Planar random walk in a stratified quasi-periodic environment

The paper’s main theorem (Theorem 4.3) precisely states: (i) if ∫ g dν_f ≠ 0 then the walk is transient for all x, and (ii) if g is a coboundary h − e^{-f}Th with h bounded and either f is even (around some x0) or f = u − Tu with e^u integrable, then the walk is recurrent for Lebesgue-a.e. x. This is proved via a quantitative structure-function criterion (Definition 4.5, Theorem 4.6) and a uniform weighted-selection lemma for ν_f (Lemma 4.7) that leverages Denjoy–Koksma bounds for BV cocycles; the statements and proofs are internally consistent and carefully justified in the paper . By contrast, the candidate solution proposes a corrector-based Lyapunov approach. While its setup matches the paper’s model and uses the same quasi-invariant measure ν_f and weighted averages convergence (matching Lemma 4.7) , it contains a critical flaw in part (i): it asserts the existence of a bounded-increment Lyapunov function after truncating the corrector, but gives no control on the vertical corrector increments Δ_n. In general Δ_n need not be uniformly bounded under the chosen A_n, so the increments of the truncated function are not bounded as claimed; hence the Foster–Lyapunov transience step is not justified. In part (ii), while the martingale coordinate when g is a coboundary is sound and matches the paper’s idea that the forcing term becomes a discrete gradient, the subsequent reduction to a divergence-free or reversible planar network is only sketched and lacks the detailed estimates the paper provides to deduce recurrence for a.e. x. Therefore, the paper’s result and proof are correct, but the model’s proof is not fully correct/complete.