A law of large numbers and a central limit theorem are proved for a locally interacting particle system. This system describes a chemical reaction with diffusion with linear creation and quadratic annihilation of particles. The deterministic limit is the solution of a nonlinear reaction-diffusion equation defined on an $n$-dimensional unit cube. The law of large numbers holds for any dimension $n$ and arbitrary times, whereas the central limit theorem holds only for dimension $n \leq 3$ and on a certain bounded time interval (depending on the initial distribution and on the creation rate). A propagation of chaos expansion of the correlation functions is used.
Peter Kotelenez. "Fluctuations in a Nonlinear Reaction-Diffusion Model." Ann. Appl. Probab. 2 (3) 669 - 694, August, 1992. https://doi.org/10.1214/aoap/1177005654