## The Annals of Probability

### A central limit theorem for reversible exclusion and zero-range particle systems

#### Abstract

We give easily verifiable conditions under which a functional central limit theorem holds for additive functionals of symmetric simple exclusion and symmetric zero-range processes. Also a reversible exclusion model with speed change is considered. Let $\eta (t)$ be the configuration of the process at time t and let $f(\eta)$ be a function on the state space. The question is: For which functions f does $\lambda^{-1/2} \int_0^{\lambda t} f(\eta(s)) ds$ converge to a Brownian motion? A general but often intractable answer is given by Kipnis and Varadhan. In this article we determine what conditions beyond a mean-zero condition on $f(\eta)$ are required for the diffusive limit above. Specifically, we characterize the $H^{-1}$ space in an applicable way.

Our method of proof relies primarily on a sharp estimate on the "spectral gap" of the process and weak regularity properties for the invariant measures.

#### Article information

Source
Ann. Probab., Volume 24, Number 4 (1996), 1842-1870.

Dates
First available in Project Euclid: 6 January 2003

https://projecteuclid.org/euclid.aop/1041903208

Digital Object Identifier
doi:10.1214/aop/1041903208

Mathematical Reviews number (MathSciNet)
MR1415231

Zentralblatt MATH identifier
0872.60079

#### Citation

Sethuraman, Sunder; Xu, Lin. A central limit theorem for reversible exclusion and zero-range particle systems. Ann. Probab. 24 (1996), no. 4, 1842--1870. doi:10.1214/aop/1041903208. https://projecteuclid.org/euclid.aop/1041903208

#### References

• 1 ANDJEL, E. D. 1982. Invariant measures for the zero range process. Ann. Probab. 10 525 547.
• 2 BILLINGSLEY, P. 1971. Weak Convergence of Probability Measures: Applications in Probability. SIAM, Philadelphia.
• 3 ESPOSITO, R., MARRA, R. and YAU, H. T. 1994. Diffusive limit of asy mmetric simple exclusion. Reviews in Mathematical physics 6 1233 1267.
• 4 KIPNIS, C. 1986. Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab. 14 397 408.
• 5 KIPNIS, C. 1987. Fluctuations des temps d'occupation d'un site dans l'exclusion simple sy mmetrique. Ann. Inst. H. Poincare Probab. Statist. 23 21 35. ´
• 6 KIPNIS, C. and VARADHAN, S. R. S. 1986. Central limit theorem for additive functionals of reversible Markov processes. Comm. Math. Phy s. 104 1 19.
• 7 KOLMOROGOV, A. N. 1962. A local limit theorem for Markov chains. Selected Translations in Mathematical Statistics and Probability 2 109 129.
• 8 LANDIM, C., SETHURAMAN, S. and VARADHAN, S. 1996. Spectral gap for zero-range dy namics. Ann. Probab. 24 1651 1682.
• 9 LEBOWITZ, J. L. and SPOHN, H. 1982. A microscopical basis for Fick's law. J. Statist. Phy s. 28 539 556.
• 10 LIGGETT, T. M. 1985. Interacting Particle Sy stems. Springer, New York.
• 11 LU, S. L. and YAU, H. T. 1993. Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dy namics. Comm. Math. Phy s. 156 399 433.
• 12 PETROV, V. V. 1975. Sums of Independent Random Variables. Springer, New York.
• 13 QUASTEL, J. 1992. Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. 45 623 679.
• 14 SPITZER, F. 1970. Interaction of Markov processes. Adv. Math. 5 142 154.
• 15 SPOHN, H. 1991. Large Scale Dy namics of Interacting Particles. Springer, New York.
• 16 VARADHAN, S. R. S. 1994. Nonlinear diffusion limit for a sy stem with nearest neighbor interactions II. In Asy mptotic Problems in Probability Theory: Stochastic Models and Z. Diffusion on Fractals K. Elworthy and N. Ikeda, eds.. Wiley, New York.