The Annals of Probability

Central limit theorems for additive functionals of the simple exclusion process

Sunder Sethuraman

Full-text: Open access

Abstract

Some invariance principles for additive functionals of simple exclusion with finite-range translation-invariant jump rates $p(i, j) = p(j - i)$ in dimensions $d \geq1$ are established. A previous investigation concentrated on the case of $p$ symmetric. The principal tools to take care of nonreversibility, when $p$ is asymmetric, are invariance principles for associated random variables and a “local balance”estimate on the asymmetric generator of the process.

As a by-product,we provide upper and lower bounds on some transition probabilities for mean-zero asymmetric second-class particles,which are not Markovian, that show they behave like their symmetric Markovian counterparts.Also some estimates with respect to second-class particles with drift are discussed.

In addition,a dichotomy between the occupation time process limits in $d =1$ and $d \geq 2$ for symmetric exclusion is shown. In the former, the limit is fractional Brownian motion with parameter 3/4, and in the latter, the usual Brownian motion.

Article information

Source
Ann. Probab., Volume 28, Number 1 (2000), 277-302.

Dates
First available in Project Euclid: 18 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1019160120

Digital Object Identifier
doi:10.1214/aop/1019160120

Mathematical Reviews number (MathSciNet)
MR1756006

Zentralblatt MATH identifier
1044.60017

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F05: Central limit and other weak theorems

Keywords
Invariance principle central limit theorem simple exclusion process FKG associated second-class particles

Citation

Sethuraman, Sunder. Central limit theorems for additive functionals of the simple exclusion process. Ann. Probab. 28 (2000), no. 1, 277--302. doi:10.1214/aop/1019160120. https://projecteuclid.org/euclid.aop/1019160120


Export citation

References

  • [1] Bertini, L. and Giacomin, G. (1997). On the long time behavior of the stochastic heat equation. Preprint.
  • [2] Ferrari, P. A. (1992) Shock fluctuations in asymmetric simple exclusion. Probab. Theory Related Fields 91 81-110.
  • [3] Kipnis, C. (1986). Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab. 14 397-408.
  • [4] Kipnis, C. (1987). Fluctuations des temps d'occupation d'un site dans l'exclusion simple symmetrique. Ann. Inst. H. Poincar´e Sec. B 23 21-35.
  • [5] Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes. Comm. Math. Phys. 104 1-19.
  • [6] Landim, C. and Yau, H. T. (1997). Fluctuation-dissipation equation of asymmetric simple exclusion processes. Probab. Theory Related Fields 108 321-356.
  • [7] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.
  • [8] Newman, C. M. (1983). A general central limit theorem for FKG systems. Comm. Math. Phys. 91 75-80.
  • [9] Newman, C. M. (1984). Asymptotic independence and limit theorems for positively and negatively dependent random variables. In Inequalities in Statistics and Probability 127- 140. IMS, Hayward, CA.
  • [10] Newman, C. M. and Wright, A. L. (1982). Associated random variables and martingale inequalities.Wahrsch. Verw. Gebiete 59 361-371.
  • [11] Rezakhanlou, F. (1995). Microscopic structure of shocks in one conservation laws. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 12 119-153.
  • [12] Saada, E. (1987). A limit theorem for the position of a tagged particle in a simple exclusion process. Ann. Probab. 15 375-381.
  • [13] Sethuraman, S., Varadhan, S. R. S. and Yau, H. T. (1999). Diffusive limit of a tagged particle in asymmetric simple exclusion processes. Comm. Pure Appl. Math. To appear.
  • [14] Sethuraman, S. and Xu, L. (1996). A central limit theorem for reversible exclusion and zero-range particle systems. Ann. Probab. 24 1842-1870.
  • [15] Varadhan, S. R. S. (1995). Self-diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion. Ann. Inst. H. Poincar´e 31 273-285.