Exponential mixing for random nonlinear wave equations: weak dissipation and localized control

The uploaded paper (Liu–Wei–Xiang–Zhang–Zhao, 2024) states and proves Theorem B: for the damped cubic wave equation with Γ-type localized damping and physically localized, decomposable, bounded noise of the form ηn(t,x)=χ(x)∑bjk θ^n_{jk} α^T_k(t) e_j(x), under T>T(D,a,χ) and the summability/nondegeneracy condition ∑ bjk λ_j^{2/7}||α_k||_{L∞(0,1)} ≤ B0 T^{1/2} with bjk≠0 on a finite block, the time-T Markov chain un+1=S(un,ηn) has a unique compactly supported invariant measure and is exponentially mixing in dual-Lipschitz distance (Theorem B, displayed as (1.11) and the ensuing inequality) . The proof does not use the Kuksin–Nersesyan–Shirikyan (KNS) 2018 criterion invoked by the model; instead, it develops and applies a new asymptotic-compactness-based criterion (Theorem 2.1 with hypotheses (AC),(I),(C)) and verifies it via: (i) asymptotic compactness (Theorem 4.1) ⇒ (AC); (ii) global stability of the unforced problem (Proposition 3.4) ⇒ (I); (iii) a frequency-based squeezing property (Theorem 5.1) ⇒ (C), together with a measure-transformation lemma for the finite-dimensional noise component (Appendix A.1.2) . By contrast, the model’s argument hinges on (a) a uniform one-step Lyapunov contraction and compactness of the one-step resolving operator, which the paper explicitly avoids by working with asymptotic compactness; (b) exact interior controllability of the linearised damped wave with the variable potential 3u^2(t,x), which is not established under the paper’s low-regularity potential (u^2∈L∞_t L^3_x) and is replaced in the paper by a bespoke low-frequency controllability plus high-frequency dissipation “squeezing” scheme (Section 5) ; and (c) a coupling step that assumes a uniform, one-step binding probability, whereas the paper’s criterion requires a quantitative coupling inequality of the form P(d(R,R′)>r d(x,x′))≤g(d(x,x′)) with g(r^n) decaying exponentially (Theorem 2.1(C)) . These mismatches indicate the model’s proof is not valid under the paper’s hypotheses, while the paper’s argument is coherent and complete within its framework.