Functional correlation bounds and deterministic homogenisation of fast-slow systems

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 $$x_{k+1}=x_k+n^{-1}{a}_n(x_k,y_k)+n^{-1\mathrm{/}2}{b}_n(x_k,y_k),y_{k+1}=T_n{y}_k\mathrm{.}$$ We then prove that these sufficient conditions are satisfied by various examples of nonuniformly hyperbolic dynamical systems $T_n$.