A Note on Diffusion Limits of Chaotic Skew Product Flows

TL;DR

This paper rigorously derives a diffusion limit in the form of a stochastic differential equation from deterministic skew-product flows with time-scale separation, including chaotic systems like Lorenz equations, without requiring strong mixing conditions.

Contribution

It provides a new proof of diffusion limits for skew-product flows that relaxes mixing assumptions, broadening applicability to chaotic systems.

Findings

Convergence to stochastic differential equations proven under mild assumptions.

Includes classical chaotic systems such as Lorenz equations.

Applicable to a large class of fast chaotic flows without strong mixing requirements.

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.