Quenched correlation decay for random splittings of some prototypical 3D flows including the ABC flow

The paper proves quenched exponential decay of correlations (Theorem 2.5) for the random splitting of three prototypical 3D shear flows using a Markov-chain approach: it establishes V-uniform geometric ergodicity of the two-point chain P^(2) (Section 6) and then applies Chebyshev + Borel–Cantelli to control Fourier-mode correlations, followed by a Sobolev-regularization argument to reach any s>0 (Section 7.1) . This yields, for any q,s>0, a random prefactor ξ with E[ξ^q]<∞ and a uniform α>0 such that |∫ f(x) g(Φ^m_τ(x)) dx| ≤ ξ(τ) e^{-α m} ||f||_{H^s}||g||_{H^s} for mean-zero f,g ∈ H^s(T^3) . By contrast, the candidate solution proposes a different proof via a per-block H^s→H^{-s} contraction and a “negative-drift” random-product argument. Two key steps are unsubstantiated or false as stated: (i) the claim that enlarging the good-block window K=[T0,2T0]^3 makes the drift E[log Y] negative overlooks that P(K) decays exponentially in T0 for Exp(h) clocks, so p log θ_s cannot be forced to dominate the O(1) positive contribution from bad blocks merely by taking T0→∞; no quantitative choice of T0 is supplied to ensure E[log Y]<0; (ii) the uniform “single–shear low-pass suppression” bound is asserted with only a sketch, and its uniformity over all (k1,k2) via (H1) is not justified. The paper’s approach avoids these pitfalls and provides a complete proof under the stated (H1) and flow definitions (Section 2) .