Euler hydrodynamics for attractive particle systems in random environment

C. Bahadoran; H. Guiol; K. Ravishankar; E. Saada

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 2, page 403-424
  • ISSN: 0246-0203

Abstract

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

How to cite

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Bahadoran, C., et al. "Euler hydrodynamics for attractive particle systems in random environment." Annales de l'I.H.P. Probabilités et statistiques 50.2 (2014): 403-424. <http://eudml.org/doc/271999>.

@article{Bahadoran2014,
abstract = {We prove quenched hydrodynamic limit under hyperbolic time scaling for bounded attractive particle systems on $\mathbb \{Z\}$ in random ergodic environment. Our result is a strong law of large numbers, that we illustrate with various examples.},
author = {Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {hydrodynamic limit; attractive particle system; scalar conservation law; entropy solution; random environment; quenched disorder; generalized misanthropes and $k$-step models; hydrodynamic limit; attractive particle system; scalar conservation law; entropy solution; random environment; quenched disorder; generalized misanthropes; -step models},
language = {eng},
number = {2},
pages = {403-424},
publisher = {Gauthier-Villars},
title = {Euler hydrodynamics for attractive particle systems in random environment},
url = {http://eudml.org/doc/271999},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Bahadoran, C.
AU - Guiol, H.
AU - Ravishankar, K.
AU - Saada, E.
TI - Euler hydrodynamics for attractive particle systems in random environment
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 2
SP - 403
EP - 424
AB - We prove quenched hydrodynamic limit under hyperbolic time scaling for bounded attractive particle systems on $\mathbb {Z}$ in random ergodic environment. Our result is a strong law of large numbers, that we illustrate with various examples.
LA - eng
KW - hydrodynamic limit; attractive particle system; scalar conservation law; entropy solution; random environment; quenched disorder; generalized misanthropes and $k$-step models; hydrodynamic limit; attractive particle system; scalar conservation law; entropy solution; random environment; quenched disorder; generalized misanthropes; -step models
UR - http://eudml.org/doc/271999
ER -

References

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  1. [1] E. D. Andjel. Invariant measures for the zero range process. Ann. Probab.10 (1982) 525–547. Zbl0492.60096MR659526
  2. [2] E. D. Andjel and M. E. Vares. Hydrodynamic equations for attractive particle systems on ℤ . J. Stat. Phys. 47 (1987) 265–288. Correction to: “Hydrodynamic equations for attractive particle systems on ℤ ”. J. Stat. Phys. 113 (2003) 379–380. Zbl0685.58043MR892931
  3. [3] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. A constructive approach to Euler hydrodynamics for attractive particle systems. Application to k -step exclusion. Stochastic Process. Appl. 99 (2002) 1–30. Zbl1058.60084MR1894249
  4. [4] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab.34 (2006) 1339–1369. Zbl1101.60075MR2257649
  5. [5] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Strong hydrodynamic limit for attractive particle systems on ℤ . Electron. J. Probab.15 (2010) 1–43. Zbl1193.60113MR2578381
  6. [6] I. Benjamini, P. A. Ferrari and C. Landim. Asymmetric processes with random rates. Stoch. Process. Appl.61 (1996) 181–204. Zbl0849.60093MR1386172
  7. [7] M. Bramson and T. Mountford. Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab.30 (2002) 1082–1130. Zbl1042.60062MR1920102
  8. [8] C. Cocozza-Thivent. Processus des misanthropes. Z. Wahrsch. Verw. Gebiete70 (1985) 509–523. Zbl0554.60097MR807334
  9. [9] P. Dai Pra, P. Y. Louis and I. Minelli. Realizable monotonicity for continuous-time Markov processes. Stochastic Process. Appl.120 (2010) 959–982. Zbl1200.60064MR2610334
  10. [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. [11] A. Faggionato. Bulk diffusion of 1D exclusion process with bond disorder. Markov Process. Related Fields13 (2007) 519–542. Zbl1144.60058MR2357386
  12. [12] A. Faggionato and F. Martinelli. Hydrodynamic limit of a disordered lattice gas. Probab. Theory Related Fields127 (2003) 535–608. Zbl1052.60083MR2021195
  13. [13] J. A. Fill and M. Machida. Stochastic monotonicity and realizable monotonicity. Ann. Probab.29 (2001) 938–978. Zbl1015.60010MR1849183
  14. [14] J. Fritz. Hydrodynamics in a symmetric random medium. Comm. Math. Phys.125 (1989) 13–25. Zbl0682.76001MR1017736
  15. [15] T. Gobron and E. Saada. Couplings, attractiveness and hydrodynamics for conservative particle systems. Ann. Inst. H. Poincaré Probab. Statist.46 (2010) 1132–1177. Zbl1252.60093MR2744889
  16. [16] P. Gonçalves and M. Jara. Scaling limits for gradient systems in random environment. J. Stat. Phys.131 (2008) 691–716. Zbl1144.82043MR2398949
  17. [17] H. Guiol. Some properties of k -step exclusion processes. J. Stat. Phys.94 (1999) 495–511. Zbl0953.60091MR1675362
  18. [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. Zbl06078295MR2883297
  19. [19] T. Kamae and U. Krengel. Stochastic partial ordering. Ann. Probab.6 (1978) 1044–1049. Zbl0392.60012MR512419
  20. [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. Zbl0927.60002MR1707314
  21. [21] A. Koukkous. Hydrodynamic behavior of symmetric zero-range processes with random rates. Stochastic Process. Appl.84 (1999) 297–312. Zbl0996.60108MR1719270
  22. [22] T. M. Liggett. Coupling the simple exclusion process. Ann. Probab.4 (1976) 339–356. Zbl0339.60091MR418291
  23. [23] T. M. Liggett. Interacting Particle Systems. Classics in Mathematics, Reprint of first edition. Springer, Berlin, 2005. Zbl1103.82016MR2108619
  24. [24] T. S. Mountford, K. Ravishankar and E. Saada. Macroscopic stability for nonfinite range kernels. Braz. J. Probab. Stat.24 (2010) 337–360. Zbl1195.82058MR2643570
  25. [25] K. Nagy. Symmetric random walk in random environment in one dimension. Period. Math. Hungar.45 (2002) 101–120. Zbl1064.60202MR1955197
  26. [26] J. Quastel. Bulk diffusion in a system with site disorder. Ann. Probab.34 (2006) 1990–2036. Zbl1104.60066MR2271489
  27. [27] F. Rezakhanlou. Hydrodynamic limit for attractive particle systems on ℤ d . Comm. Math. Phys.140 (1991) 417–448. Zbl0738.60098MR1130693
  28. [28] T. Seppäläinen. Existence of hydrodynamics for the totally asymmetric simple K -exclusion process. Ann. Probab.27 (1999) 361–415. Zbl0947.60088MR1681094
  29. [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. Zbl0964.82041MR1700871
  30. [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. Zbl0930.35001MR1707279
  31. [31] V. Strassen. The existence of probability measures with given marginals. Ann. Math. Statist.36 (1965) 423–439. Zbl0135.18701MR177430

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