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.

Étant donné un polynôme Φ de la forme Φ(α) = α + ∑2≤jmajαk=1j respectant Φ'(1) > 0, nous démontrons que l’évolution, sur une échelle diffusive, de la densité empirique des processus d’exclusion sur 𝕋 d , 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=∑dxkWkΦ(ρ). Nous dérivons également certaines propriétés de l'opérateur ∑k=1dxkWk.

Fix a polynomial Φ of the form Φ(α) = α + ∑2≤jmajαk=1j with Φ'(1) > 0. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes on 𝕋 d , 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 = ∑dxkWkΦ(ρ). We also derive some properties of the operator ∑k=1dxkWk.

DOI : 10.1214/10-AIHP397
Classification : 60K35, 26A24, 35K55
Mots clés : exclusion processes, random conductances, hydrodynamic limit
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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/

[1] E. B. Dynkin. Markov Processes, Vol. II. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 122. Springer, Berlin, 1965. | Zbl

[2] A. Faggionato. 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 | Zbl

[3] A. Faggionato, M. Jara and C. Landim. 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 | Zbl

[4] W. Feller. On second order differential operators. Ann. Math. (2) 61 (1955) 90-105. | MR | Zbl

[5] W. Feller. Generalized second order differential operators and their lateral conditions. Illinois J. Math. 1 (1957) 459-504. | MR | Zbl

[6] T. Franco and C. Landim. Hydrodynamic limit of gradient exclusion processes with conductances. Arch. Ration. Mech. Anal. 195 (2010) 409-439. | MR | Zbl

[7] M. Jara and C. Landim. 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 | Zbl

[8] C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 320. Springer, Berlin, 1999. | MR | Zbl

[9] P. Mandl. Analytical Treatment of One-Dimensional Markov Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 151. Springer, Berlin, 1968. | MR | Zbl

[10] A. B. Simas and F. J. Valentim. W-Sobolev spaces: Theory, homogenization and applications. Preprint, 2009. Available at arXiv:0911.4177. | MR | Zbl

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