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

The paper’s Theorem 3.4 states that, for a uniform family of nonuniformly expanding maps with Σn → Σ and En → E, the pair (Wn, W̄n) converges in distribution to (W, ∫0^t W ⊗ dW + Et), where W is Brownian with covariance Σ . The proof reduces to a martingale–coboundary decomposition on a Young tower, applies an iterated WIP for martingale difference arrays (Appendix A), and then identifies the drift by showing sup|An − Bn − tEn| → 0 . The candidate’s solution follows the same program in spirit: it uses a Gordin decomposition, a martingale functional CLT, identifies W̄n as a discrete Itô integral plus a drift term, and uses stability of stochastic integrals (Kurtz–Protter) to pass to the limit. The main divergence is that the candidate assumes a uniform L2 bound for the coboundary χn to control remainders, whereas the paper only assumes τ ∈ Lp with p ≥ 2 and uses moment/Ui estimates on the tower; this is a stronger-than-needed hypothesis in the model’s outline, but the overall argument aligns and reaches the same conclusion.