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Time-periodic measures, random periodic orbits, and the linear response for dissipative non-autonomous stochastic differential equations
Research in the Mathematical Sciences ( IF 1.2 ) Pub Date : 2021-06-22 , DOI: 10.1007/s40687-021-00256-5
Michał Branicki , Kenneth Uda

We consider a class of dissipative stochastic differential equations (SDE’s) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE’s to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations, and it is often very useful for sensitivity analysis, uncertainty quantification, and improving probabilistic predictions of nonlinear dynamical systems, especially in high dimensions. Here, we are concerned with the linear response to small perturbations in the case when the time-asymptotic probability measures are time-periodic. First, we establish sufficient conditions for the existence of stable random time-periodic orbits generated by the underlying SDE. Ergodicity of time-periodic probability measures supported on these random periodic orbits is subsequently discussed. Then, we derive the so-called fluctuation–dissipation relations which allow to describe the linear response of statistical observables to small perturbations away from the time-periodic ergodic regime in a manner which only exploits the unperturbed dynamics. The results are formulated in an abstract setting, but they apply to problems ranging from aspects of climate modelling, to molecular dynamics, to the study of approximation capacity of neural networks and robustness of their estimates.



中文翻译:

时间周期度量、随机周期轨道和耗散非自治随机微分方程的线性响应

我们考虑一类具有有限维时间周期系数的耗散随机微分方程 (SDE),以及由此类 SDE 引起的时间渐近概率度量对基本动力学的足够规则的小扰动的响应。了解这样的响应提供了一种系统的方法来研究统计可观测量响应扰动的变化,它通常对灵敏度分析、不确定性量化和改进非线性动力系统的概率预测非常有用,尤其是在高维中。在这里,我们关注在时间渐近概率度量是时间周期的情况下对小扰动的线性响应。第一的,我们为底层 SDE 产生的稳定随机时间周期轨道的存在建立了充分条件。随后讨论了这些随机周期轨道上支持的时间周期概率度量的遍历性。然后,我们推导出所谓的波动-耗散关系,该关系允许以仅利用未扰动动态的方式描述统计观测值对远离时间周期遍历机制的小扰动的线性响应。结果是在抽象的环境中制定的,但它们适用于气候建模、分子动力学、神经网络近似能力研究及其估计稳健性等问题。随后讨论了这些随机周期轨道上支持的时间周期概率度量的遍历性。然后,我们推导出所谓的波动-耗散关系,该关系允许以仅利用未扰动动态的方式描述统计观测值对远离时间周期遍历机制的小扰动的线性响应。结果是在抽象环境中制定的,但它们适用于气候建模、分子动力学、神经网络近似能力研究及其估计稳健性等问题。随后讨论了这些随机周期轨道上支持的时间周期概率度量的遍历性。然后,我们推导出所谓的波动-耗散关系,该关系允许以仅利用未扰动动态的方式描述统计观测值对远离时间周期遍历机制的小扰动的线性响应。结果是在抽象环境中制定的,但它们适用于气候建模、分子动力学、神经网络近似能力研究及其估计稳健性等问题。我们推导出所谓的波动-耗散关系,它允许以仅利用未扰动动态的方式描述统计观测值对远离时间周期遍历机制的小扰动的线性响应。结果是在抽象环境中制定的,但它们适用于气候建模、分子动力学、神经网络近似能力研究及其估计稳健性等问题。我们推导出所谓的波动-耗散关系,它允许以仅利用未扰动动态的方式描述统计观测值对远离时间周期遍历机制的小扰动的线性响应。结果是在抽象环境中制定的,但它们适用于气候建模、分子动力学、神经网络近似能力研究及其估计稳健性等问题。

更新日期:2021-06-23
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