Nous démontrons les fluctuations hors d'équilibre d'une particule marquée pour une classe de systèmes de particules à portée nulle uni-dimensionels de moyenne nulle dont le taux de sauts croit de manière sous-linéaire. Dans Jara-Landim-Sethuraman (Probab. Theory Related Fields 145 (2009) 565-590), ce résutat a été démontré pour des processus dont le taux croit au moins linéairement. La démonstration du lemme de remplacement dans le cas sous-linéaire exige une nouvelle approche en conséquence des différences entre les propriétés de mélanges des deux processus. La méthode présentée permet également de démontrer les fluctuations d’une particule de deuxième classe dans le modèle à portée nulle symmétrique dont le taux de sauts est égal à .
Nonequilibrium fluctuations of a tagged, or distinguished particle in a class of one dimensional mean-zero zero-range systems with sublinear, increasing rates are derived. In Jara-Landim-Sethuraman (Probab. Theory Related Fields 145 (2009) 565-590), processes with at least linear rates are considered. A different approach to establish a main “local replacement” limit is required for sublinear rate systems, given that their mixing properties are much different. The method discussed also allows to capture the fluctuations of a “second-class” particle in unit rate, symmetric zero-range models.
Mots clés : interacting, particle system, zero-range, tagged, nonequilibrium, diffusion
@article{AIHPB_2013__49_3_611_0, author = {Jara, Milton and Landim, Claudio and Sethuraman, Sunder}, title = {Nonequilibrium fluctuations for a tagged particle in one-dimensional sublinear zero-range processes}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {611--637}, publisher = {Gauthier-Villars}, volume = {49}, number = {3}, year = {2013}, doi = {10.1214/12-AIHP478}, mrnumber = {3112428}, language = {en}, url = {http://www.numdam.org/articles/10.1214/12-AIHP478/} }
TY - JOUR AU - Jara, Milton AU - Landim, Claudio AU - Sethuraman, Sunder TI - Nonequilibrium fluctuations for a tagged particle in one-dimensional sublinear zero-range processes JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2013 SP - 611 EP - 637 VL - 49 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/12-AIHP478/ DO - 10.1214/12-AIHP478 LA - en ID - AIHPB_2013__49_3_611_0 ER -
%0 Journal Article %A Jara, Milton %A Landim, Claudio %A Sethuraman, Sunder %T Nonequilibrium fluctuations for a tagged particle in one-dimensional sublinear zero-range processes %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 611-637 %V 49 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/12-AIHP478/ %R 10.1214/12-AIHP478 %G en %F AIHPB_2013__49_3_611_0
Jara, Milton; Landim, Claudio; Sethuraman, Sunder. Nonequilibrium fluctuations for a tagged particle in one-dimensional sublinear zero-range processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 611-637. doi : 10.1214/12-AIHP478. http://www.numdam.org/articles/10.1214/12-AIHP478/
[1] Mathematical Methods for Hydrodynamic Limits. Lecture Notes in Mathematics 1501. Springer, Berlin, 1991. | MR | Zbl
and .[2] Nonequilibrium statistical mechanics of the zero-range process and related models. J. Phys. A: Math. Gen. 38 (2005) 195-240. | MR | Zbl
and .[3] Self-diffusion for Brownian motions with local interaction. Ann. Probab. 27 (1999) 1208-1267. | MR | Zbl
.[4] Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps. Commun. Pure Appl. Math. 62 (2009) 198-214. | MR | Zbl
.[5] Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion. Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 567-577. | Numdam | MR | Zbl
and .[6] Nonequilibrium fluctuations for a tagged particle in mean-zero one dimensional zero-range processes. Probab. Theory Related Fields 145 (2009) 565-590. | MR | Zbl
, and .[7] Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 320. Springer, Berlin, 1999. | MR | Zbl
and .[8] Central limit theorem for additive functionals of reversible markov processes. Comm. Math. Phys. 104 (1986) 1-19. | MR | Zbl
and .[9] Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation. Grundlehren der Mathematischen Wissenschaften 345. Springer, Berlin, 2012. | MR
, and .[10] Spectral gap for zero range dynamics. Ann. Probab. 24 (1996) 1871-1902. | MR | Zbl
, and .[11] Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 276. Springer, New York, 1985. | MR | Zbl
.[12] Spectral gap for the zero range process with constant rate. Ann. Probab. 34 (2006) 1645-1664. | MR | Zbl
.[13] Spectral gap for zero-range processes with jump rate . Stochastic Process Appl. 120 (2010) 949-958. | MR | Zbl
.[14] Infinite particle systems. Trans. Amer. Math. Soc. 178 (1973) 307-340. | MR | Zbl
and .[15] Propagation of chaos for symmetric simple exclusions. Commun. Pure Appl. Math. 47 (1994) 943-957. | MR | Zbl
.[16] Processus de zero-range avec particule marquée. Ann. Inst. Henri Poincaré Probab. Stat. 26 (1990) 5-17. | Numdam | MR | Zbl
.[17] On diffusivity of a tagged particle in asymmetric zero-range dynamics. Ann. Inst. Henri Poincaré Probab. Stat. 43 (2007) 215-232. | Numdam | MR | Zbl
.[18] Large Scale Dynamics of Interacting Particles. Springer, Berlin, 1991. | Zbl
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