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

Ann. Inst. H. Poincaré (B) Probab. Statist. 58 (2022) 1328-1350.

Ilya Chevyrev, Peter Friz, Alexey Korepanov, Ian Melbourne and Huilin Zhang

Abstract

We consider deterministic homogenization for discrete-time fast-slow systems of the form

X_k+1 = X_k + n^-1a_n(X_k,Y_k) + n^-1/2b_n(X_k,Y_k),     Y_k+1 = T_nY_k

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