An invariance principle for reversible Markov processes. Applications to random motions in random environments. (English) Zbl 0713.60041

Summary: 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 a d-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in a d-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in a d- dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.


60F17 Functional limit theorems; invariance principles
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
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[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). · Zbl 0529.60099 · doi:10.1007/BF01206313
[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). · Zbl 0642.60102
[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). · Zbl 0465.76083 · doi:10.1103/RevModPhys.53.175
[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). · Zbl 0512.60059 · doi:10.1007/BF01010797
[5] V. V. Anshelevich, K. M. Khanin, and J. Ya. Sinai, Symmetric random walks in random environments,Commun. Math. Phys. 85:449-470 (1982). · Zbl 0512.60058 · doi:10.1007/BF01208724
[6] R. Arratia, The motion of a tagged particle in the simple exclusion system on ?,Ann. Prob. 11:362 (1983). · Zbl 0515.60097 · doi:10.1214/aop/1176993602
[7] P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968). · Zbl 0172.21201
[8] L. Breiman,Probability (Addison-Wesley, Reading, Massachusetts, 1968).
[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). · Zbl 0624.60117 · doi:10.1007/BF01010480
[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). · Zbl 0571.60044
[11] P. Doyle and J. L. Snell, Random walk and electrical networks, Dartmouth College preprint (1982).
[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).
[13] W. G. Faris,Self-Adjoint Operators (Lecture Notes in Mathematics 433, 1975).
[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).
[15] R. Figari, E. Orlandi, and G. Papanicolaou, Diffusive behavior of a random walk in a random medium, inProceedings Kyoto Conference (1982). · Zbl 0549.60075
[16] J. Fritz, Gradient dynamics of infinite point systems, preprint (1984). · Zbl 0591.60098
[17] K. Golden and G. Papanicolaou, Bounds for the effective parameters of heterogenous media by analytic continuation,Commun. Math. Phys. 90:473-491 (1983). · doi:10.1007/BF01216179
[18] G. Grimmett and H. Kesten, First passage percolation, network flows and electrical resistances,Z. Wahrsch. Verw. Geb. 66:335-366 (1984). · Zbl 0525.60098 · doi:10.1007/BF00533701
[19] M. Guo, Limit theorems for interacting particle systems, Ph.D. Thesis, New York University (1984).
[20] T. E. Harris, Diffusion with ?collision? between particles,J. Appl. Prob. 2:323-338 (1965). · Zbl 0139.34804 · doi:10.2307/3212197
[21] I. S. Heiland, On weak convergence to Brownian motion,Z. Wahrsch. Verw. Geb. 52:251-265 (1980). · Zbl 0411.60037 · doi:10.1007/BF00538890
[22] I. S. Helland, Central limit theorems for martingales with discrete or continuous time,Scand. J. Stat. 1982:979-994 (1982).
[23] K. Kawazu and H. Kesten, On birth and death processes in symmetric random environments,J. Stat. Phys. 37:561 (1984). · Zbl 0587.60088 · doi:10.1007/BF01010495
[24] H. Kesten,Percolation Theory for Mathematicians (Birkhauser, 1982).
[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). · Zbl 0388.60069
[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). · Zbl 0588.60058 · doi:10.1007/BF01210789
[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). · doi:10.1007/BF01010870
[28] W. Kohler and G. Papanicolaou, Bounds for the effective conductivity of random media, inLecture Notes in Physics, Vol. 154 (1982), pp. 111-130. · Zbl 0496.73002 · doi:10.1007/3-540-11202-2_9
[29] R. Kunnemann, The diffusion limit for reversible jump processes in ? d with ergodic random bond conductivities,Commun. Math. Phys. 90:27-68 (1983). · Zbl 0523.60097 · doi:10.1007/BF01209386
[30] R. Lang,Z. Wahrsch. Verw. Geb. 38:55 (1977). · Zbl 0349.60103 · doi:10.1007/BF00534170
[31] R. Lang, Stochastic models of many-particle systems and their time evolution, Habilitationsschrift, Universität Heidelberg (1982).
[32] J. L. Lebowitz and H. Spohn, Microscopic basis for Fick’s law for self-diffusion,J. Stat. Phys. 28:539-555 (1982). · Zbl 0512.60075 · doi:10.1007/BF01008323
[33] T. Liggett,Interacting Particle Systems (Springer-Verlag, 1984). · Zbl 0557.60087
[34] Gy. Lippner,Coll. Math. Soc. Janos Bolyai 24:277-290 (North-Holland, 1981).
[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). · Zbl 0589.60097 · doi:10.1007/BF01018553
[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.
[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.
[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.
[39] M. Reed and B. Simon,Methods of Mathematical Physics II (Academic Press, New York, 1975). · Zbl 0308.47002
[40] M. Rosenblatt,Markov Processes, Structure and Asymptotic Behavior (Springer-Verlag, Berlin, 1970).
[41] D. Ruelle, Superstable interactions in classical statistical mechanics,Commun. Math. Phys. 18:127 (1970). · Zbl 0198.31101 · doi:10.1007/BF01646091
[42] H. Rost, inLecture Notes in Control and Information Sciences, Vol. 25 (1980), pp. 297-302. · doi:10.1007/BFb0004020
[43] T. Shiga,Z. Wahrsch. Verw. Geb. 47:299 (1979). · Zbl 0407.60098 · doi:10.1007/BF00535165
[44] F. Solomon, Random walks in a random environment,Ann. Prob. 3(1):1-31 (1975). · Zbl 0305.60029 · doi:10.1214/aop/1176996444
[45] H. Spohn, Equilibrium fluctuations for interacting Brownian particles,Commun. Math. Phys. 103:1-33 (1986). · Zbl 0605.60092 · doi:10.1007/BF01464280
[46] D. W. Stroock and S. R. S. Varadhan,Multidimensional Diffusion Processes (Springer-Verlag, Berlin, 1979). · Zbl 0426.60069
[47] N. Ikeda and S. Watanabe,Stochastic Differential Equations and Diffusion Process (North-Holland, New York, 1981). · Zbl 0495.60005
[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). · Zbl 0578.60094 · doi:10.1002/cpa.3160380508
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