Averaging principle and normal deviations for multi-scale stochastic hyperbolic–parabolic equations

Abstract

We study the asymptotic behavior of stochastic hyperbolic–parabolic equations with slow–fast time scales. Both the strong and weak convergence in the averaging principle are established. Then we study the stochastic fluctuations of the original system around its averaged equation. We show that the normalized difference converges weakly to the solution of a linear stochastic wave equation. An extra diffusion term appears in the limit which is given explicitly in terms of the solution of a Poisson equation. Furthermore, sharp rates for the above convergence are obtained, which are shown not to depend on the regularity of the coefficients in the equation for the fast variable.