Time inhomogeneous Poisson equations and non-autonomous multi-scale stochastic systems

TL;DR

This paper introduces a novel approach using time inhomogeneous Poisson equations to analyze the asymptotic behavior of non-autonomous multi-scale stochastic systems with irregular, oscillating coefficients, establishing convergence results and rates.

Contribution

It develops a new analytical tool for non-autonomous stochastic systems with irregular coefficients, enabling convergence analysis without regularity assumptions.

Findings

Proves strong convergence of double averaging principle.

Establishes a functional central limit theorem with homogenized diffusion.

Provides convergence rates independent of coefficient regularity.

Abstract

We develop a new tool, the time inhomogeneous Poisson equation in the whole space and with a terminal condition at infinity, to study the asymptotic behavior of the non-autonomous multi-scale stochastic system with irregular coefficients, where both the fast and the slow equation depend on the highly oscillating time component. In particular, periodic, quasi-periodic and almost periodic coefficients are allowed. The strong convergence of double averaging principle as well as the functional central limit theorem with homogenized-averaged diffusion coefficient are established. Moreover, we also obtain rates of convergence, which do not depend on the regularities of the coefficients with respect to the time component and the fast variable.