EXPONENTIAL MIXING AND LIMIT THEOREMS OF QUASI-PERIODICALLY FORCED 2D STOCHASTIC NAVIER-STOKES EQUATIONS IN THE HYPOELLIPTIC SETTING

The paper proves existence, uniqueness, and exponential mixing of a quasi-periodic invariant path for the time-inhomogeneous 2D NSE with quasi-periodic forcing, via a uniform contraction of the two-parameter Markov family in a weighted 1-Wasserstein metric, followed by a fixed-point construction of an invariant graph Γ over the torus and a Hölder-regularity estimate (Theorems 2.6–2.7 and Theorem 3.7). The candidate solution follows the same scheme: (i) Lyapunov drift with V(w)=e^{η||w||^2}, (ii) smoothing/irreducibility (via Hörmander A∞=H) to obtain asymptotic strong Feller and a small-set condition, and (iii) a Harris/weak-Harris style argument to deduce uniform Wasserstein contraction, then a fixed-point on C(T^n,P1(H)) for Γ and Hölder regularity with the same exponent ζ = ϖγ/(r+ϖ) where r is taken from the paper’s estimate (6.3). This matches the paper’s statements and proof structure, including the homogenization/skew-product viewpoint and the uniform-in-time fiber-wise estimates leading to Theorem 2.6 and (2.14) .