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Constructive Euler hydrodynamics for one-dimensional attractive particle systems. (English) Zbl 1446.82048

Sidoravicius, Vladas (ed.), Sojourns in probability theory and statistical physics. III. Interacting particle systems and random walks, a festschrift for Charles M. Newman. Singapore: Springer; Shanghai: NYU Shanghai. Springer Proc. Math. Stat. 300, 43-89 (2019).
Summary: We review a (constructive) approach first introduced in [C. Bahadoran et al., Stochastic Processes Appl. 99, No. 1, 1–30 (2002; Zbl 1058.60084)] and further developed in [C. Bahadoran et al., Ann. Probab. 34, No. 4, 1339–1369 (2006; Zbl 1101.60075); Electron. J. Probab. 15, Paper No. 1, 1–43 (2010; Zbl 1193.60113); Ann. Inst. Henri Poincaré, Probab. Stat. 50, No. 2, 403–424 (2014; Zbl 1294.60116); T. S. Mountford et al., Braz. J. Probab. Stat. 24, No. 2, 337–360 (2010; Zbl 1195.82058)] for hydrodynamic limits of asymmetric attractive particle systems, in a weak or in a strong (that is, almost sure) sense, in an homogeneous or in a quenched disordered setting.
For the entire collection see [Zbl 1429.60005].

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
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[1] Andjel, E.D.: Invariant measures for the zero range process. Ann. Probab. 10(3), 525-547 (1982) · Zbl 0492.60096 · doi:10.1214/aop/1176993765
[2] Andjel, E., Ferrari, P.A., Siqueira, A.: Law of large numbers for the asymmetric exclusion process. Stoch. Process. Appl. 132(2), 217-233 (2004) · Zbl 1080.60089 · doi:10.1016/j.spa.2004.04.003
[3] Andjel, E.D., Kipnis, C.: Derivation of the hydrodynamical equation for the zero range interaction process. Ann. Probab. 12, 325-334 (1984) · Zbl 0536.60097 · doi:10.1214/aop/1176993293
[4] Andjel, E.D., Vares, M.E.: Hydrodynamic equations for attractive particle systems on \(\mathbb{Z} \). J. Stat. Phys. 47(1/2), 265-288 (1987) Correction to: “Hydrodynamic equations for attractive particle systems on \(Z\). J. Stat. Phys. 113(1-2), 379-380 (2003) · Zbl 0685.58043
[5] Bahadoran, C.: Blockage hydrodynamics of driven conservative systems. Ann. Probab. 32(1B), 805-854 (2004) · Zbl 1079.60076 · doi:10.1214/aop/1079021465
[6] Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: A constructive approach to Euler hydrodynamics for attractive particle systems. Application to \(k\)-step exclusion. Stoch. Process. Appl. 99(1), 1-30 (2002) · Zbl 1058.60084
[7] Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab. 34(4), 1339-1369 (2006) · Zbl 1101.60075 · doi:10.1214/009117906000000115
[8] Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Strong hydrodynamic limit for attractive particle systems on \(\mathbb{Z} \). Electron. J. Probab. 15(1), 1-43 (2010) · Zbl 1193.60113
[9] Bahadoran, C., Guiol, H., Ravishankar, K., Saada, E.: Euler hydrodynamics for attractive particle systems in random environment. Ann. Inst. H. Poincaré Probab. Statist. 50(2), 403-424 (2014) · Zbl 1294.60116 · doi:10.1214/12-AIHP510
[10] Bahadoran, C., Mountford, T.S., Ravishankar, K., Saada, E.: Supercriticality conditions for the asymmetric zero-range process with sitewise disorder. Braz. J. Probab. Stat. 29(2), 313-335 (2015) · Zbl 1319.60179 · doi:10.1214/14-BJPS273
[11] Bahadoran, C., Mountford, T.S., Ravishankar, K., Saada, E.: Supercritical behavior of zero-range process with sitewise disorder. Ann. Inst. H. Poincaré Probab. Statist. 53(2), 766-801 (2017) · Zbl 1369.60065 · doi:10.1214/15-AIHP736
[12] Bahadoran, C., Mountford, T.S., Ravishankar, K., Saada, E.: Hydrodynamics in a condensation regime: the disordered asymmetric zero-range process. Ann. Probab. (to appear). arXiv:1801.01654 · Zbl 1453.60153
[13] Bahadoran, C., Mountford, T.S., Ravishankar, K., Saada, E.: Quenched convergence and strong local equilibrium for asymmetric zero-range process with site disorder. Probab. Theory Relat. Fields. published online 17 May 2019. https://doi.org/10.1007/s00440-019-00916-2 · Zbl 1434.60269 · doi:10.1007/s00440-019-00916-2
[14] Ballou, D.P.: Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions. Trans. Am. Math. Soc. 152, 441-460 (1970) · Zbl 0207.40401 · doi:10.1090/S0002-9947-1970-0435615-3
[15] Benassi, A., Fouque, J.P.: Hydrodynamical limit for the asymmetric exclusion process. Ann. Probab. 15, 546-560 (1987) · Zbl 0623.60120 · doi:10.1214/aop/1176992158
[16] Benjamini, I., Ferrari, P.A., Landim, C.: Asymmetric processes with random rates. Stoch. Process. Appl. 61(2), 181-204 (1996) · Zbl 0849.60093
[17] Bramson, M., Mountford, T.: Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30(3), 1082-1130 (2002) · Zbl 1042.60062 · doi:10.1214/aop/1029867122
