带退化噪声的三维阻尼三次波动方程的谱间隙
Spectral gap for the three-dimensional damped cubic wave equation with degenerate noise
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中文总结 AI 辅助
该研究针对带有限秩布朗强迫的三维阻尼三次波动方程，通过稳定-紧渐近耦合机制建立加权Wasserstein谱间隙，推导得唯一遍历性与指数混合性，还给出饱和条件的精确几何刻画。
中文摘要 AI 辅助
我们针对带真正有限秩布朗强迫的三维阻尼三次波动方程，建立了加权Wasserstein谱间隙。在饱和条件下，该间隙对每个0<s<1/2都关于负相拓扑$\boldsymbol{\textit{E}_s=H^{-s}\times H^{-1-s}}$成立，由此我们推导出能量拓扑下的唯一遍历性和指数混合性，还得到了饱和条件的精确几何刻画。我们提出的方法是针对非抛物型 hypoelliptic 耗散SPDE的稳定-紧渐近耦合机制。该方法不依赖正时间平滑或渐近梯度估计，而是将无穷维收缩障碍归约为弱耦合拓扑中的紧缺陷；稠密Malliavin范围允许该缺陷被有限维Cameron–Martin位移补偿，在有界Lyapunov核上产生有限距离收缩，结合耗散带来的独立高能收缩，最终得到全局加权Wasserstein谱间隙。
英文摘要
We establish a weighted Wasserstein spectral gap for the three-dimensional damped cubic wave equation with genuinely finite rank Brownian forcing. Under a saturation condition, the gap holds with respect to the negative phase topology $\mathcal E_s=H^{-s}\times H^{-1-s}$ for every $0<s<1/2$, from which we deduce unique ergodicity and exponential mixing in the energy topology. Sharp geometric characterizations of the saturation condition are also obtained. The method we develop is a stable--compact asymptotic coupling mechanism for hypoelliptic dissipative SPDEs beyond the parabolic setting. Instead of relying on positive time smoothing or asymptotic gradient estimates, it reduces the infinite-dimensional obstruction to contraction to a compact defect in a weaker coupling topology. Dense Malliavin range then permits this defect to be compensated by a finite dimensional Cameron--Martin shift, producing a finite distance contraction on bounded Lyapunov cores. Together with a separate high energy contraction from dissipation, this yields a global weighted Wasserstein spectral gap.
