Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Non-equilibrium fluctuations for the SSEP with a slow bond

D. Erhard, T. Franco, P. Gonçalves, A. Neumann, and M. Tavares

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Abstract

We prove the non-equilibrium fluctuations for the one-dimensional symmetric simple exclusion process with a slow bond. This generalizes a result of [Stochastic Process. Appl. 123 (2013) 4156–4185; Stochastic Process. Appl. 126 (2016) 3235–3242], which dealt with the equilibrium fluctuations. The foundation stone of our proof is a precise estimate on the correlations of the system, and that is by itself one of the main novelties of this paper. To obtain these estimates, we first deduce a spatially discrete PDE for the covariance function and we relate it to the local times of a random walk in a non-homogeneous environment via Duhamel’s principle. Projection techniques and coupling arguments reduce the analysis to the problem of studying the local times of the classical random walk. We think that the method developed here can be applied to a variety of models, and we provide a discussion on this matter.

Résumé

Nous décrivons les fluctuations hors-équilibre du processus d’exclusion simple symétrique en dimension 1 avec des liens lents. Ceci étend un résultat de [Stochastic Process. Appl. 123 (2013) 4156–4185; Stochastic Process. Appl. 126 (2016) 3235–3242], qui traitait des fluctuations à l’équilibre. La pierre de touche de notre preuve est une estimée précise des corrélations du système, qui est en elle-même une des nouveautés principales de cet article. Pour obtenir ces estimées, nous obtenons dans un premier temps une EDP discrète en espace pour la fonction de covariance et nous la relions aux temps locaux d’une marche aléatoire dans un environnement non-homogène, par le principe de Duhamel. Des techniques de projection et des arguments de couplage permettent de réduire l’analyse à l’étude des temps locaux de la marche aléatoire classique. Nous pensons que cette méthode peut être appliquée à une variété de modèles, et nous argumentons ce point.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 1099-1128.

Dates
Received: 13 September 2018
Revised: 12 March 2019
Accepted: 23 April 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1584345631

Digital Object Identifier
doi:10.1214/19-AIHP995

Mathematical Reviews number (MathSciNet)
MR4076777

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Non-equilibrium fluctuations Slowed exclusion Local times of random walks Two point correlation function

Citation

Erhard, D.; Franco, T.; Gonçalves, P.; Neumann, A.; Tavares, M. Non-equilibrium fluctuations for the SSEP with a slow bond. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 1099--1128. doi:10.1214/19-AIHP995. https://projecteuclid.org/euclid.aihp/1584345631


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