一维随机 Ginzburg-Landau 方程长时间统计量的无粘极限
The inviscid limit for long time statistics of the one-dimensional stochastic Ginzburg-Landau equation
AI总结：
本文研究一维加性高斯噪声驱动的三次随机 Ginzburg-Landau 方程，利用定制 Foias-Prodi 耦合证明与黏性无关的多项式混合，并在 Wasserstein 距离下建立其不变测度向随机 Schrödinger 方程的无粘极限。
摘要：
我们研究具有三次非线性的一维随机 Ginzburg-Landau 方程在通过加性高斯噪声受到随机扰动时的长时间统计量。在相空间中有足够多方向受到随机强迫的假设下，我们发现动力学以独立于趋于零的黏性的多项式速率吸引到唯一的不变概率测度。这依赖于一种耦合技术，该技术利用了专门为该系统定制的 Foias-Prodi 论证。随后，在无粘区域，我们证明不变测度序列在适当的 Wasserstein 距离下收敛于随机 Schrödinger 方程的不变测度。结合一致多项式混合性质，我们以 log log 速率得到了无限时间区间上解的无粘极限的有效性。
英文摘要：
We study the long time statistics of a one-dimensional Ginzburg-Landau equation with cubic nonlinearity while being subjected to random perturbations via an additive Gaussian noise. Under the assumption that sufficiently many directions of the phase space are stochastically forced, we find that in the inviscid limit, the solutions are well approximated by the stochastic Schrodinger equation on the infinite time horizon. Moreover, the convergence is established to exhibit a quantitative order of log log rate with respect to the vanishing viscosity. Our proof relies on the uniform polynomial mixing of the Ginzburg-Landau equation as well as the convergence of the sequence of invariant probability measures in suitable Wasserstein distances.
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