Asymptotic behavior of multiscale stochastic partial differential equations

TL;DR

This paper investigates the long-term behavior of multiscale stochastic partial differential equations, establishing strong convergence and fluctuation limits, with results applicable regardless of the regularity of fast-variable coefficients.

Contribution

It provides the first rigorous analysis of asymptotic behavior, including averaging and fluctuation results, for semi-linear multiscale SPDEs with singular coefficients.

Findings

Strong convergence in averaging principle established

Weak convergence to Ornstein-Uhlenbeck process shown

Convergence rates independent of coefficient regularity

Abstract

In this paper, we study the asymptotic behavior of a semi-linear slow-fast stochastic partial differential equation with singular coefficients. Using the Poisson equation in Hilbert space, we first establish the strong convergence in the averaging principe, which can be 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 can be viewed as a functional central limit theorem. Furthermore, rates of convergence both for the strong convergence and the normal deviation are obtained, and these convergence 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.