Edgeworth expansions for slow-fast systems with finite time scale separation

TL;DR

This paper develops Edgeworth expansions to improve the approximation of slow-fast systems' behavior beyond the classical Gaussian limit, using a semi-group formalism and diagrammatic methods to account for finite time scale separation.

Contribution

It introduces a novel semi-group and diagrammatic approach to derive higher-order corrections for slow-fast systems with finite time scale separation.

Findings

Explicit formulas for first two correction orders

Numerical simulations confirm analytical predictions

Method improves classical homogenization results

Abstract

We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of slow-fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel-Dyson series is used to asymptotically determine the corrections at any desired order of the time scale parameter $\varepsilon$. The corrections involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in $\varepsilon$ and provide explicit expressions for the first two orders. At a formal level, the expressions derived are valid in the case when the fast dynamics is stochastic as well as when the fast dynamics is entirely deterministic. We corroborate our analytical results with numerical simulations and show that our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time scale separation.