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

Abstract

We consider deterministic homogenization for discrete-time fast-slow systems of the form $$ X_{k+1} = X_k + n^{-1}a_n(X_k,Y_k) + n^{-1/2}b_n(X_k,Y_k)\;, \quad Y_{k+1} = T_nY_k\;$$ and give conditions under which the dynamics of the slow equations converge weakly to an It\^o diffusion $X$ as $n\to\infty$. 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.