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.
Citation
Sunder Sethuraman. Lin Xu. "A central limit theorem for reversible exclusion and zero-range particle systems." Ann. Probab. 24 (4) 1842 - 1870, October 1996. https://doi.org/10.1214/aop/1041903208
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