Étant donné un polynôme Φ de la forme Φ(α) = α + ∑2≤j≤m aj αk=1j respectant Φ'(1) > 0, nous démontrons que l’évolution, sur une échelle diffusive, de la densité empirique des processus d’exclusion sur , dont les conductances sont données par une classe spéciale de fonctions W, est décrite par l'unique solution faible de l'équation aux dérivées partielles parabolique : ∂tρ=∑d ∂xk ∂Wk Φ(ρ). Nous dérivons également certaines propriétés de l'opérateur ∑k=1d∂xk ∂Wk.
Fix a polynomial Φ of the form Φ(α) = α + ∑2≤j≤m aj αk=1j with Φ'(1) > 0. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes on , with conductances given by special class of functions W, is described by the unique weak solution of the non-linear parabolic partial differential equation ∂tρ = ∑d ∂xk ∂Wk Φ(ρ). We also derive some properties of the operator ∑k=1d ∂xk ∂Wk.
Mots clés : exclusion processes, random conductances, hydrodynamic limit
@article{AIHPB_2012__48_1_188_0,
author = {Valentim, F\'abio J\'ulio},
title = {Hydrodynamic limit of a $d$-dimensional exclusion process with conductances},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
pages = {188--211},
publisher = {Gauthier-Villars},
volume = {48},
number = {1},
year = {2012},
doi = {10.1214/10-AIHP397},
zbl = {1254.60093},
language = {en},
url = {http://www.numdam.org/articles/10.1214/10-AIHP397/}
}
TY - JOUR AU - Valentim, Fábio Júlio TI - Hydrodynamic limit of a $d$-dimensional exclusion process with conductances JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2012 SP - 188 EP - 211 VL - 48 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/10-AIHP397/ DO - 10.1214/10-AIHP397 LA - en ID - AIHPB_2012__48_1_188_0 ER -
%0 Journal Article %A Valentim, Fábio Júlio %T Hydrodynamic limit of a $d$-dimensional exclusion process with conductances %J Annales de l'I.H.P. Probabilités et statistiques %D 2012 %P 188-211 %V 48 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/10-AIHP397/ %R 10.1214/10-AIHP397 %G en %F AIHPB_2012__48_1_188_0
Valentim, Fábio Júlio. Hydrodynamic limit of a $d$-dimensional exclusion process with conductances. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 1, pp. 188-211. doi : 10.1214/10-AIHP397. http://www.numdam.org/articles/10.1214/10-AIHP397/
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