Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Euler hydrodynamics for attractive particle systems in random environment

C. Bahadoran, H. Guiol, K. Ravishankar, and E. Saada

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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.


Nous obtenons la limite hydrodynamique trempée, sous un changement d’échelle hyperbolique, pour un système de particules attractif sur $\mathbb{Z}$ 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.

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Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2 (2014), 403-424.

First available in Project Euclid: 26 March 2014

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35]

Hydrodynamic limit Attractive particle system Scalar conservation law Entropy solution Random environment Quenched disorder Generalized misanthropes and $k$-step models


Bahadoran, C.; Guiol, H.; Ravishankar, K.; Saada, E. Euler hydrodynamics for attractive particle systems in random environment. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 403--424. doi:10.1214/12-AIHP510.

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  • [1] E. D. Andjel. Invariant measures for the zero range process. Ann. Probab. 10 (1982) 525–547.
  • [2] E. D. Andjel and M. E. Vares. Hydrodynamic equations for attractive particle systems on $\mathbb{Z}$. J. Stat. Phys. 47 (1987) 265–288. Correction to: “Hydrodynamic equations for attractive particle systems on $\mathbb{Z}$”. J. Stat. Phys. 113 (2003) 379–380.
  • [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.
  • [4] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab. 34 (2006) 1339–1369.
  • [5] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Strong hydrodynamic limit for attractive particle systems on $\mathbb{Z}$. Electron. J. Probab. 15 (2010) 1–43.
  • [6] I. Benjamini, P. A. Ferrari and C. Landim. Asymmetric processes with random rates. Stoch. Process. Appl. 61 (1996) 181–204.
  • [7] M. Bramson and T. Mountford. Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30 (2002) 1082–1130.
  • [8] C. Cocozza-Thivent. Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70 (1985) 509–523.
  • [9] P. Dai Pra, P. Y. Louis and I. Minelli. Realizable monotonicity for continuous-time Markov processes. Stochastic Process. Appl. 120 (2010) 959–982.
  • [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.
  • [12] A. Faggionato and F. Martinelli. Hydrodynamic limit of a disordered lattice gas. Probab. Theory Related Fields 127 (2003) 535–608.
  • [13] J. A. Fill and M. Machida. Stochastic monotonicity and realizable monotonicity. Ann. Probab. 29 (2001) 938–978.
  • [14] J. Fritz. Hydrodynamics in a symmetric random medium. Comm. Math. Phys. 125 (1989) 13–25.
  • [15] T. Gobron and E. Saada. Couplings, attractiveness and hydrodynamics for conservative particle systems. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010) 1132–1177.
  • [16] P. Gonçalves and M. Jara. Scaling limits for gradient systems in random environment. J. Stat. Phys. 131 (2008) 691–716.
  • [17] H. Guiol. Some properties of $k$-step exclusion processes. J. Stat. Phys. 94 (1999) 495–511.
  • [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.
  • [19] T. Kamae and U. Krengel. Stochastic partial ordering. Ann. Probab. 6 (1978) 1044–1049.
  • [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.
  • [21] A. Koukkous. Hydrodynamic behavior of symmetric zero-range processes with random rates. Stochastic Process. Appl. 84 (1999) 297–312.
  • [22] T. M. Liggett. Coupling the simple exclusion process. Ann. Probab. 4 (1976) 339–356.
  • [23] T. M. Liggett. Interacting Particle Systems. Classics in Mathematics, Reprint of first edition. Springer, Berlin, 2005.
  • [24] T. S. Mountford, K. Ravishankar and E. Saada. Macroscopic stability for nonfinite range kernels. Braz. J. Probab. Stat. 24 (2010) 337–360.
  • [25] K. Nagy. Symmetric random walk in random environment in one dimension. Period. Math. Hungar. 45 (2002) 101–120.
  • [26] J. Quastel. Bulk diffusion in a system with site disorder. Ann. Probab. 34 (2006) 1990–2036.
  • [27] F. Rezakhanlou. Hydrodynamic limit for attractive particle systems on $\mathbb{Z}^{d}$. Comm. Math. Phys. 140 (1991) 417–448.
  • [28] T. Seppäläinen. Existence of hydrodynamics for the totally asymmetric simple $K$-exclusion process. Ann. Probab. 27 (1999) 361–415.
  • [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.
  • [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.
  • [31] V. Strassen. The existence of probability measures with given marginals. Ann. Math. Statist. 36 (1965) 423–439.