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.
@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},
volume = {48},
year = {2012},
pages = {188-211},
doi = {10.1214/10-AIHP397},
zbl = {1254.60093},
language = {en},
url = {http://dml.mathdoc.fr/item/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) pp. 188-211. doi : 10.1214/10-AIHP397. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_1_188_0/
[1] . Markov Processes, Vol. II. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 122. Springer, Berlin, 1965. | Zbl 0132.37901
[2] . Random walks and exclusion processs among random conductances on random infinite clusters: Homogenization and hydrodynamic limit. Electron. J. Probab. 13 (2008) 2217-2247. Available at arXiv:0704.3020v3. | MR 2469609 | Zbl 1189.60172
[3] , and . Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances. Probab. Theory Related Fields 144 (2009) 633-667. Available at arXiv:0709.0306. | MR 2496445 | Zbl 1169.60326
[4] . On second order differential operators. Ann. Math. (2) 61 (1955) 90-105. | MR 68082 | Zbl 0064.11301
[5] . Generalized second order differential operators and their lateral conditions. Illinois J. Math. 1 (1957) 459-504. | MR 92046 | Zbl 0077.29102
[6] and . Hydrodynamic limit of gradient exclusion processes with conductances. Arch. Ration. Mech. Anal. 195 (2010) 409-439. | MR 2592282 | Zbl 1192.82062
[7] and . Quenched nonequilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 341-361. Available at arXiv:math/0603653. | Numdam | MR 2446327 | Zbl 1195.60124
[8] and . Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin, 1999. | MR 1707314 | Zbl 0927.60002
[9] . Analytical Treatment of One-Dimensional Markov Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 151. Springer, Berlin, 1968. | MR 247667 | Zbl 0179.47802
[10] and . W-Sobolev spaces: Theory, homogenization and applications. Preprint, 2009. Available at arXiv:0911.4177. | MR 2805508 | Zbl 1221.35017