Homogenization results for a linear dynamics in random Glauber type environment
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 3, pp. 792-818.

On considère un système d'équations differentielles linéaires couplées conservant une certaine énergie et l'on perturbe ce système par une dynamique de type Glauber dont l'intensité varie aléatoirement site par site. Nous prouvons les limites hydrodyanmiques pour ce système non réversible en milieu aléatoire. Le coefficient de diffusion dépend de l'aléa uniquement par sa loi. Nous étudions aussi le coefficient de diffusion défini par la formule de Green-Kubo et montrons la convergence de celle-ci vers un coefficient de diffusion homogénéisé.

We consider an energy conserving linear dynamics that we perturb by a Glauber dynamics with random site dependent intensity. We prove hydrodynamic limits for this non-reversible system in random media. The diffusion coefficient turns out to depend on the random field only by its statistics. The diffusion coefficient defined through the Green-Kubo formula is also studied and its convergence to some homogenized diffusion coefficient is proved.

DOI : 10.1214/11-AIHP424
Classification : 60K35, 82C22, 82C44
Mots-clés : hydrodynamic limits, random media, Green-Kubo formula, homogenization 
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     author = {Bernardin, C\'edric},
     title = {Homogenization results for a linear dynamics in random {Glauber} type environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {792--818},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {3},
     year = {2012},
     doi = {10.1214/11-AIHP424},
     mrnumber = {2976564},
     zbl = {1279.60123},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/11-AIHP424/}
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Bernardin, Cédric. Homogenization results for a linear dynamics in random Glauber type environment. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 3, pp. 792-818. doi : 10.1214/11-AIHP424. http://www.numdam.org/articles/10.1214/11-AIHP424/

[1] C. Bernardin. Hydrodynamics for a system of harmonic oscillators perturbed by a conservative noise. Stochastic Process. Appl. 117 (2007) 487-513. | MR | Zbl

[2] C. Bernardin. Thermal conductivity for a noisy disordered harmonic chain. J. Stat. Phys. 133 (2008) 417-433. | MR | Zbl

[3] C. Bernardin and S. Olla. Non-equilibrium macroscopic dynamics of chains of anharmonic oscillators

[4] F. Bonetto, J. L. Lebowitz, J. Lukkarinen and S. Olla. Heat conduction and entropy production in anharmonic crystals with self-consistent stochastic reservoirs. J. Stat. Phys. 134 (2009) 1097-1119. | MR | Zbl

[5] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge, 1992. | MR | Zbl

[6] A. Faggionato. Bulk diffusion of 1D exclusion process with bond disorder. Markov Process. Related Fields 13 (2007) 519-542. | MR | Zbl

[7] A. Faggionato. Random walks and exclusion processes among random conductances on random infinite clusters: Homogenization and hydrodynamic limit. Electron. J. Probab. 13 (2008) 2217-2247. | MR | Zbl

[8] A. Faggionato and F. Martinelli. Hydrodynamic limit of a disordered lattice gas. Probab. Theory Related Fields 127 (2003) 535-608. | MR | Zbl

[9] A. Faggionato, M. Jara and C. Landim. Hydrodynamic behavior of 1D subdiffusive exclusion processes with random conductances. Probab. Theory Related Fields 144 (2009) 633-667. | MR | Zbl

[10] J. Fritz. Hydrodynamics in a symmetric random medium. Comm. Math. Phys. 125 (1989) 13-25. | MR | Zbl

[11] J. Fritz, T. Funaki and J. L. Lebowitz. Stationary states of random Hamiltonian systems. Probab. Theory Related Fields 99 (1994) 211-236. | MR | Zbl

[12] P. Gonçalves and M. Jara. Scaling limits for gradient systems in random environment. J. Stat. Phys. 131 (2008) 691-716. | MR | Zbl

[13] E. Hsu. Characterization of Brownian motion on manifolds through integration by parts. In Stein's Method and Applications 195-208. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5. Singapore Univ. Press, Singapore, 2005. | MR

[14] 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. | Numdam | MR | Zbl

[15] C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems. Springer, Berlin, 1999. | MR | Zbl

[16] K. Nagy. Symmetric random walk in random environment in one dimension. Period. Math. Hungar. 45 (2002) 101-120. | MR | Zbl

[17] S. Olla. Central limit theorems for tagged particles and for diffusions in random environment. In Milieux Aléatoires 75-100. Panor. Synthèses 12. Soc. Math. France, Paris, 2001. | MR | Zbl

[18] S. Olla, S. R. S. Varadhan and H. T. Yau. Hydrodynamic limit for a Hamiltonian system with weak noise. Comm. Math. Phys. 155 (1993) 523-560. | MR | Zbl

[19] J. Quastel. Diffusion in disordered media. In Nonlinear Stochastic PDEs (Minneapolis, MN, 1994) 65-79. IMA Vol. Math. Appl. 77. Springer, New York, 1996. | MR | Zbl

[20] J. Quastel. Bulk diffusion in a system with site disorder. Ann. Probab. 34 (2006) 1990-2036. | MR | Zbl

[21] M. Reed and B. Simon. Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd edition. Academic Press, New York, 1980. | MR | Zbl

[22] S. R. S. Varadhan. Nonlinear diffusion limit for a system with nearest neighbor interactions. II. In Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals (Sanda/Kyoto, 1990) 75-128. Pitman Res. Notes Math. Ser. 283. Longman Sci. Tech., Harlow, 1993. | MR | Zbl

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