Normal deviations from the averaged motion for some reaction–diffusion equations with fast oscillating perturbation

We study the normalized difference between the solution u of a reaction–diffusion equation in a bounded interval [0,L], perturbed by a fast oscillating term arising as the solution of a stochastic reaction–diffusion equation with a strong mixing behavior, and the solution of the corresponding averaged equation. We assume the smoothness of the reaction coefficient and we prove that a central limit type theorem holds. Namely, we show that the normalized difference converges weakly in C([0,T];L²(0,L)) to the solution of the linearized equation, where an extra Gaussian term appears. Such a term is explicitly given.