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Large Deviation Properties of the Empirical Measure of a Metastable Small Noise Diffusion
Journal of Theoretical Probability ( IF 0.8 ) Pub Date : 2021-01-29 , DOI: 10.1007/s10959-020-01072-3
Paul Dupuis , Guo-Jhen Wu

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.



中文翻译:

亚稳态小噪声扩散经验测度的大偏差性质

本文的目的是为小噪声扩散的经验度量开发可处理的大偏差近似值。起点是Freidlin-Wentzell理论,该理论显示了如何通过大偏差原理近似估计这种扩散的不变分布。不变量度的速率函数是用准势表示的,准量是衡量从一个亚稳态集的邻域到另一个亚稳态集过渡的难度的量。该理论为不变性测度提供了一种直观且有用的近似值,并且在此过程中,还开发了许多有用的相关结果(例如,亚稳态之间的跃迁速率)。考虑到设计蒙特卡洛方案的特定目标,我们证明了相对于经验测度的积分的大偏差极限,在一个时间间隔内考虑该过程,该时间间隔的长度随着噪声减小到零而增加。特别是,我们展示了如何用准势来表示这些积分的第一和第二矩。当过程的动力学取决于参数时,这些近似值可用于算法设计,并且此类应用程序将出现在其他地方。最好使用较小的噪声限制,因为在此限制下,状态空间的良好采样变得最具挑战性。证明采用了一种再生结构,需要许多新技术才能将再生周期内的大偏差估计值转换为经验测度及其矩的估计值。我们展示了这些积分的第一和第二矩如何用准势来表示。当过程的动力学取决于参数时,这些近似值可用于算法设计,并且此类应用程序将出现在其他地方。最好使用较小的噪声限制,因为在此限制下,状态空间的良好采样变得最具挑战性。证明采用了一种再生结构,需要许多新技术才能将再生周期内的大偏差估计值转换为经验测度及其矩的估计值。我们展示了这些积分的第一和第二矩如何用准势来表示。当过程的动力学取决于参数时,这些近似值可用于算法设计,并且此类应用程序将出现在其他地方。最好使用较小的噪声限制,因为在此限制下,状态空间的良好采样变得最具挑战性。证明采用了一种再生结构,需要许多新技术才能将再生周期内的大偏差估计值转换为经验测度及其矩的估计值。最好使用较小的噪声限制,因为在此限制下,状态空间的良好采样变得最具挑战性。证明采用了一种再生结构,需要许多新技术才能将再生周期内的大偏差估计值转换为经验测度及其矩的估计值。最好使用较小的噪声限制,因为在此限制下,状态空间的良好采样变得最具挑战性。证明采用了一种再生结构,需要许多新技术才能将再生周期内的大偏差估计值转换为经验测度及其矩的估计值。

更新日期:2021-01-29
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