Uniform-in-time diffusion approximations for multiscale stochastic systems

TL;DR

This paper develops a rigorous, uniform-in-time diffusion approximation for multiscale stochastic systems, providing new insights into their long-term behavior and stationary distributions.

Contribution

It introduces a novel uniform-in-time framework for diffusion approximations, accommodating complex coefficients and structures, with applications to averaging, Langevin systems, and homogenization.

Findings

First quantitative identification of the limiting stationary distribution.

Proved the commutativity of limits $\\eps\to0$ and $t\to\infty$ for many observables.

Established a uniform-in-time averaging principle and homogenization results.

Abstract

This paper establishes a quantitative, uniform-in-time diffusion approximation for the joint law of a broad class of fully coupled multiscale stochastic systems. We derive a precise characterization of the limiting joint distribution as a specific skew-product of the conditional equilibrium of the fast process and the homogenized law of the slow component, thereby providing a rigorous uniform-in-time formulation of the adiabatic elimination principle. The convergence rate explicitly separates the initial relaxation of the fast dynamics from the long-time homogenized evolution and depends only on the regularity of the coefficients in the slow variable. As a consequence, we obtain the first quantitative identification of the limiting stationary distribution of the original multiscale system and prove the commutativity of the limits $\eps\to0$ and $t\to\infty$ for a large class of observables. Our framework accommodates unbounded and irregular coefficients, degenerate structures, and weakly mixing dynamics. We illustrate its scope with three applications: {\it (i)} a uniform-in-time averaging principle for fast-slow systems; {\it (ii)} a uniform Smoluchowski--Kramers approximation for degenerate Langevin systems, yielding convergence of the joint position-scaled velocity law and global-in-time asymptotics of key thermodynamic functionals (e.g., total energy, entropy production, free energy); and {\it (iii)} the first uniform-in-time periodic homogenization result for SDEs with distributional drifts.