Deterministic homogenization under optimal moment assumptions for fast-slow systems. Part 2

We consider deterministic homogenization for discrete-time fast-slow systems of the form $$X_{k+1}=X_k+n^{-1}{a}_n(X_k,Y_k)+n^{-1\mathrm{/}2}{b}_n(X_k,Y_k),Y_{k+1}=T_n{Y}_k$$ and give conditions under which the dynamics of the slow equations converge weakly to an It^o diffusion $X$ as $n\to \mathrm{\infty }$. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by $X$ are given explicitly. This extends the results of [Kelly-Melbourne, J. Funct. Anal. 272 (2017) 4063–4102] from the continuous-time case to the discrete-time case. Moreover, our methods (c`adl`ag $p$-variation rough paths) work under optimal moment assumptions. Combined with parallel developments on martingale approximations for families of nonuniformly expanding maps in Part 1 by Korepanov, Kosloff & Melbourne, we obtain optimal homogenization results when $T_n$ is such a family of maps.