Edgeworth expansions for slow-fast systems and their application to model reduction for finite time scale separation

We show that transition probabilities of the slow variable of a multi-scale dynamics can be expanded in orders of the time scale separation parameter. The resulting Edgeworth corrections characterise deviations from Gaussianity due to the finite time scale separation. Explicit expressions for the first two orders of correction are provided, valid for stochastic as well as deterministic chaotic fast dynamics. The corrections are then used to construct a stochastic reduced system which reliably approximates the effective diffusive behaviour of the slow dynamics. Our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time scale separation. We corroborate our analytical results with numerical simulations, demonstrating improvement over homogenization.