Abstract
It is well known that the hydrodynamic limit of an interacting particle system satisfying a gradient condition (such as the zero-range process or the symmetric simple exclusion process) is given by a possibly non-linear parabolic equation and the equilibrium fluctuations from this limit are given by a generalized Ornstein-Uhlenbeck process.
We prove that in the presence of a symmetric random environment, these scaling limits also hold for almost every choice of the random environment, with an homogenized diffusion coefficient that does not depend on the realization of the random environment.
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References
Braess, D.: Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics, 2nd edn. Cambridge University Press, Cambridge (2001)
Chang, C.C.: Equilibrium fluctuations of gradient reversible particle systems. Probab. Theory Relat. Fields 100(3), 269–283 (1994)
Faggionato, A.: Bulk diffusion of 1D exclusion process with bond disorder. Markov Process. Relat. Fields 13(3), 519–542 (2007)
Faggionato, A., Martinelli, F.: Hydrodynamic limit of a disordered lattice gas. Probab. Theory Relat. Fields 127(4), 535–608 (2003)
Faggionato, A., Jara, M., Landim, C.: Hydrodynamic limit of a one-dimensional subdiffusive exclusion process with random conductances. Probab. Theory Relat. Fields (2008, in press)
Ferrari, P.A., Presutti, E., Vares, M.E.: Nonequilibrium fluctuations for a zero range process. Ann. Inst. H. Poincaré Probab. Stat. 24(2), 237–268 (1988)
Fritz, J.: Hydrodynamics in a symmetric random medium. Commun. Math. Phys. 125(1), 13–25 (1989)
Gonçalves, P., Landim, C., Toninelli, C.: Hydrodynamic limit for a particle system with degenerate rates. Preprint. Available online at http://arxiv.org/abs/0704.2242
Guo, M., Papanicolaou, G., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys. 118(1), 31–59 (1988)
Jara, M., Landim, C.: Quenched nonequilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder. Ann. Inst. H. Poincaré Probab. Stat. (2008, in press)
Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer, New York (1999)
Mitoma, I.: Tightness of probabilities on \(C([0,1];{\mathcal{S}}')\) and \(D([0,1];{\mathcal{S}}')\) . Ann. Probab. 11(4), 989–999 (1983)
Olla, S., Siri, P.: Homogenization of a bond diffusion in a locally ergodic random environment. Stoch. Process. Appl. 109(2), 317–326 (2004)
Piatnitski, A., Remy, E.: Homogenization of elliptic difference operators. SIAM J. Math. Anal. 33(1), 53–83 (2001)
Quastel, J.: Diffusion of color in the simple exclusion process. Commun. Pure Appl. Math. 45(6), 623–679 (1992)
Quastel, J.: Bulk diffusion in a system with site disorder. Ann. Probab. 34(5), 1990–2036 (2006)
Stroock, D., Zheng, W.: Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Stat. 33(5), 619–649 (1997)
Varadhan, S.R.S.: 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). Pitman Res. Notes Math. Ser., vol. 283, pp. 75–128. Longman Sci. Tech., Harlow (1993)
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P. Gonçalves wants to thank F.C.T. (Portugal) for supporting her Phd with the grant/SFRH/BD/11406/2002.
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Gonçalves, P., Jara, M. Scaling Limits for Gradient Systems in Random Environment. J Stat Phys 131, 691–716 (2008). https://doi.org/10.1007/s10955-008-9509-z
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DOI: https://doi.org/10.1007/s10955-008-9509-z