论文标题
Navier的随机方程式 - Stokes类型
Ergodicity for stochastic equation of Navier--Stokes type
论文作者
论文摘要
在注释的第一部分中，我们分析了二维随机导航器的长时间行为 - 在带有退化的，一维噪声的圆环上Stokes方程系统。特别是，对于某些初始数据和噪声，我们确定了系统的不变概率度量，并提供了足够的条件，在该条件下它是独特且随机稳定的。在注释的第二部分中，我们考虑了一个由一维维也纳过程驱动的随机微分方程的有限维系统的简单示例，它与随机的N.S.E.显示出一定的相似性，并根据漂移的强度研究了其ergodic属性。如果后者足够小，并且位于临界阈值以下，则该系统将接受一种独特的不变概率度量，即高斯。另一方面，如果噪声漂移的强度大于阈值，则除了高斯不变的概率度量外，还有另一种。特别是，系统的发电机不是低纤维化的。
In the first part of the note we analyze the long time behaviour of a two dimensional stochastic Navier--Stokes equations system on a torus with a degenerate, one dimensional noise. In particular, for some initial data and noises we identify the invariant probability measure for the system and give a sufficient condition under which it is unique and stochastically stable. In the second part of the note, we consider a simple example of a finite-dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic N.S.E., and investigate its ergodic properties depending on the strength of the drift. If the latter is sufficiently small and lies below a critical threshold, then the system admits a unique invariant probability measure which is Gaussian. If, on the other hand, the strength of the noise drift is larger than the threshold, then in addition to a Gaussian invariant probability measure, there exist another one. In particular, the generator of the system is not hypoelliptic.
