On the exit behaviors of non-Gaussian stochastic dynamical systems based on large deviation theory
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摘要: 本文介绍了大偏差理论的基本思想及其在非高斯随机动力系统的离出问题研究中的应用. 依据不同的非高斯噪声类型, 本文分别评述了随机混合系统、指数轻跳跃过程和
$\alpha $ 稳定Lévy噪声驱动的随机动力系统的离出问题的主要研究方法和近期研究进展. 针对随机混合系统, 本文介绍了利用随机微分方程对其进行近似的拟稳态扩散近似方法, 计算拟势和最优离出路径的WKB近似方法与细致平衡条件的研究, 以及求解随机混合系统的简化版本 (即生灭过程) 的离出问题的研究进展. 对于指数轻跳跃过程驱动的随机动力系统, 本文介绍了其大偏差原理和中度偏差原理的泛函极值问题的建立, 拟势概念的定义和平均离出时间的估计. 针对具有$\alpha $ 稳定Lévy噪声的随机动力系统, 本文介绍了计算平均首次离出时间和离出概率的理论和数值方法, 计算最优离出路径的Onsager-Machlup理论、机器学习方法、最大似然法和数据驱动方法. 最后, 给出了非高斯随机动力系统的离出现象相关的一些开放性问题.Abstract: This paper introduces the basic ideas of large deviation theory and its applications in the study of exit problems of non-Gaussian stochastic dynamical systems. According to different types of non-Gaussian noise, the main research methods and recent progresses of exit problems are reviewed for stochastic hybrid systems, stochastic dynamical systems with exponentially light jump fluctuations, and stochastic systems with$\alpha $ -stable Lévy noises. For the stochastic hybrid systems, the quasi-steady-state diffusion approximation which is approximated by stochastic differential equations, the WKB approximation for computing quasi-potential and optimal exit paths, the research on detailed balance conditions, and progresses in exit problems of the simplified version of stochastic hybrid systems (i.e. birth-and-death processes) are introduced. For the stochastic dynamical systems driven by the exponential light jump processes, the establishment of the functional extremum problems of large deviation principle and moderate deviation principle, the definition of the quasi-potential concept and the estimation of the mean exit time are discussed. For stochastic systems with stable Lévy noises, the theoretical and numerical methods for calculating the mean exit time and exit probability, and Onsager-Machlup theory, machine learning method, maximum likelihood method and data-driven method for computing the optimal exit paths are illustrated. Finally, some open problems related to the exit phenomena of non-Gaussian stochastic dynamical systems are given. -
图 1 平均场方程关于
$\theta $ 从单稳态到双稳态的分岔图.$\theta = 0.95$ 时, 存在两个稳定不动点${x_1}$ ,${x_2}$ 和一个不稳定不动点${x_0}$ 
图 2 哈密顿系统的相图和离出路径. 粉色和亮蓝曲线分别代表从
${x_1}$ 到${x_2}$ 和${x_2}$ 到${x_1}$ 的离出路径. 绿色和棕色虚线表示扩散近似的离出路径
图 5 具有
$L$ 层隐藏层的神经网络的结构.${x_{fi}}$ 和${\lambda _i}$ ,$i = 1,\;2,\; \cdots ,\;n$ 分别是神经网络的输入和输出.$a_j^{\left( l \right)}$ 表示第$l$ 层第$j$ 个神经元的值, 其中$j = 1,\;2,\; \cdots ,\;{n_l}$ ,$l = 1,\;2,\;\cdots,\;L$ 表 1 几种典型随机过程的细致平衡条件
随机过程 细致平衡条件 文献 扩散过程 加性高斯、
有势系统满足细致平衡 朱位秋 (1992) 一般情况 $\begin{array}{c} {b_i}\left( {\boldsymbol{x} } \right){p_s}\left( {\boldsymbol{x} } \right) + {\varepsilon _i}{b_i}\left( {\varepsilon {\boldsymbol{x} } } \right){p_s}\left( {\boldsymbol{x} } \right) - \displaystyle\sum\limits_j {\frac{\partial }{ {\partial {x_j} } }\left[ { {a_{ij} }\left( {\boldsymbol{x} } \right){p_s}\left( {\boldsymbol{x} } \right)} \right]} = 0 \\ {\varepsilon _i}{\varepsilon _j}{a_{ij} }\left( {\varepsilon{\boldsymbol{ x} } } \right) - {a_{ij} }\left( {\boldsymbol{x} } \right) = 0 \end{array}$
($b$为漂移系数, $a$为扩散矩阵, ${p_s}$为平稳概率分布,
${\varepsilon _i} = \pm 1$依赖于变量奇偶性)跳跃Markov过程 ${W_{ji}}{P_i} = {W_{ij}}{P_j}$
(${W_{ij}}$为转移率, ${P_i}$为平稳概率)Dykman 等 (1994) 随机混合系统 不满足细致平衡 Li 和 Liu (2019) 
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