A limit theorem for quadratic fluctuations in symmetric simple exclusion

We consider quadratic fluctuations VεH(ηs)=ε∑x∈ZH(εx)ηs(x)ηs(x+x0) in the centered symmetric simple exclusion process in dimension d=1d=1. Although the order of divergence of E[∫0ε−2dsVεH(ηs)]2 is known to be ε−3/2ε−3/2 if ε↓0ε↓0, the corresponding limit theorem was so far not explored. We now show that ε3/2∫0tε−2dsVεH(ηs) 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 tt. We remark that in symmetric exclusion in dimensions d≥3d≥3 the corresponding functional central limit theorem is known to produce Gaussian martingales in tt. The case d=2d=2 remains open.