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 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 (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 and Melbourne, we obtain optimal homogenization results when Tn is such a family of maps.