Quantitative fluctuation analysis of multiscale diffusion systems via Malliavin calculus

TL;DR

This paper analyzes the fluctuations of multiscale diffusion systems with small noise, providing quantitative convergence rates to Gaussian limits using Malliavin calculus, which advances understanding of stochastic homogenization.

Contribution

It introduces a detailed Malliavin calculus framework to derive explicit convergence rates for fluctuations in multiscale diffusions, including fully coupled systems.

Findings

Quantitative convergence rates to Gaussian limits in Wasserstein metric

Detailed estimates of first and second Malliavin derivatives

Analysis of coupled multiscale diffusion systems

Abstract

We study fluctuations of small noise multiscale diffusions around their homogenized deterministic limit. We derive quantitative rates of convergence of the fluctuation processes to their Gaussian limits in the appropriate Wasserstein metric requiring detailed estimates of the first and second order Malliavin derivative of the slow component. We study a fully coupled system and the derivation of the quantitative rates of convergence depends on a very careful decomposition of the first and second Malliavin derivatives of the slow and fast component to terms that have different rates of convergence depending on the strength of the noise and timescale separation parameter.