Strong hydrodynamic limit for attractive particle systems on ℤ

Electronic Journal of Probability

Volumn 15, Issue , 2010, Pages 1-43

  Bahadoran, C ^a   Guiol, H ^b   Ravishankar, K ^c   Saada, E ^d  

^a UNIVERSITÉ CLERMONT AUVERGNE   (France)

^b GRENOBLE INSTITUTE OF TECHNOLOGY   (France)

^c STATE UNIVERSITY OF NEW YORK   (United States)

^d UNIVERSITÉ DE ROUEN   (France)

Author keywords

Attractive particle system; Entropy solution; Glimm scheme; Graphical construction; Non convex or non concave flux; Non explicit invariant measures; Strong (a.s.) hydrodynamics

Indexed keywords

EID: 77952825933     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v15-728     Document Type: Article

References (45)

1. Andjel, E., Ferrari, PA., Siqueira, A. Law of large numbers for the asymmetric exclusion process. Stoch. Process. Appl. 132 (2004) no. 2, 217-233.
2. Andjel, E.D., Kipnis, C. Derivation of the hydrodynamical equation for the zero range interaction process. Ann. Probab. 12 (1984), 325-334.
3. Andjel, E.D., Vares, M.E. Hydrodynamic equations for attractive particle systems on Z. J. Stat. Phys. 47 (1987) no. 1/2, 265-288.
4. Andjel, E.D., Vares, M.E. Correction to: “Hydrodynamic equations for attractive particle systems on Z”. J. Stat. Phys. 113 (2003) 379-380.
5. Belletini, G., Bertini, L., Mariani, M., Novaga, M. T-entropy cost for scalar conservation laws. Archive Rational Mechanics Anal. 195 (2010), 261-309.
6. Benassi, A., Fouque, J.P Hydrodynamical limit for the asymmetric exclusion process. Ann. Probab. 15 (1987), 546-560.
7. Bahadoran, C., Guiol, H., Ravishankhar, K., Saada, E. A constructive approach to Euler hydrodynamics for attractive particle systems. Application to k-step exclusion. Stoch. Process. Appl. 99 (2002) no. 1, 1-30.
8. Bahadoran, C., Guiol, H., Ravishankhar, K., Saada, E. Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab. 34 (2006), 1339-1369.
9. Biane, P, Durrett, R. Lectures on Probability Theory, Ecole d'Eté de Probabilités de Saint-Flour XXUI-1993. LN in Math. 1608, Springer, 1995.
10. Bramson, M., Mountford, T. Existence of blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30 (2002), 1082-1130.
11. Breiman, L. Probability. Addison-Wesley, 1968.
12. Cocozza-Thivent, C. Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70 (1985) no. 4, 509-523.
13. De Lellis, C., Otto, F., Westdickenberg, M. Minimal entropy conditions for Burgers Equa-tion.Quarterly of Applied Math. 62 (2004), 687-700.
14. De Masi A., Presutti E. Mathematical Methods for Hydrodynamic Limits. LN in Math. 1501, Springer, 1991.
15. Donsker, M.D., Varadhan, S.R.S. Asymptotic evaluation of certain Markov process expectations for large time I. Commun. Pure Appl. Math. 28 (1975), 1-47.
16. Ferrari, PA. Shock fluctuations in asymmetric simple exclusion. Probab. Theory Rel. Fields 91 (1992), 81-101.
17. Ferrari, PA. Shocks in the Burgers equation and the asymmetric simple exclusion process; in Automata Networks, Dynamical Systems and Statistical Physics, edited by E. Goles, S. Martinez, Kluwer Math. and its Appl. (1992), 25-64.
18. Guiol, H. Some properties of k-step exclusion processes. J. Stat. Phys. 94 (1999) no. (3/4), 495-511.
19. Harris, T.E. Nearest neighbor Markov interaction processes on multidimensional lattices. Adv. in Math. 9 (1972), 66-89.
