Ian Melbourne and Andrew Stuart
Abstract
We provide an explicit rigorous derivation of a diffusion
limit - a stochastic differential equation with
additive noise - from a deterministic skew-product flow. This
flow is assumed to exhibit time-scale separation and has
the form of a slowly evolving system driven by a
fast chaotic flow. Under mild assumptions on the fast flow,
we prove convergence to a stochastic differential equation as
the time-scale separation grows.
In contrast to existing work, we do not require the flow to have good
mixing properties. As a consequence, our results incorporate a large class
of fast flows, including the classical Lorenz equations.