论文标题
通过Freidlin-Wentzell大偏差原理的弱KAM解决方案的选择原理
A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures
论文作者
论文摘要
本文从弱的KAM角度重新诠释了Freidlin-Wentzell在大偏差原理中的速率函数的变异构建。通过在圆环上的一维不可逆扩散过程，我们明确地表征了弱KAM理论中的基本概念，例如PEIERLS屏障和预计的Mather/Aubry/Mañé集。弗雷德林·韦兹尔（Freidlin-Wentzell）的差异量较弱的KAM表示，根据对Aubry集合的边界数据的全局调整以及从提起的PEIERLS屏障的局部修剪讨论了速率函数的变化构建。此速率函数可为相应的固定汉密尔顿 - 雅各比方程（HJE）提供最大的Lipschitz连续粘度解决方案，从而满足了边界数据的Freidlin-Wentzell的变化公式。在每个本地吸引子上选择有意义的自洽边界数据对于为固定HJE选择独特的弱KAM解决方案至关重要。该选定的粘度解决方案也充当原始随机过程的全球能量局势。可以通过首先采取长时间限制然后采取零噪声限制来描述固定HJES的这种选择，这也提供了消失的粘度近似的特殊结构。
This paper reinterprets Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. Through a one-dimensional irreversible diffusion process on a torus, we explicitly characterize essential concepts in the weak KAM theory, such as the Peierls barrier and the projected Mather/Aubry/Mañé sets. The weak KAM representation of Freidlin-Wentzell's variational construction of the rate function is discussed based on the global adjustment for the boundary data on the Aubry set and the local trimming from the lifted Peierls barriers. This rate function gives the maximal Lipschitz continuous viscosity solution to the corresponding stationary Hamilton-Jacobi equation (HJE), satisfying Freidlin-Wentzell's variational formula for the boundary data. Choosing meaningful self-consistent boundary data at each local attractor are essential to select a unique weak KAM solution to stationary HJE. This selected viscosity solution also serves as the global energy landscape of the original stochastic process. This selection for stationary HJEs can be described by first taking the long time limit and then taking the zero noise limit, which also provides a special construction of vanishing viscosity approximation.
