We consider a lattice system of linear interacting diffusion processes with infinitely many invariant distributions. We first prove a nonstandard central limit theorem and identify the equation of the fluctuation field. We then derive dimension dependent large deviation results for the empirical mean.
"Invariance Principle and Empirical Mean Large Deviations of the Critical Ornstein-Uhlenbeck Process." Ann. Probab. 17 (1) 74 - 90, January, 1989. https://doi.org/10.1214/aop/1176991495