Deterministic homogenization under optimal moment assumptions for fast-slow systems. Part 1.

Ann. Inst. H. Poincaré (B) Probab. Statist. 58 (2022) 1305-1327.

Alexey Korepanov, Zemer Kosloff and Ian Melbourne

Abstract

We consider deterministic homogenization (convergence to a stochastic differential equation) for multiscale 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

where the fast dynamics is given by a family T_n of nonuniformly expanding maps. Part 1 builds on our recent work on martingale approximations for families of nonuniformly expanding maps. We prove an iterated weak invariance principle and establish optimal iterated moment bounds for such maps. (The iterated moment bounds are new even for a fixed nonuniformly expanding map T.) The homogenization results are a consequence of this together with parallel developments on rough path theory in Part 2 by Chevyrev, Friz, Korepanov, Melbourne and Zhang.