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