Functional correlation bounds and deterministic homogenisation of fast-slow systems

TL;DR

This paper establishes explicit conditions, including a functional correlation bound, under which discrete-time fast-slow systems converge to stochastic differential equations, with applications to nonuniformly hyperbolic dynamical systems.

Contribution

It provides elementary, explicit criteria for deterministic homogenisation in fast-slow systems, extending to nonuniformly hyperbolic dynamical systems.

Findings

Sufficient conditions for homogenisation are given.

Conditions are verified for various nonuniformly hyperbolic systems.

Convergence to stochastic differential equations is demonstrated.

Abstract

We give elementary and explicit sufficient conditions (in particular, a functional correlation bound) for deterministic homogenisation (convergence to a stochastic differential equation) for discrete-time fast-slow systems of the form \[ x_{k+1} = x_k + n^{-1}a_n(x_k,y_k) + n^{-1/2}b_n(x_k,y_k), \quad y_{k+1}=T_n y_k. \] We then prove that these sufficient conditions are satisfied by various examples of nonuniformly hyperbolic dynamical systems $ T_n $.