The Annals of Probability

Central limit theorems for additive functionals of the simple exclusion process

Sunder Sethuraman

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

https://projecteuclid.org/euclid.aop/1019160120

Digital Object Identifier
doi:10.1214/aop/1019160120

Mathematical Reviews number (MathSciNet)
MR1756006

Zentralblatt MATH identifier
1044.60017

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

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.