Euler hydrodynamics for attractive particle systems in random environment
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, p. 403-424

We prove quenched hydrodynamic limit under hyperbolic time scaling for bounded attractive particle systems on in random ergodic environment. Our result is a strong law of large numbers, that we illustrate with various examples.

Nous obtenons la limite hydrodynamique trempée, sous un changement d’échelle hyperbolique, pour un système de particules attractif sur en milieu aléatoire ergodique, avec un nombre borné de particules par site. Notre résultat est une loi forte des grands nombres. Nous l’illustrons sur différents exemples.

DOI : https://doi.org/10.1214/12-AIHP510
Classification:  60K35,  82C22
Keywords: hydrodynamic limit, attractive particle system, scalar conservation law, entropy solution, random environment, quenched disorder, generalized misanthropes and k-step models
@article{AIHPB_2014__50_2_403_0,
     author = {Bahadoran, C. and Guiol, H. and Ravishankar, K. and Saada, Ellen},
     title = {Euler hydrodynamics for attractive particle systems in random environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {2014},
     pages = {403-424},
     doi = {10.1214/12-AIHP510},
     zbl = {1294.60116},
     mrnumber = {3189077},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_2_403_0}
}
Bahadoran, C.; Guiol, H.; Ravishankar, K.; Saada, E. Euler hydrodynamics for attractive particle systems in random environment. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 403-424. doi : 10.1214/12-AIHP510. http://www.numdam.org/item/AIHPB_2014__50_2_403_0/

[1] E. D. Andjel. Invariant measures for the zero range process. Ann. Probab. 10 (1982) 525-547. | MR 659526 | Zbl 0492.60096

[2] E. D. Andjel and M. E. Vares. Hydrodynamic equations for attractive particle systems on . J. Stat. Phys. 47 (1987) 265-288. Correction to: “Hydrodynamic equations for attractive particle systems on ”. J. Stat. Phys. 113 (2003) 379-380. | MR 892931 | Zbl 0685.58043

[3] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. A constructive approach to Euler hydrodynamics for attractive particle systems 99 (2002) 1-30. | MR 1894249 | Zbl 1058.60084

[4] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab. 34 (2006) 1339-1369. | MR 2257649 | Zbl 1101.60075

[5] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Strong hydrodynamic limit for attractive particle systems on . Electron. J. Probab. 15 (2010) 1-43. | MR 2578381 | Zbl 1193.60113

[6] I. Benjamini, P. A. Ferrari and C. Landim. Asymmetric processes with random rates. Stoch. Process. Appl. 61 (1996) 181-204. | MR 1386172 | Zbl 0849.60093

[7] M. Bramson and T. Mountford. Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30 (2002) 1082-1130. | MR 1920102 | Zbl 1042.60062

[8] C. Cocozza-Thivent. Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70 (1985) 509-523. | MR 807334 | Zbl 0554.60097

[9] P. Dai Pra, P. Y. Louis and I. Minelli. Realizable monotonicity for continuous-time Markov processes. Stochastic Process. Appl. 120 (2010) 959-982. | MR 2610334 | Zbl 1200.60064

[10] M. R. Evans. Bose-Einstein condensation in disordered exclusion models and relation to traffic flow. Europhys. Lett. 36 (1996) 13-18. DOI:10.1209/epl/i1996-00180-y.

[11] A. Faggionato. Bulk diffusion of 1D exclusion process with bond disorder. Markov Process. Related Fields 13 (2007) 519-542. | MR 2357386 | Zbl 1144.60058

[12] A. Faggionato and F. Martinelli. Hydrodynamic limit of a disordered lattice gas. Probab. Theory Related Fields 127 (2003) 535-608. | MR 2021195 | Zbl 1052.60083

[13] J. A. Fill and M. Machida. Stochastic monotonicity and realizable monotonicity. Ann. Probab. 29 (2001) 938-978. | MR 1849183 | Zbl 1015.60010

[14] J. Fritz. Hydrodynamics in a symmetric random medium. Comm. Math. Phys. 125 (1989) 13-25. | MR 1017736 | Zbl 0682.76001

[15] T. Gobron and E. Saada. Couplings, attractiveness and hydrodynamics for conservative particle systems. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010) 1132-1177. | Numdam | MR 2744889 | Zbl 1252.60093

[16] P. Gonçalves and M. Jara. Scaling limits for gradient systems in random environment. J. Stat. Phys. 131 (2008) 691-716. | MR 2398949 | Zbl 1144.82043

[17] H. Guiol. Some properties of k-step exclusion processes. J. Stat. Phys. 94 (1999) 495-511. | MR 1675362 | Zbl 0953.60091

[18] M. Jara. Hydrodynamic limit of the exclusion process in inhomogeneous media. In Dynamics, Games and Science II 449-465. M. M. Peixoto, A. A. Pinto and D. A. Rand (Eds). Springer Proceedings in Mathematics 2. Springer, Heidelberg, 2011. | MR 2883297 | Zbl pre06078295

[19] T. Kamae and U. Krengel. Stochastic partial ordering. Ann. Probab. 6 (1978) 1044-1049. | MR 512419 | Zbl 0392.60012

[20] C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin, 1999. | MR 1707314 | Zbl 0927.60002

[21] A. Koukkous. Hydrodynamic behavior of symmetric zero-range processes with random rates. Stochastic Process. Appl. 84 (1999) 297-312. | MR 1719270 | Zbl 0996.60108

[22] T. M. Liggett. Coupling the simple exclusion process. Ann. Probab. 4 (1976) 339-356. | MR 418291 | Zbl 0339.60091

[23] T. M. Liggett. Interacting Particle Systems. Classics in Mathematics, Reprint of first edition. Springer, Berlin, 2005. | MR 2108619 | Zbl 1103.82016

[24] T. S. Mountford, K. Ravishankar and E. Saada. Macroscopic stability for nonfinite range kernels. Braz. J. Probab. Stat. 24 (2010) 337-360. | MR 2643570 | Zbl 1195.82058

[25] K. Nagy. Symmetric random walk in random environment in one dimension. Period. Math. Hungar. 45 (2002) 101-120. | MR 1955197 | Zbl 1064.60202

[26] J. Quastel. Bulk diffusion in a system with site disorder. Ann. Probab. 34 (2006) 1990-2036. | MR 2271489 | Zbl 1104.60066

[27] F. Rezakhanlou. Hydrodynamic limit for attractive particle systems on d . Comm. Math. Phys. 140 (1991) 417-448. | MR 1130693 | Zbl 0738.60098

[28] T. Seppäläinen. Existence of hydrodynamics for the totally asymmetric simple K-exclusion process. Ann. Probab. 27 (1999) 361-415. | MR 1681094 | Zbl 0947.60088

[29] T. Seppäläinen and J. Krug. Hydrodynamics and Platoon formation for a totally asymmetric exclusion model with particlewise disorder. J. Stat. Phys. 95 (1999) 525-567. | MR 1700871 | Zbl 0964.82041

[30] D. Serre. Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves. Cambridge University Press, Cambridge, 1999. Translated from the 1996 French original by I. N. Sneddon. | MR 1707279 | Zbl 0930.35001

[31] V. Strassen. The existence of probability measures with given marginals. Ann. Math. Statist. 36 (1965) 423-439. | MR 177430 | Zbl 0135.18701