Diffusion approximation for multi-scale stochastic reaction-diffusion equations

Abstract: In this paper, we study the diffusion approximation for singularly perturbed stochastic reaction-diffusion equation with a fast oscillating term. The asymptotic limit for the original system is obtained, where an extra Gaussian term appears. Such a term is explicitly given in terms of the solution of Poisson equation in Hilbert space. Moreover, we also obtain the optimal rate of convergence, and the convergence rate is shown not to depend on the regularity of the coefficients of the original system with respect to the fast variable, which coincides with the intuition since the fast component has been totally homogenized out in the limit equation.