A limit theorem for quadratic fluctuations in symmetric simple exclusion

We consider quadratic fluctuations V ε H ( η s ) = ε ∑ x ∈ Z H ( ε x ) η s ( x ) η s ( x + x 0 ) in the centered symmetric simple exclusion process in dimension d = 1 . Although the order of divergence of E [ ∫ 0 ε − 2 d s V ε H ( η s ) ] 2 is known to be ε − 3 / 2 if ε ↓ 0 , the corresponding limit theorem was so far not explored. We now show that ε 3 / 2 ∫ 0 t ε − 2 d s V ε H ( η 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.