and C. Toninelli. Finite size scaling of the dynamical free-energy in a kinetically constrained model. J. Stat. Phys. 147 (2012) 1–17.
  • [12] T. Bodineau and C. Toninelli. Activity phase transition for constrained dynamics. Comm. Math. Phys. 311 (2012) 357–396.
  • [13] M. F. Chen. From Markov Chains to Nonequilibrium Particle Systems. World Scientific, Singapore, 1992.
  • [14] M. F. Chen. Eigenvalues, Inequalities and Ergodic Theory. Springer, Berlin, 2005.
  • [15] A. de La Fortelle. The large-deviation principle for Markov chains with continuous time (Russian). Problemy Peredachi Informatsii 37 (2) (2001) 40–61. Translation in Probl. Inf. Transm. 37 (2) (2001) 120–139.
  • [16] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998.
  • [17] F. den Hollander. Large Deviations. Fields Institute Monographs. Amer. Math. Soc., Providence, RI, 2000.
  • [18] J.-D. Deuschel and D. W. Stroock. Large Deviations. Academic Press, Boston, MA, 1989.
  • [19] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. Comm. Pure Appl. Math. (I) 28 (1975) 1–47. (II) 28 (1975) 279–301. (III) 29 (1976) 389–461. (IV) 36 (1983) 183–212.
  • [20] J. Dugundji. Topology. Allyn and Bacon, Boston, 1966.
  • [21] P. Eichelsbacher and U. Schmock. Exponential approximations in completely regular topological spaces and extensions of Sanov’s theorem. Stochastic Process. Appl. 77 (1998) 233–251.
  • [22] S. N. Ethier and T. G. Kurtz. Markov Processes. Characterization and Convergence. Wiley, New York, 1986.
  • [23] A. Faggionato and D. Di Pietro. Gallavotti–Cohen–Type symmetry related to cycle decompositions for Markov chains and biochemical applications. J. Stat. Phys. 143 (2011) 11–32.
  • [24] D. Gabrielli and C. Valente. Which random walks are cyclic? ALEA, Lat. Am. J Probab. Math. Stat. 9 (2012) 231–267.
  • [25] R. J. Harris, A. Rákos and G. M. Schütz. Current fluctuations in the zero-range process with open boundaries. J. Stat. Mech. Theory Exp. (2005) P08003.
  • [26] L. H. Jensen. Large deviations of the asymmetric simple exclusion process in one dimension. Ph.D. thesis, Courant Institute NYU, 2000.
  • [27] G. Kesidis and J. Walrand. Relative entropy between Markov transition rate matrices. IEEE Trans. Inform. Theory 39 (3) (1993) 1056–1057.
  • [28] C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems. Springer, Berlin, 1999.
  • [29] S. Kusuoka, K. Kuwada and Y. Tamura. Large deviation for stochastic line integrals as $L^{p}$-currents. Probab. Theory Related Fields 147 (2010) 649–674.
  • [30] K. Kuwada. On large deviations for random currents induced from stochastic line integrals. Forum Math. 18 (2006) 639–676.
  • [31] D. Lacoste and K. Mallick. Fluctuation Relations for Molecular Motors. B. Duplantier and V. Rivasseau (Eds). Biological Physics. Poincaré Seminar 2009, Progress in Mathematical Physics 60. Birkhäuser, Basel, 2011.
  • [32] J. L. Lebowitz and H. Spohn. A Gallavotti–Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95 (1999) 333–365.
  • [33] J. MacQueen. Circuit processes. Ann. Probab. 9 (1981) 604–610.
  • [34] M. Mariani. A $\varGamma $-convergence approach to large deviations. Preprint, 2012. Available at arXiv:1204.0640.
  • [35] M. Mariani, Y. Shen and L. Zambotti. Large deviations for the empirical measure of Markov renewal processes. Preprint, 2012. Available at arXiv:1203.5930.
  • [36] R. E. Megginson. An Introduction to Banach Space Theory. Springer, New York, 1998.
  • [37] S. Smirnov. Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents. St. Petersburg Math. J. 5 (4) (1994) 841–867.
  • [38] H. Spohn. Large Scale Dynamics of Interacting Particles. Springer, Berlin, 1991.
  • [39] J. R. Norris. Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press, Cambridge, 1999.
  • [40] S. R. S. Varadhan. Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics 46. SIAM, Philadelphia, PA, 1984.