QUADRATIC RESPONSE AND SPEED OF CONVERGENCE OF INVARIANT MEASURES IN THE ZERO-NOISE LIMIT.

The paper proves that for a C^8 expanding circle map T and additive i.i.d. noise with kernel ρ ∈ BV having zero mean and variance σ^2, the stationary densities h_δ of L_δ = ρ_δ ∗ L_T admit a second-order expansion: ∥(h_δ − h_0)/δ^2 − (σ^2/2)(Id − L_T)^{-1} h_0''∥_{W^{1,1}} → 0 (Theorem 1) , with L_δ and ρ_δ defined as in (7)–(8) . The proof in the paper proceeds via abstract linear/quadratic response results, verifying hypotheses including a uniform Lasota–Yorke inequality and resolvent boundedness for L_T on W^{k,1} (Proposition 7 and Lemma 8) , and showing the second-order expansion of the noise operator applied to h_0 (Proposition 14) . The model’s solution reaches the same conclusion by a direct resolvent-based expansion: it expands K_δ to second order with an O(δ^3) remainder, uses h_δ − h_0 = (Id − L_δ)^{-1}(L_δ − L_T)h_0, controls the remainder via uniform bounds on (Id − L_δ)^{-1}, and then replaces (Id − L_δ)^{-1} with (Id − L_T)^{-1} using the resolvent identity and an O(δ^2) bound on (L_δ − L_T)g for suitable g. This argument is consistent with the paper’s assumptions and results. Thus both the paper and the model are correct; the approaches differ (abstract response theory vs. an explicit resolvent manipulation).