论文标题
长时间统计的质量限制很小，随机非线性阻尼波方程
The small mass limit for long time statistics of a stochastic nonlinear damped wave equation
论文作者
论文摘要
我们研究了具有多项式非线性的一类半线性阻尼波方程的长时间统计数据，并在维度2和3中受到加斯高斯噪声的扰动。我们发现，如果相位空间中足够多的方向非常强迫，则该系统对与质量相差的独特的融合速率非常有吸引力。然后，在较小的质量限制中，我们证明了在适当的瓦斯汀距离距离随机反应方程的独特不变度度量的情况下，不变度度量的第一个边缘的收敛性。这与统一的几何形状界限意味着无限时间范围$ [0，\ infty）$在质量上的质量极限的有效性，从而扩展了先前已知的结果，该结果是在Lipschitz非线性下为阻尼波动方程式建立的。
We study the long time statistics of a class of semi--linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase space are stochastically forced, the system is exponentially attractive toward its unique invariant measure with a convergent rate that is uniform with respect to the mass. Then, in the small mass limit, we prove the convergence of the first marginal of the invariant measures in a suitable Wasserstein distance toward the unique invariant measure of a stochastic reaction--diffusion equation. This together with uniform geometric ergodcity implies the validity of the small mass limit for the solutions on the infinite time horizon $[0,\infty)$, thereby extending previously known results established for the damped wave equations under Lipschitz nonlinearities.
