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

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 ε . 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 ε 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.