Journal of Statistical Physics

, Volume 55, Issue 3–4, pp 787–855 | Cite as

An invariance principle for reversible Markov processes. Applications to random motions in random environments

  • A. De Masi
  • P. A. Ferrari
  • S. Goldstein
  • W. D. Wick
Articles

Abstract

We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.

Key words

Symmetric random environment random potential reversible Markov process central limit theorem invariance principle interacting Brownian particles 

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References

  1. 1.
    M. Aizenman, J. T. Chayes, L. Chayes, J. Frohlich, and L. Russo, On a sharp transition from area law to perimeter law in a system of random surfaces,Commun. Math. Phys. 92:19–69 (1983).Google Scholar
  2. 2.
    M. Aizenman, H. Kesten, and C. M. Newman, Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation, inPercolation Theory and Ergodic Theory of Infinite Particle Systems, H. Kesten, ed. (IMA Volumes in Math and its Applications, Vol. 8, 1978).Google Scholar
  3. 3.
    S. Alexander, J. Bernasconi, W. R. Schneider, and R. Orbach, Excitation dynamics in random one-dimensional system,Rev. Mod. Phys. 53:175–198 (1981).Google Scholar
  4. 4.
    V. V. Anshelevich and A. V. Vologodskii, Laplace operator and random walk on one-dimensional non-homogenous lattice,J. Stat. Phys. 25:419–430 (1981).Google Scholar
  5. 5.
    V. V. Anshelevich, K. M. Khanin, and J. Ya. Sinai, Symmetric random walks in random environments,Commun. Math. Phys. 85:449–470 (1982).Google Scholar
  6. 6.
    R. Arratia, The motion of a tagged particle in the simple exclusion system on ℤ,Ann. Prob. 11:362 (1983).Google Scholar
  7. 7.
    P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).Google Scholar
  8. 8.
    L. Breiman,Probability (Addison-Wesley, Reading, Massachusetts, 1968).Google Scholar
  9. 9.
    A. De Masi and P. A. Ferrari, Self diffusion in one dimensional lattice gas in the presence of an external field,J. Stat. Phys. 38:603–613 (1985).Google Scholar
  10. 10.
    A. De Masi, P. A. Ferrari, S. Goldstein, and D. W. Wick, Invariance principle for reversible Markov processes with application to diffusion in the percolation regime,Contemp. Math. 41:71–85 (1985).Google Scholar
  11. 11.
    P. Doyle and J. L. Snell, Random walk and electrical networks, Dartmouth College preprint (1982).Google Scholar
  12. 12.
    D. Dürr and S. Goldstein, Remarks on the central limit theorem for weakly dependent random variables, inStochastic Process—Mathematics and Physics (Proceedings, Bielefeld 1984; Lecture Notes in Mathematics 1158, Springer, 1985).Google Scholar
  13. 13.
    W. G. Faris,Self-Adjoint Operators (Lecture Notes in Mathematics 433, 1975).Google Scholar
  14. 14.
    P. A. Ferrari, S. Goldstein, and J. L. Lebowitz, Diffusion, mobility and the Einstein relation, inStatistical Physics and Dynamical Systems: Rigorous Results, J. Fritz, A. Jaffe, and D. Szasz, eds. (Birkhauser, 1985).Google Scholar
  15. 15.
    R. Figari, E. Orlandi, and G. Papanicolaou, Diffusive behavior of a random walk in a random medium, inProceedings Kyoto Conference (1982).Google Scholar
  16. 16.
    J. Fritz, Gradient dynamics of infinite point systems, preprint (1984).Google Scholar
  17. 17.
    K. Golden and G. Papanicolaou, Bounds for the effective parameters of heterogenous media by analytic continuation,Commun. Math. Phys. 90:473–491 (1983).Google Scholar
  18. 18.
    G. Grimmett and H. Kesten, First passage percolation, network flows and electrical resistances,Z. Wahrsch. Verw. Geb. 66:335–366 (1984).Google Scholar
  19. 19.
    M. Guo, Limit theorems for interacting particle systems, Ph.D. Thesis, New York University (1984).Google Scholar
  20. 20.
    T. E. Harris, Diffusion with “collision” between particles,J. Appl. Prob. 2:323–338 (1965).Google Scholar
  21. 21.
    I. S. Heiland, On weak convergence to Brownian motion,Z. Wahrsch. Verw. Geb. 52:251–265 (1980).Google Scholar
  22. 22.
    I. S. Helland, Central limit theorems for martingales with discrete or continuous time,Scand. J. Stat. 1982:979–994 (1982).Google Scholar
  23. 23.
    K. Kawazu and H. Kesten, On birth and death processes in symmetric random environments,J. Stat. Phys. 37:561 (1984).Google Scholar
  24. 24.
