Large Deviation Properties of the Empirical Measure of a Metastable Small Noise Diffusion
Large Deviation Properties of the Empirical Measure of a Metastable Small Noise Diffusion
复制标题
亚稳态小噪声扩散经验测量的大偏差特性
DOI：
10.1007/s10959-020-01072-3
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发表时间：
2022
影响因子：
0.8
通讯作者：
Wu, Guo-Jhen
中科院分区：
文献类型：
--
作者：
Dupuis, Paul;Wu, Guo-Jhen
本文的目的是为小噪声扩散的经验度量建立易于处理的大偏差近似。本文的出发点是Freidlin-Wentzell理论，它展示了如何通过大偏差原理来近似这种扩散的不变分布。不变度量的速率函数是根据准势来表示的，准势是衡量从一个亚稳态集到另一个亚稳态集的邻域转换的难度的量。该理论为不变测量提供了一个直观和有用的近似，同时也发展了许多有用的相关结果(例如亚稳态之间的跃迁速率)。考虑到蒙特卡罗格式设计的具体目标，我们证明了积分相对于经验度量的大偏差极限，其中该过程被考虑在一个时间间隔上，该时间间隔的长度随着噪声减小到零而增长。特别地，我们展示了这些积分的一阶矩和二阶矩是如何用准势表示的。当过程的动态依赖于参数时，这些近似可以用于算法设计，并且这种类型的应用将出现在其他地方。使用较小的噪声限制是有良好动机的，因为在这个限制下，状态空间的良好采样变得最具挑战性。该证明利用了一种再生结构，需要一些新的技术来将再生周期中的大偏差估计转化为对经验度量及其矩的估计。
The aim of this paper is to develop tractable large deviation approximations for the empirical measure of a small noise diffusion. The starting point is the Freidlin–Wentzell theory, which shows how to approximate via a large deviation principle the invariant distribution of such a diffusion. The rate function of the invariant measure is formulated in terms of quasipotentials, quantities that measure the difficulty of a transition from the neighborhood of one metastable set to another. The theory provides an intuitive and useful approximation for the invariant measure, and along the way many useful related results (e.g., transition rates between metastable states) are also developed. With the specific goal of design of Monte Carlo schemes in mind, we prove large deviation limits for integrals with respect to the empirical measure, where the process is considered over a time interval whose length grows as the noise decreases to zero. In particular, we show how the first and second moments of these integrals can be expressed in terms of quasipotentials. When the dynamics of the process depend on parameters, these approximations can be used for algorithm design, and applications of this sort will appear elsewhere. The use of a small noise limit is well motivated, since in this limit good sampling of the state space becomes most challenging. The proof exploits a regenerative structure, and a number of new techniques are needed to turn large deviation estimates over a regenerative cycle into estimates for the empirical measure and its moments.
DOI：
10.1080/17442508308833244
发表时间：
1983
期刊：
Stochastics An International Journal of Probability and Stochastic Processes
影响因子：
--
作者：
M. Day
通讯作者：
M. Day
影响因子：
1.6
作者：
Dupuis, Paul;Liu, Yufei;Doll, J. D.
通讯作者：
Doll, J. D.
平行回火和无限交换算法某些定性性质的大偏差分析
影响因子：
1.8
作者：
J. D. Doll;P. Dupuis;P. Nyquist
通讯作者：
P. Nyquist