Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Random walks on quasi one dimensional lattices: Large deviations and fluctuation theorems

Alessandra Faggionato and Vittoria Silvestri

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Several stochastic processes modeling molecular motors on a linear track are given by random walks (not necessarily Markovian) on quasi 1d lattices and share a common regenerative structure. Analyzing this abstract common structure, we derive information on the large fluctuations of the stochastic process by proving large deviation principles for the first-passage times and for the position. We focus our attention on the Gallavotti–Cohen-type symmetry of the position rate function (fluctuation theorem), showing its equivalence with the independence of suitable random variables. In the special case of Markov random walks, we show that this symmetry is universal only inside a suitable class of quasi 1d lattices.


Nous considérons différents processus stochastiques modélisant des moteurs moléculaires : il s’agit de marches aléatoires, non nécessairement markoviennes, le long d’un rail linéaire, presque un réseau unidimensionnel, qui partagent une même structure de régénération. En analysant cette structure abstraite commune nous contrôlons les grandes déviations du processus stochastique, nous établissons des principes de grandes déviations pour les temps de premier passage et pour la variable de position. Nous nous concentrons sur les symétries de type Gallavotti–Cohen de la fonction de taux positionnelle (théorème de fluctuations), en montrant son équivalence avec l’indépendance de certaines variables aléatoires. Dans le cas particulier des marches aléatoires markoviennes, nous montrons que cette symétrie n’est universelle qu’au sein d’une classe particulière de réseaux presque unidimensionnels.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 1 (2017), 46-78.

Received: 14 September 2014
Revised: 14 June 2015
Accepted: 31 July 2015
First available in Project Euclid: 8 February 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Markov chain Random time change Large deviation principle Gallavotti–Cohen-type symmetry Fluctuation theorem Molecular motor


Faggionato, Alessandra; Silvestri, Vittoria. Random walks on quasi one dimensional lattices: Large deviations and fluctuation theorems. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 1, 46--78. doi:10.1214/15-AIHP708.

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  • [1] D. Andrieux. Nonequilibrium Statistical Thermodynamics at the Nanoscale: From Maxwell Demon to Biological Information Processing. VDM Verlag, Saarbrücken, 2009.
  • [2] H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Heidelberg, 2011.
  • [3] H. Cartan. Elementary Theory of Analytic Functions of One or Several Complex Variables. Dover Publications Inc., New York, 1995.
  • [4] R. K. Das and A. B. Kolomeisky. Dynamic properties of molecular motors in the divided-pathway model. Phys. Chem. Chem. Phys. 11 (2009) 4815–4820.
  • [5] A. Dembo, N. Gantert and O. Zeitouni. Large deviations for random walk in random environment with holding times. Ann. Probab. 32 (2004) 996–1029.
  • [6] A. Dembo and O. Zeitouni. Large Deviation Techniques and Applications, 2nd edition. Springer, New York, 1998.
  • [7] K. Duffy and M. Rodgers-Lee. Some useful functions for functional large deviations. Stoch. Stoch. Reports 76 (2004) 267–279.
  • [8] A. Faggionato and V. Silvestri. Discrete kinetic models for molecular motors: Asymptotic velocity and Gaussian fluctuations. J. Stat. Phys. 157 (2014) 1062–1096.
  • [9] A. Faggionato and V. Silvestri. Fluctuation theorems for discrete kinetic models of molecular motors. Preprint, 2017. Available at arXiv:1701.01721.
  • [10] 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.
  • [11] M. E. Fisher and A. B. Kolomeisky. The force exerted by a molecular motor. Proc. Natl. Acad. Sci. USA 96 (1999) 6597–6602.
  • [12] M. E. Fisher and A. B. Kolomeisky. Molecular motors and the force they exert. Phys. A 274 (1999) 241–266.
  • [13] N. Gantert. Private communication.
  • [14] R. C. Gunning and H. Rossi. Analytic Functions of Several Complex Variables. Prentice-Hall Inc., Englewood Cliffs, NJ, 1965.
  • [15] F. den Hollander. Large Deviations. Fields Institute Monographs 14. Amer. Math. Soc., Providence, RI, 2000.
  • [16] F. Jülicher, A. Ajdari and J. Prost. Modeling molecular motors. Rev. Modern Phys. 69 (1997) 1269–1281.
  • [17] A. B. Kolomeisky. Exact results for parallel-chain kinetic models of biological transport. J. Chem. Phys. 115 (2001) 7523.
  • [18] A. B. Kolomeisky and M. E. Fisher. Extended kinetic models with waiting-time distributions: Exact results. J. Chem. Phys. 113 (2000) 10867–10877.
  • [19] A. B. Kolomeisky and M. E. Fisher. Periodic sequential kinetic models with jumping, branching and deaths. Phys. A 279 (2000) 1–20.
  • [20] A. B. Kolomeisky and M. E. Fisher. Molecular motors: A theorist’s perspective. Annu. Rev. Phys. Chem. 58 (2007) 675–695.
  • [21] D. Lacoste, A. W. C. Lau and K. Mallick. Fluctuation theorem and large deviation function for a solvable model of a molecular motor. Phys. Rev. E 78 (2008) 011915.
  • [22] 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 61–88. Birkhäuser Verlag, Basel, 2011.
  • [23] 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.
  • [24] R. Lefevere, M. Mariani and L. Zambotti. Large deviations for renewal processes. Stochastic Process. Appl. 121 (2011) 2243–2271.
  • [25] T. Kato. A Short Introduction to Perturbation Theory for Linear Operators. Springer, New York, 1982.
  • [26] M. Mariani and L. Zambotti. A renewal version of the Sanov theorem. Electron. Commun. Probab. 19 (2014) 1–13.
  • [27] J. R. Norris. Markov Chains. Cambridge Univ. Press, Cambridge, 1997.
  • [28] R. Russell. The large deviations of random time changes. Ph.D. thesis, Trinity College, Dublin, 1997.
  • [29] U. Seifert. Stochastic thermodynamics, fluctuation theorems, and molecular machines. Rep. Progr. Phys. 75 (2012) 126001.
  • [30] D. Tsygankov and M. E. Fisher. Kinetic models for mechanoenzymes: Structural aspects under large loads. J. Chem. Phys. 128 (2008) 015102.