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

TL;DR

This paper investigates the asymptotic behavior and fluctuations of multi-scale stochastic hyperbolic-parabolic equations, establishing averaging principles, convergence rates, and a central limit theorem using Hilbert space techniques.

Contribution

It provides a unified proof of averaging and fluctuation results for multi-scale stochastic hyperbolic-parabolic equations, including convergence rates independent of coefficient regularity.

Findings

Established strong and weak convergence in averaging principles.

Proved the normalized difference converges to a stochastic wave equation.

Derived sharp convergence rates unaffected by coefficient regularity.

Abstract

We study the asymptotic behavior of stochastic hyperbolic parabolic equations with slow and fast time scales. Both the strong and weak convergence in the averaging principe are established, which can be viewed as a functional law of large numbers. 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, which is a form of functional central limit theorem. We provide a unified proof for the above convergence by using the Poisson equation in Hilbert spaces. Moreover, sharp rates of convergence are obtained, which are shown not to depend on the regularity of the coefficients in the equation for the fast variable.