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

We study the normalized difference between the solution u(epsilon) 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 (u) over bar 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 (u epsilon - (u) over bar)/root epsilon converges weakly in C([0, T]: L-2(0, L)) to the solution of the linearized equation, where an extra Gaussian term appears. Such a term is explicitly given. (C) 2009 Elsevier Masson SAS. All fights reserved.