Fluctuation analysis and short time asymptotics for multiple scales diffusion processes

TL;DR

This paper analyzes the fluctuations of multi-scale small noise diffusions around their deterministic limit, providing new insights into their short time asymptotics without relying on periodicity assumptions.

Contribution

It introduces a comprehensive fluctuation analysis framework for multi-scale diffusions with full dependence on slow and fast variables, including explicit characterization of additional drift terms.

Findings

Different behaviors depending on the interaction order between fast scale and noise

Explicit characterization of additional drift terms in the limiting process

Provides improved approximation over classical homogenization results

Abstract

We consider the limiting behavior of fluctuations of small noise diffusions with multiple scales around their homogenized deterministic limit. We allow full dependence of the coefficients on the slow and fast motion. These processes arise naturally when one is interested in short time asymptotics of multiple scale diffusions. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. Depending on the order of interaction between the fast scale and the size of the noise we get different behavior. In certain cases additional drift terms arise in the limiting process, which are explicitly characterized. These results provide a better approximation to the limiting behavior of such processes when compared to the law of large numbers homogenization limit.