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.
@article{AIHPB_2015__51_3_901_0, author = {Avena, Luca and Franco, Tertuliano and Jara, Milton and V\"ollering, Florian}, title = {Symmetric exclusion as a random environment: {Hydrodynamic} limits}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {901--916}, publisher = {Gauthier-Villars}, volume = {51}, number = {3}, year = {2015}, doi = {10.1214/14-AIHP607}, mrnumber = {3365966}, zbl = {1359.82012}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/14-AIHP607/} }
TY - JOUR AU - Avena, Luca AU - Franco, Tertuliano AU - Jara, Milton AU - Völlering, Florian TI - Symmetric exclusion as a random environment: Hydrodynamic limits JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 901 EP - 916 VL - 51 IS - 3 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/14-AIHP607/ DO - 10.1214/14-AIHP607 LA - en ID - AIHPB_2015__51_3_901_0 ER -
%0 Journal Article %A Avena, Luca %A Franco, Tertuliano %A Jara, Milton %A Völlering, Florian %T Symmetric exclusion as a random environment: Hydrodynamic limits %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 901-916 %V 51 %N 3 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/14-AIHP607/ %R 10.1214/14-AIHP607 %G en %F AIHPB_2015__51_3_901_0
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://archive.numdam.org/articles/10.1214/14-AIHP607/
[1] Large deviation principle for one-dimensional random walk in dynamic random environment: Attractive spin-flips and simple symmetric exclusion. Markov Process. Related Fields 16 (2010) 139–168. | MR | Zbl
, and .[2] R. dos Santos and F. Völlering. A transient random walk driven by an exclusion process: Regenerations, limit theorems and an Einstein relation. ALEA Lat. Am. J. Probab. Math. Stat. 10 (2) (2013) 693–709. | MR | Zbl
,[3] Continuity and anomalous fluctuations in random walks in dynamic random environments: Numerics, phase diagrams and conjectures. J. Stat. Phys. 147 (2012) 1041–1067. | DOI | MR | Zbl
and .[4] Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab. Theory Related Fields 118 (2000) 65–114. | DOI | MR | Zbl
, and .[5] Random walk in Markovian environment. Ann. Probab. 36 (2008) 1676–1710. | DOI | MR | Zbl
, and .[6] Large deviations for a random walk in random environment. Ann. Probab. 22 (1994) 1381–1428. | DOI | MR | Zbl
and .[7] Random walks in dynamic random environments: A transference principle. Ann. Probab. 41 (2013) 3157–3180. | DOI | MR | Zbl
and .[8] Hydrodynamic limit of gradient exclusion processes with conductances. Arch. Ration. Mech. Anal. 195 (2010) 409–439. | DOI | MR | Zbl
and .[9] Nonlinear diffusion limit for a system with nearest neighbor interactions. Comm. Math. Phys. 118 (1988) 31–59. | DOI | MR | Zbl
, and .[10] Nonequilibrium fluctuations for a tagged particle in mean-zero one dimensional zero-range processes. Probab. Theory Related Fields 145 (2009) 565–590. | DOI | MR | Zbl
, and .[11] Scaling Limits of Particle Systems. Grundlehren der Mathematischen Wissenschaften 320. Springer, Berlin, 1999. | MR | Zbl
and .[12] Asymptotic behavior of a tagged particle in simple exclusion processes. Bol. Soc. Brasil. Mat. 31 (3) (2000) 241–275. | MR | Zbl
, and .[13] Lectures on Random Motions in Random Media. Ten Lectures on Random Media. DMV-Lectures 32. Birkhäuser, Basel, 2002. | MR | Zbl
.[14] Random walks in random environments. J. Phys. A: Math. Gen. 39 (2006) 433–464. | MR | Zbl
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