Large deviations of the empirical flow for continuous time Markov chains
Bertini, Lorenzo ; Faggionato, Alessandra ; Gabrielli, Davide
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 867-900 / Harvested from Numdam
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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.

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.

Publié le : 2015-01-01
DOI : https://doi.org/10.1214/14-AIHP601 
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     author = {Bertini, Lorenzo and Faggionato, Alessandra and Gabrielli, Davide},
     title = {Large deviations of the empirical flow for continuous time Markov chains},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {51},
     year = {2015},
     pages = {867-900},
     doi = {10.1214/14-AIHP601},
     mrnumber = {3365965},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2015__51_3_867_0}
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Bertini, Lorenzo; Faggionato, Alessandra; Gabrielli, Davide. Large deviations of the empirical flow for continuous time Markov chains. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) pp. 867-900. doi : 10.1214/14-AIHP601. http://gdmltest.u-ga.fr/item/AIHPB_2015__51_3_867_0/

[1] M. Baiesi, C. Maes and K. Netočný. Computation of current cumulants for small nonequilibrium systems. J. Stat. Phys. 135 (1) (2009) 57–75. | MR 2505725 | Zbl 1173.82028

[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. | MR 1665260 | Zbl 0915.60045

[3] G. Basile and L. Bertini. Donsker–Varadhan asymptotics for degenerate jump Markov processes. Preprint, 2013. Available at arXiv:1310.5829. | MR 3333733

[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. | MR 2227084 | Zbl 1097.82016

[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. | MR 2324172 | Zbl 1190.82027

[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. | MR 3289132 | Zbl 06390872

[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. | MR 2227085 | Zbl 1115.82020

[10] T. Bodineau, B. Derrida and J. L. Lebowitz. Vortices in the two-dimensional simple exclusion process. J. Stat. Phys. 131 (2008) 821–841. | MR 2398955 | Zbl 1214.82068

[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. | MR 2922756 | Zbl 1245.82049

[12] T. Bodineau and C. Toninelli. Activity phase transition for constrained dynamics. Comm. Math. Phys. 311 (2012) 357–396. | MR 2902193 | Zbl 1241.82053

[13] M. F. Chen. From Markov Chains to Nonequilibrium Particle Systems. World Scientific, Singapore, 1992. | MR 1168209 | Zbl 0753.60055

[14] M. F. Chen. Eigenvalues, Inequalities and Ergodic Theory. Springer, Berlin, 2005. | MR 2105651 | Zbl 1079.60005

[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. | MR 2099897 | Zbl 0996.60086

[16] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. | MR 1619036 | Zbl 1177.60035

[17] F. Den Hollander. Large Deviations. Fields Institute Monographs. Amer. Math. Soc., Providence, RI, 2000. | MR 1739680 | Zbl 0949.60001

[18] J.-D. Deuschel and D. W. Stroock. Large Deviations. Academic Press, Boston, MA, 1989. | MR 997938 | Zbl 0705.60029

[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. | Zbl 0348.60031

[20] J. Dugundji. Topology. Allyn and Bacon, Boston, 1966. | MR 193606 | Zbl 0397.54003

[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. | MR 1649006 | Zbl 0934.60023

[22] S. N. Ethier and T. G. Kurtz. Markov Processes. Characterization and Convergence. Wiley, New York, 1986. | MR 838085 | Zbl 1089.60005

[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. | MR 2787971 | Zbl 1219.82127

[24] D. Gabrielli and C. Valente. Which random walks are cyclic? ALEA, Lat. Am. J Probab. Math. Stat. 9 (2012) 231–267. | MR 2923192 | Zbl 1277.82028

[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. | MR 2700635

[27] G. Kesidis and J. Walrand. Relative entropy between Markov transition rate matrices. IEEE Trans. Inform. Theory 39 (3) (1993) 1056–1057. | MR 1237729 | Zbl 0784.60034

[28] C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems. Springer, Berlin, 1999. | MR 1707314 | Zbl 0927.60002

[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. | MR 2639718 | Zbl 1196.60039

[30] K. Kuwada. On large deviations for random currents induced from stochastic line integrals. Forum Math. 18 (2006) 639–676. | MR 2254389 | Zbl 1112.60005

[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. | MR 1705590 | Zbl 0934.60090

[33] J. Macqueen. Circuit processes. Ann. Probab. 9 (1981) 604–610. | MR 624686 | Zbl 0464.60070

[34] M. Mariani. A 𝛤-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. | MR 1650235 | Zbl 0910.46008

[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. | MR 1246427 | Zbl 0832.49024

[38] H. Spohn. Large Scale Dynamics of Interacting Particles. Springer, Berlin, 1991. | Zbl 0742.76002

[39] J. R. Norris. Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ. Press, Cambridge, 1999. | MR 1600720 | Zbl 0938.60058

[40] S. R. S. Varadhan. Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics 46. SIAM, Philadelphia, PA, 1984. | MR 758258 | Zbl 0549.60023