Asymptotic behavior of multiscale stochastic partial differential equations with Hölder coefficients

Abstract

In this paper, we establish a quantified asymptotic analysis for a semi-linear slow-fast stochastic partial differential equation with Hölder coefficients. By studying the Poisson equation in Hilbert space, we first prove the strong convergence in the averaging principle, which is viewed as a functional law of large numbers. Then we study the stochastic fluctuations between the original system and its averaged equation. We show that the normalized difference converges weakly to an Ornstein-Uhlenbeck type process, which is viewed as a functional central limit theorem. Rates of convergence both for the strong convergence and the normal deviation are obtained, and these convergences are shown not to depend on the regularity of the coefficients in the equation for the fast variable, which coincides with the intuition, since in the limit systems the fast component has been totally averaged or homogenized out.