A limit theorem for quadratic fluctuations in symmetric simple exclusion

We consider quadratic fluctuations V-epsilon(H) (eta(s)) = root epsilon Sigma(x is an element of Z) H(epsilon x)eta(s)(x)eta(s) (x + x(0)) in the centered symmetric simple exclusion process in dimension d = 1. Although the order of divergence of root E[(epsilon-2)(0) ds v(epsilon)(h) (eta(s))](2) is known to be epsilon(-3/2) if epsilon down arrow 0 the corresponding limit theorem was so far not explored. We now show that epsilon(3/2) integral(t epsilon)(0)(-2) ds V-epsilon(H) (eta(s)) converges in law to a non-Gaussian singular functional of an infinite-dimensional Ornstein-Uhlenbeck process. Despite the singularity of the limiting functional we find enough structure to conclude that it is continuous but not a martingale in t. We remark that in symmetric exclusion in dimensions d >= 3 the corresponding functional central limit theorem is known to produce Gaussian martingales in t. The case d = 2 remains open. (C) 2006 Elsevier B.V. All rights reserved