Symmetric exclusion as a random environment: Hydrodynamic limits
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 901-916.

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.

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.

DOI : 10.1214/14-AIHP607
Mots-clés : random walks in random environments, macroscopic speed, hydrodynamic limits, particle systems, exclusion process 
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Avena, Luca; Franco, Tertuliano; Jara, Milton; Völlering, Florian. Symmetric exclusion as a random environment: Hydrodynamic limits. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 901-916. doi : 10.1214/14-AIHP607. http://www.numdam.org/articles/10.1214/14-AIHP607/

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