    H. Kesten,Percolation Theory for Mathematicians (Birkhauser, 1982).Google Scholar
  25. 25.
    H. Kesten, M. V. Kozlov, and F. Spitzer, A limit law for random walk in a random environment,Compositio Mathematica 30(2):145–168 (1975).Google Scholar
  26. 26.
    C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functional of reversible Markov processes and applications to simple exclusion,Commun. Math. Phys. 104:1–19 (1986).Google Scholar
  27. 27.
    C. Kipnis, J. L. Lebowitz, E. Presutti, and H. Spohn, Self-diffusion for particles with stochastic collisions in one dimension,J. Stat. Phys. 30:107–121 (1983).Google Scholar
  28. 28.
    W. Kohler and G. Papanicolaou, Bounds for the effective conductivity of random media, inLecture Notes in Physics, Vol. 154 (1982), pp. 111–130.Google Scholar
  29. 29.
    R. Kunnemann, The diffusion limit for reversible jump processes in ℤdwith ergodic random bond conductivities,Commun. Math. Phys. 90:27–68 (1983).Google Scholar
  30. 30.
    R. Lang,Z. Wahrsch. Verw. Geb. 38:55 (1977).Google Scholar
  31. 31.
    R. Lang, Stochastic models of many-particle systems and their time evolution, Habilitationsschrift, Universität Heidelberg (1982).Google Scholar
  32. 32.
    J. L. Lebowitz and H. Spohn, Microscopic basis for Fick's law for self-diffusion,J. Stat. Phys. 28:539–555 (1982).Google Scholar
  33. 33.
    T. Liggett,Interacting Particle Systems (Springer-Verlag, 1984).Google Scholar
  34. 34.
    Gy. Lippner,Coll. Math. Soc. Janos Bolyai 24:277–290 (North-Holland, 1981).Google Scholar
  35. 35.
    R. B. Pandey, D. Stauffer, A. Margolina, and J. G. Zabolitzky, Diffusion on random systems above, below and at their percolation threshold in two and three dimensions,J. Stat. Phys. 34:427 (1984).Google Scholar
  36. 36.
    G. Papanicolaou and S. R. S. Varadhan, Diffusion with random coefficients, inStatistics and Probability: Essays in Honor of C. R. Rao, G. Kallianpur, P. R. Krishaniah, and J. K. Ghosh, eds. (North-Holland, 1982), pp. 547–552.Google Scholar
  37. 37.
    G. Papanicolaou, Diffusion and random walks in random media, inMathematics and Physics of Disordered Media, B. D. Auges and B. Nihara, eds. (Springer Lecture Notes in Mathematics No. 1035, 1983), p. 391.Google Scholar
  38. 38.
    G. Papanicolaou, Macroscopic properties of composities, bubbly fluids, suspensions and related problems, inLes méthodes de l'homogénisation théorie et applications en physique (CEA-EDF-INRIA École d'été d'analyse numérique, 1985), pp. 229–317.Google Scholar
  39. 39.
    M. Reed and B. Simon,Methods of Mathematical Physics II (Academic Press, New York, 1975).Google Scholar
  40. 40.
    M. Rosenblatt,Markov Processes, Structure and Asymptotic Behavior (Springer-Verlag, Berlin, 1970).Google Scholar
  41. 41.
    D. Ruelle, Superstable interactions in classical statistical mechanics,Commun. Math. Phys. 18:127 (1970).Google Scholar
  42. 42.
    H. Rost, inLecture Notes in Control and Information Sciences, Vol. 25 (1980), pp. 297–302.Google Scholar
  43. 43.
    T. Shiga,Z. Wahrsch. Verw. Geb. 47:299 (1979).Google Scholar
  44. 44.
    F. Solomon, Random walks in a random environment,Ann. Prob. 3(1):1–31 (1975).Google Scholar
  45. 45.
    H. Spohn, Equilibrium fluctuations for interacting Brownian particles,Commun. Math. Phys. 103:1–33 (1986).Google Scholar
  46. 46.
    D. W. Stroock and S. R. S. Varadhan,Multidimensional Diffusion Processes (Springer-Verlag, Berlin, 1979).Google Scholar
  47. 47.
    N. Ikeda and S. Watanabe,Stochastic Differential Equations and Diffusion Process (North-Holland, New York, 1981).Google Scholar
  48. 48.
    D. Dürr, S. Goldstein, and J. L. Lebowitz, Asymptotics of particle trajectories in infinite one-dimensional systems with collisions,Commun. Pure Appl. Math. 38:573–597 (1985).Google Scholar

Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • A. De Masi
    • 1
  • P. A. Ferrari
    • 2
  • S. Goldstein
    • 3
  • W. D. Wick
    • 4
  1. 1.Dipartmento di Matematica Pura e ApplicataUniversità dell'AquilaL'AquilaItaly
  2. 2.Instituto de Matemàtica e EstatisticaUniversidade de São PauloSão PauloBrazil
  3. 3.Department of MathematicsRutgers UniversityNew Brunswick
  4. 4.Department of MathematicsUniversity of ColoradoBoulder

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