Nous considérons le processus d’exclusion dans le tore discret uni-dimensionnel avec points, où tous les liens ont conductance un, sauf pour un nombre fini de liens lents qui ont conductance , avec . Nous prouvons que l’évolution en temps de la densité empirique de particules, après un changement d’échelle diffusif, a un comportement différent selon la valeur du paramètre . Si , la limite hydrodynamique est donnée par l’équation de la chaleur usuelle. Si , la limite est donnée par une équation parabolique avec un opérateur , où est la mesure de Lebesgue sur le tore plus la somme des masses de Dirac en chaque point macroscopique relatif à un lien lent. Si , la limite est donnée par l’équation de la chaleur avec conditions au bord de Neumann, et ceci traduit l’absence de passage par les liens lents dans le continu.
We consider the exclusion process in the one-dimensional discrete torus with points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance , with . We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter . If , the hydrodynamic limit is given by the usual heat equation. If , it is given by a parabolic equation involving an operator , where is the Lebesgue measure on the torus plus the sum of the Dirac measure supported on each macroscopic point related to the slow bond. If , it is given by the heat equation with Neumann’s boundary conditions, meaning no passage through the slow bonds in the continuum.
Mots clés : hydrodynamic limit, exclusion process, slow bonds
@article{AIHPB_2013__49_2_402_0, author = {Franco, Tertuliano and Gon\c{c}alves, Patr{\'\i}cia and Neumann, Adriana}, title = {Hydrodynamical behavior of symmetric exclusion with slow bonds}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {402--427}, publisher = {Gauthier-Villars}, volume = {49}, number = {2}, year = {2013}, doi = {10.1214/11-AIHP445}, mrnumber = {3088375}, zbl = {1282.60095}, language = {en}, url = {http://www.numdam.org/articles/10.1214/11-AIHP445/} }
TY - JOUR AU - Franco, Tertuliano AU - Gonçalves, Patrícia AU - Neumann, Adriana TI - Hydrodynamical behavior of symmetric exclusion with slow bonds JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2013 SP - 402 EP - 427 VL - 49 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/11-AIHP445/ DO - 10.1214/11-AIHP445 LA - en ID - AIHPB_2013__49_2_402_0 ER -
%0 Journal Article %A Franco, Tertuliano %A Gonçalves, Patrícia %A Neumann, Adriana %T Hydrodynamical behavior of symmetric exclusion with slow bonds %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 402-427 %V 49 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/11-AIHP445/ %R 10.1214/11-AIHP445 %G en %F AIHPB_2013__49_2_402_0
Franco, Tertuliano; Gonçalves, Patrícia; Neumann, Adriana. Hydrodynamical behavior of symmetric exclusion with slow bonds. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 402-427. doi : 10.1214/11-AIHP445. http://www.numdam.org/articles/10.1214/11-AIHP445/
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