On the long-time statistical behavior of smooth solutions of the weakly damped, stochastically-driven KdV equation

The paper proves exactly the claimed uniqueness (at most one invariant/ergodic measure) for weakly damped, additively forced KdV under Range(σ) ⊃ P_N H^2, with N depending on (γ, ‖f‖_{H^1}, ‖σ‖_{L^2}), via an asymptotic coupling built from a nudged, cut-off modified SPDE and a weak (in-expectation) Foias–Prodi estimate; see Theorem 6.1 and its proof, which use a stopping-time cutoff ensuring Novikov’s condition and absolute continuity under Girsanov, and synchronization along discrete times via Borel–Cantelli (Proposition 6.4) . The candidate solution, by contrast, attempts a pathwise Foias–Prodi contraction on a blockwise “good event” and inserts a non-adapted indicator 1_{sup_{t∈[0,T]}‖u‖_{H^2}∨‖v‖_{H^2}≤R} into the feedback. This breaks progressive measurability and invalidates the Girsanov step, and the pathwise high/low estimates invoke ε‖h_x‖^2 without any positive control of ‖h_x‖^2 on the left (KdV lacks viscous dissipation). The paper explicitly relies on a weak Foias–Prodi mechanism instead of a pathwise one for precisely these reasons .