Symmetric exclusion as a random environment: Hydrodynamic limits
Avena, Luca ; Franco, Tertuliano ; Jara, Milton ; Völlering, Florian
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 901-916 / Harvested from Numdam
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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.

Publié le : 2015-01-01
DOI : https://doi.org/10.1214/14-AIHP607 
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     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},
     volume = {51},
     year = {2015},
     pages = {901-916},
     doi = {10.1214/14-AIHP607},
     mrnumber = {3365966},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2015__51_3_901_0}
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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) pp. 901-916. doi : 10.1214/14-AIHP607. http://gdmltest.u-ga.fr/item/AIHPB_2015__51_3_901_0/

[1] L. Avena, F. Den Hollander and F. Redig. 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 2664339 | Zbl 1198.82049

[2] L. Avena, 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 3108811 | Zbl 1277.60185

[3] L. Avena and P. Thomann. Continuity and anomalous fluctuations in random walks in dynamic random environments: Numerics, phase diagrams and conjectures. J. Stat. Phys. 147 (2012) 1041–1067. | MR 2949519 | Zbl 1246.82040

[4] F. Comets, N. Gantert and O. Zeitouni. Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab. Theory Related Fields 118 (2000) 65–114. | MR 1785454 | Zbl 0965.60098

[5] D. Dolgopyat, G. Keller and C. Liverani. Random walk in Markovian environment. Ann. Probab. 36 (2008) 1676–1710. | MR 2440920 | Zbl 1192.60110

[6] A. Greven and F. Den Hollander. Large deviations for a random walk in random environment. Ann. Probab. 22 (1994) 1381–1428. | MR 1303649 | Zbl 0820.60054

[7] F. Redig and F. Völlering. Random walks in dynamic random environments: A transference principle. Ann. Probab. 41 (2013) 3157–3180. | MR 3127878 | Zbl 1277.82051

[8] T. Franco and C. Landim. Hydrodynamic limit of gradient exclusion processes with conductances. Arch. Ration. Mech. Anal. 195 (2010) 409–439. | MR 2592282 | Zbl 1192.82062

[9] M. Z. Guo, G. C. Papanicolau and S. R. S. Varadhan. Nonlinear diffusion limit for a system with nearest neighbor interactions. Comm. Math. Phys. 118 (1988) 31–59. | MR 954674 | Zbl 0652.60107

[10] M. Jara, C. Landim and S. Sethuraman. Nonequilibrium fluctuations for a tagged particle in mean-zero one dimensional zero-range processes. Probab. Theory Related Fields 145 (2009) 565–590. | MR 2529439 | Zbl 1185.60113

[11] C. Kipnis and C. Landim. Scaling Limits of Particle Systems. Grundlehren der Mathematischen Wissenschaften 320. Springer, Berlin, 1999. | MR 1707314 | Zbl 0927.60002

[12] C. Landim, S. Olla and S. R. S. Varadhan. Asymptotic behavior of a tagged particle in simple exclusion processes. Bol. Soc. Brasil. Mat. 31 (3) (2000) 241–275. | MR 1817088 | Zbl 0983.60100

[13] A. S. Sznitman. Lectures on Random Motions in Random Media. Ten Lectures on Random Media. DMV-Lectures 32. Birkhäuser, Basel, 2002. | MR 1890289 | Zbl 1075.60128

[14] O. Zeitouni. Random walks in random environments. J. Phys. A: Math. Gen. 39 (2006) 433–464. | MR 2261885 | Zbl 1108.60085