Quantitative fluctuation analysis of multiscale diffusion systems via Malliavin calculus

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.