具有遍历分支的Hamilton流的平均
Averaging of Hamiltonian flows with an ergodic component
摘要：
考虑$\mathbb{T}^2$上的一个过程，它由不可压缩周期向量场在白噪声扰动下沿流线的快速运动组成。它在图上产生一个过程，该过程自然地与未扰动流的流线的结构相关联。Freidlin和Wentzell [Random Perturbations of动力系统，2nd ed.Springer，New York（1998）]和[Mem.Amer.Math.Soc.109（1994）]已经表明，如果流的流函数是周期性的，则图上的相应过程弱收敛于马尔可夫过程。我们考虑的情况下，流函数不是周期性的，和流（当考虑的环面）有一个遍历组件的正措施。我们表明，如果旋转数是丢番图，然后在图上的过程仍然收敛到一个马尔可夫过程，花了一个积极的时间比例的顶点对应的遍历组件的流量。
英文摘要：
We consider a process on $\mathbb{T}^2$, which consists of fast motion along the stream lines of an incompressible periodic vector field perturbed by white noise. It gives rise to a process on the graph naturally associated to the structure of the stream lines of the unperturbed flow. It has been shown by Freidlin and Wentzell [Random Perturbations of Dynamical Systems, 2nd ed. Springer, New York (1998)] and [Mem. Amer. Math. Soc. 109 (1994)] that if the stream function of the flow is periodic, then the corresponding process on the graph weakly converges to a Markov process. We consider the situation where the stream function is not periodic, and the flow (when considered on the torus) has an ergodic component of positive measure. We show that if the rotation number is Diophantine, then the process on the graph still converges to a Markov process, which spends a positive proportion of time in the vertex corresponding to the ergodic component of the flow.
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