[18] Bressan, A.: Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem. Oxford Lecture Series in Mathematics, vol. 20. Oxford University Press, Oxford (2000) · Zbl 0997.35002
[19] Cocozza-Thivent, C.: Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70(4), 509-523 (1985) · Zbl 0554.60097 · doi:10.1007/BF00531864
[20] Dai Pra, P., Louis, P.Y., Minelli, I.: Realizable monotonicity for continuous-time Markov processes. Stoch. Process. Appl. 120(6), 959-982 (2010) · Zbl 1200.60064 · doi:10.1016/j.spa.2010.03.002
[21] De Masi, A., Presutti, E.: Mathematical Methods for Hydrodynamic Limits. Lecture Notes in Mathematics, vol. 1501. Springer, Heidelberg (1991) · Zbl 0754.60122 · doi:10.1007/BFb0086457
[22] Evans, M.R.: Bose-Einstein condensation in disordered exclusion models and relation to traffic flow. Europhys. Lett. 36(1), 13-18 (1996) · doi:10.1209/epl/i1996-00180-y
[23] Fajfrovà, L., Gobron, T., Saada, E.: Invariant measures of Mass Migration Processes. Electron. J. Probab. 21(60), 1-52 (2016) · Zbl 1348.60137
[24] Fill, J.A., Machida, M.: Stochastic monotonicity and realizable monotonicity. Ann. Probab. 29(2), 938-978 (2001) · Zbl 1015.60010 · doi:10.1214/aop/1008956698
[25] Fristedt, B., Gray, L.: A Modern Approach to Probability Theory. Probability and its Applications. Birkhäuser, Boston (1997) · Zbl 0869.60001 · doi:10.1007/978-1-4899-2837-5
[26] Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18, 697-715 (1965) · Zbl 0141.28902 · doi:10.1002/cpa.3160180408
[27] Gobron, T., Saada, E.: Couplings, attractiveness and hydrodynamics for conservative particle systems. Ann. Inst. H. Poincaré Probab. Statist. 46(4), 1132-1177 (2010) · Zbl 1252.60093 · doi:10.1214/09-AIHP347
[28] Godlewski, E., Raviart, P.A.: Hyperbolic systems of conservation laws. Mathématiques & Applications, Ellipses (1991) · Zbl 0768.35059
[29] Guiol, H.: Some properties of \(k\)-step exclusion processes. J. Stat. Phys. 94(3-4), 495-511 (1999) · Zbl 0953.60091
[30] Harris, T.E.: Nearest neighbor Markov interaction processes on multidimensional lattices. Adv. Math. 9, 66-89 (1972) · Zbl 0267.60107 · doi:10.1016/0001-8708(72)90030-8
[31] Harris, T.E.: Additive set-valued Markov processes and graphical methods. Ann. Probab. 6, 355-378 (1978) · Zbl 0378.60106 · doi:10.1214/aop/1176995523
[32] Kamae, T., Krengel, U.: Stochastic partial ordering. Ann. Probab. 6(6), 1044-1049 (1978) · Zbl 0392.60012 · doi:10.1214/aop/1176995392
[33] Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 320. Springer-Verlag, Berlin, (1999) · Zbl 0927.60002
[34] Kružkov, S.N.: First order quasilinear equations with several independent variables. Math. URSS Sb. 10, 217-243 (1970) · Zbl 0215.16203 · doi:10.1070/SM1970v010n02ABEH002156
[35] Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. 4(3), 339-356 (1976) · Zbl 0339.60091 · doi:10.1214/aop/1176996084
[36] Liggett, T.M.: Interacting Particle Systems. Classics in Mathematics (Reprint of first edition). Springer-Verlag, New York (2005) · doi:10.1007/b138374
[37] Liggett, T.M., Spitzer, F.: Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. Verw. Gebiete 56(4), 443-468 (1981) · Zbl 0444.60096 · doi:10.1007/BF00531427
[38] Mountford, T.S., Ravishankar, K., Saada, E.: Macroscopic stability for nonfinite range kernels. Braz. J. Probab. Stat. 24(2), 337-360 (2010) · Zbl 1195.82058 · doi:10.1214/09-BJPS034
[39] Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on \(\mathbb{Z}^d\). Commun. Math. Phys. 140(3), 417-448 (1991) · Zbl 0738.60098
[40] Rezakhanlou, F.: Continuum limit for some growth models. II. Ann. Probab. 29(3), 1329-1372 (2001) · Zbl 1081.82016 · doi:10.1214/aop/1015345605
[41] Rost, H.: Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58(1), 41-53 (1981) · Zbl 0451.60097 · doi:10.1007/BF00536194
[42] Seppäläinen, T., Krug, J.: Hydrodynamics and Platoon formation for a totally asymmetric exclusion model with particlewise disorder. J. Stat. Phys. 95(3-4), 525-567 (1999) · Zbl 0964.82041
[43] Seppäläinen, T.: Existence of hydrodynamics for the totally asymmetric simple \(K\)-exclusion process. Ann. Probab. 27(1), 361-415 (1999) · Zbl 0947.60088
[44] Serre, D.: Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves. Cambridge University Press, Cambridge (1999). Translated from the 1996 French original by I. Sneddon
[45] Spohn, H.: Large Scale Dynamics of Interacting Particles. Theoretical and Mathematical Physics. Springer, Heidelberg (1991) · Zbl 0742.76002 · doi:10.1007/978-3-642-84371-6
[46] Strassen, V.: The existence of probability measures with given marginals. Ann. Math. Stat. 36(2), 423-439 (1965) · Zbl 0135.18701 · doi:10.1214/aoms/1177700153
[47] Swart, J.M.: A course in interacting particle systems. staff.utia.cas.cz/swart/lecture_notes/partic15_1.pdf
[48] Vol’pert, A.
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