关于弱阻尼随机驱动KdV方程光滑解的长时间统计行为
On the long-time statistical behavior of smooth solutions of the weakly damped, stochastically-driven KdV equation
AI总结：
本文研究弱阻尼随机驱动KdV方程的长时间统计行为，利用运动积分建立Lyapunov结构与弱Foias-Prodi估计，证明不变测度的存在性、光滑支集及不同阻尼下的遍历性质。
摘要：
本文研究了含时间白噪声、空间光滑随机源项的阻尼周期Korteweg-de Vries（KdV）方程，探讨其解的长时间行为。我们证明可利用KdV的运动积分来证明阻尼随机KdV方程不变测度的正则性与遍历性质。首先，通过对运动积分进行非平凡修正，我们证明了解的任意正则阶Sobolev范数的矩在全局时间上有界，从而建立了Lyapunov结构；不变测度的存在性是其直接推论。接下来，我们证明了阻尼随机KdV方程的弱Foias-Prodi型估计，其同步性在期望意义下成立，该估计在后续分析中起到关键作用。作为首个创新性应用，我们将Foias-Prodi估计与Lyapunov结构相结合，证明了当外部驱动力光滑时，不变测度的支集包含于光滑函数空间。随后我们分别针对任意阻尼和大阻尼情形，建立了不变测度的遍历性质。对于任意阻尼，我们证明只要相空间中有足够多的方向受到随机驱动，就可应用“渐近耦合”框架给出不变测度唯一性的简洁证明，该证明方法对满足弱Foias-Prodi型性质的随机偏微分方程（SPDE）具有范式意义。最后，对于大阻尼情形，我们证明了在类Wasserstein距离下存在谱隙，由此可得不变测度的指数混合性与唯一性。
英文摘要：
This paper considers the damped periodic Korteweg-de Vries (KdV) equation in the presence of a white-in-time and spatially smooth stochastic source term and studies the long-time behavior of solutions. We show that the integrals of motion for KdV can be exploited to prove regularity and ergodic properties of invariant measures for damped stochastic KdV. First, by considering non-trivial modifications of the integrals of motion, we establish Lyapunov structure by proving that moments of Sobolev norms of solutions at all orders of regularity are bounded globally-in-time; existence of invariant measures follows as an immediate consequence. Next, we prove a weak Foias-Prodi type estimate for damped stochastic KdV, for which the synchronization occurs in expected value. This estimate plays a crucial role throughout our subsequent analysis. As a first novel application, we combine the Foias-Prodi estimate with the Lyapunov structure to establish that invariant measures are supported on smooth functions provided that the external driving forces are smooth. We then establish ergodic properties of invariant measures, treating the regimes of arbitrary damping and large damping separately. For arbitrary damping, we demonstrate that the framework of `asymptotic coupling' can be implemented for a compact proof of uniqueness of the invariant measure provided that sufficiently many directions in phase space are stochastically forced. Our proof is paradigmatic for SPDEs for which a weak Foias-Prodi type property holds. Lastly, for large damping, we establish the existence of a spectral gap with respect to a Wasserstein-like distance, and exponential mixing and uniqueness of the invariant measure follows.
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