Deterministic homogenization under optimal moment assumptions for fast-slow systems

We consider deterministic homogenization for discrete-time fast-slow systems of the form Xk+1=Xk+n−1an(Xk,Yk)+n−1/2bn(Xk,Yk),Yk+1=TnYk and give conditions under which the dynamics of the slow equations converge weakly to an Itô diffusion X as n→∞. 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 (càdlàg 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 Tn is such a family of maps.