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.
Jean-Dominique Deuschel. "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