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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Abstract

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.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1486544884

Digital Object Identifier
doi:10.1214/15-AIHP708

Mathematical Reviews number (MathSciNet)
MR3606734

Zentralblatt MATH identifier
1364.60036

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

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

Citation

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. https://projecteuclid.org/euclid.aihp/1486544884


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