Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Symmetric exclusion as a random environment: Hydrodynamic limits

Luca Avena, Tertuliano Franco, Milton Jara, and Florian Völlering

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Abstract

We consider a one-dimensional continuous time random walk with transition rates depending on an underlying autonomous simple symmetric exclusion process starting out of equilibrium. This model represents an example of a random walk in a slowly non-uniform mixing dynamic random environment. Under a proper space–time rescaling in which the exclusion is speeded up compared to the random walk, we prove a hydrodynamic limit theorem for the exclusion as seen by this walk and we derive an ODE describing the macroscopic evolution of the walk. The main difficulty is the proof of a replacement lemma for the exclusion as seen from the walk without explicit knowledge of its invariant measures. We further discuss how to obtain similar results for several variants of this model.

Résumé

Nous considérons une marche aléatoire unidimensionnelle à temps continu, avec des taux des sauts dépendants d’un processus d’exclusion autonome et hors équilibre. Ce modèle répresente un exemple de marche aléatoire en milieu aléatoire dynamique, où le milieu n’a pas des bonnes proprietés de mélange. Sous la bonne échelle spatio-temporelle, où le processus d’exclusion est accéléré de plus en plus par rapport à la marche, nous démonstrons un théorème de limite hydrodynamique pour le processus d’exclusion vu par la marche aléatoire, et nous dérivons une EDO qui décrit l’évolution macroscopique de la marche. La difficulté principale est la démonstration d’un lemme de remplacement pour le processus d’exclusion vu par la marche aléatoire, sans une connaissance explicite de ses mesures invariantes. Nous discutons comment obtenir des résultats similaires pour des variantes du modèle en question.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 3 (2015), 901-916.

Dates
Received: 10 July 2013
Revised: 25 November 2013
Accepted: 23 January 2014
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1435759233

Digital Object Identifier
doi:10.1214/14-AIHP607

Mathematical Reviews number (MathSciNet)
MR3365966

Zentralblatt MATH identifier
1359.82012

Subjects
Primary: 82C05: Classical dynamic and nonequilibrium statistical mechanics (general) 60K37: Processes in random environments

Keywords
Random walks in random environments Macroscopic speed Hydrodynamic limits Particle systems Exclusion process

Citation

Avena, Luca; Franco, Tertuliano; Jara, Milton; Völlering, Florian. Symmetric exclusion as a random environment: Hydrodynamic limits. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 3, 901--916. doi:10.1214/14-AIHP607. https://projecteuclid.org/euclid.aihp/1435759233


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