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

TL;DR

This paper establishes deterministic homogenization for multiscale systems with nonuniformly expanding fast dynamics, proving an iterated weak invariance principle and optimal moment bounds, advancing the mathematical understanding of such systems.

Contribution

It introduces new iterated moment bounds and an iterated weak invariance principle for nonuniformly expanding maps, extending homogenization theory.

Findings

Proved an iterated weak invariance principle.

Established optimal iterated moment bounds for nonuniformly expanding maps.

Demonstrated homogenization results for multiscale systems with fast-slow dynamics.

Abstract

We consider deterministic homogenization (convergence to a stochastic differential equation) for multiscale systems of the form \[ x_{k+1} = x_k + n^{-1} a_n(x_k,y_k) + n^{-1/2} b_n(x_k,y_k), \quad y_{k+1} = T_n y_k, \] 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 & Zhang.