20. Harris, T.E. Additive set-valued Markov processes and graphical methods. Ann. Probab. 6 (1978), 355-378.
21. Hayes, B.T., le Floch, EG. Non-Classical Shocks and Kinetic Relations: Scalar Conservation Laws. Archive for Rational Mech. Anal. 139 (1997), 1-56.
22. Jensen, L. Large deviations of the asymmetric simple exclusion process in one dimension. PhD dissertation, New York University (2000).
23. Kamae, T., Krengel, U. Stochastic partial ordering. Ann. Probab. 6 (1978), 1044-1049.
24. Kamae, T., Krengel, U., O'Brien, L.G. Stochastic inequalities on partially ordered spaces. Ann. Probab. 5 (1977), 899-912.
25. Kipnis, C., Landim, C. Scaling limits of interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 320. SpringerVerlag, Berlin, 1999.
26. Kipnis, C., Olla, S., Varadhan, S.R.S. Hydrodynamics and large deviations for simple exclusion process. Comm. Pure. Appl. Math. Vol XLII (1989), 115-137.
27. Krengel, U. Ergodic theory. De Gruyter Studies in Mathematics 6, 1985.
28. Kruzkov, N. First order quasilinear equations in several independent variables. Math. USSR Sb. 10 (1970), 217-243.
29. Lax, PD. Hyperbolic systems of conservation laws II. Comm. Pure. Appl. Math. 10 (1957), 537-566.
30. Liggett, T.M. Interacting particle systems. Classics in Mathematics (Reprint of first edition), Springer-Verlag, New York, 2005.
31. Liggett, T.M. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 324, Springer-Verlag, New York, 1999.
32. Mariani, M. Large deviations principle for stochastic conservation laws. M. Large deviations principle for stochastic scalar conservation laws. Probab. Theory Related Fields (2009) (Published online)
33. Mountford, T.S., Ravishankar, K., Saada, E. Macroscopic stability for nonfinite range kernels. http://fr.arxiv.org/abs/0903.0498, Braz. J. Probab. Stat. (2009).
34. Rezakhanlou, F Hydrodynamic limit for attractive particle systems on Zd. Comm. Math. Phys. 140 (1991), no. 3, 417-448.
35. Rezakhanlou, F Continuum limit for some growth models. II. Ann. Probab. 29 (2001) no. 3, 1329-1372.
36. Rost, H. Nonequilibrium behaviour of a many particle process: density profile and local equilibria, Z. Wahrsch. Verw. Gebiete 58 (1981) no. 1, 41-43.
37. Seppäläinen, T. Existence of hydrodynamics for the totally asymmetric simple K-exclusion process. Ann. Probab. 27 (1999) no. 1, 361-415.
38. Seppäläinen, T. Translation Invariant Exclusion Processes. In preparation. http://www.math. wise.edu/∼seppalai/exel-book/etusivu.html.
39. Serre, D. Systèmes de lois de conservation. Diderot Ed., 1996.
40. Spohn, H. Large Scale Dynamics of Interacting Particles. Springer, 1991.
41. Strassen, V The existence of probability measures with given marginals. Ann. Math. Statist. 36 (1965), 423-439.
42. Varadhan, S.R.S. Lectures on Hydrodynamic Scaling; in Hydrodynamic Limits and Related Topics; edited by S. Feng, A. Lawniczak, S.R.S. Varadhan, Fields Institute Communications, AMS, 2000.
43. Varadhan, S.R.S. Large deviations for the asymmetric simple exclusion process. Stochastic analysis on large scale interacting systems, 1-27, Adv. Stud. Pure Math. 39, Math. Soc. Japan, Tokyo, 2004.
44. Vol'pert, A.I. The spaces BV and quasilinear equations. Math. USSR Sbornik 2 (1967) no. 2, 225-267. Transl. from (Russian) Mat. Sb. (N.S.) 73 (1967) no. 115, 255-302.
45. Wiener, N. The ergodic theorem. Duke Math. 15 (1939) no. 1, 1-18.