Estimating Probabilities of Rare Events in Interacting Particle Systems

稀有事件的概率估计在许多不同领域都是一个重要的课题。例如，极端结果的风险量化是金融、经济、环境科学、物理学和工程学中许多问题的核心关注点。研究此类问题的数学框架由大偏差理论给出，该理论涉及具有不确定性的系统偏离其标称期望行为的概率衰减率的表征。在过去的二十年中，一种新的方法来研究大偏差问题，这使得承担技术从随机控制理论，已成为突出。本研究的目标是开发一个系统的框架，随机控制思想的基础上，在相互作用的粒子系统的研究中出现的一类重要的和具有挑战性的大偏差问题。这里考虑的相互作用粒子系统是由自然科学中的现象激发的。一些例子包括生物粒子的趋化性模型，生态模型产生的反应扩散系统，以及化学晶体生长模型。除了发展提供稀有事件概率衰减率渐近界的数学机制外，本研究将发展加速的蒙特-卡罗方法，利用重要性抽样方法估算感兴趣的概率，该方法是受基础系统的大偏差分析的启发。（B）Patlak-Keller-Segel方程的微观粒子模型;（C）反应扩散方程的布朗粒子系统;（D）局部相互作用跳跃扩散。在主题（A）下，重点将放在粒子数量变大和公共噪声强度变小时的大偏差渐近性上。目标是表征两个参数以不同的相对速率接近极限时不同形式的大偏差行为。题目（B）中的Patlak-Keller-Segal方程是一种非线性非局部偏微分方程，它模拟了生物粒子的主动趋化性。相互作用的扩散粒子系统，是完全耦合的基础化学场的演变已被用来给一个介观描述的现象。这里提出的研究目标是研究大偏差问题，旨在了解这种粒子系统的长期行为。特别是，这项工作将研究不变测度渐近性和亚稳态行为附近的休息点。主题（C）研究反应扩散型质点系的大偏差行为。特别感兴趣的是零化布朗粒子近似反应扩散方程的多项式反应项，其中的相互作用成为奇异的限制。最后，主题（D）是关于离散格点上的局部相互作用粒子系统，具有适当的时间和空间尺度。这样的系统例如作为化学中的晶体生长模型而出现。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Estimation of probabilities of rare events is an important topic in many different areas. For example, quantification of risk of extreme outcomes is of central concern in many problems in finance, economics, environmental science, geophysics, and engineering. The mathematical framework for studying such problems is given by the theory of large deviations which is concerned with the characterization of decay rate of probabilities of deviations of a system with uncertainties from its nominal expected behavior. In the last twenty years a new approach to the study of large deviations problems, that brings to bear techniques from the theory of stochastic control, has become prominent. The goal of this research is to develop a systematic framework based on stochastic control ideas for an important and challenging class of large deviation problems that arise in the study of interacting particle systems. Interacting particle systems considered here are motivated by phenomena in Natural Sciences. Some examples include models for chemotaxis of biological particles, reaction-diffusion systems arising from ecological models, and crystal growth models from chemistry. In addition to developing the mathematical machinery for providing asymptotic bounds on decay rate of rare event probabilities, this work will develop accelerated Monte-Carlo methods for estimating probabilities of interest using methods of importance sampling that are inspired by a large deviation analysis of the underlying systems.This project will study four different families of particle systems: (A) Weakly interacting diffusions with a small common noise; (B) Microscopic particle models for Patlak-Keller-Segel equations; (C) Brownian particle systems for reaction-diffusion equations; (D) Locally interacting jump-diffusions. Under topic (A), the focus will be on large deviation asymptotics as the number of particles becomes large and the intensity of the common noise becomes small. The goal is to characterize different forms of large deviation behavior as the two parameters approach limits at different relative rates. Patlak-Keller-Segal equations in topic (B) are nonlinear non-local PDE that model active chemotaxis of biological particles. Interacting diffusive particle systems that are fully coupled with the evolution of the underlying chemical field have been used to give a mesoscopic description of the phenomenon. Goal of the proposed research here is to study large deviation problems aimed at understanding the long time behavior of such particle systems. In particular, this work will study invariant measure asymptotics and metastability behavior near a rest point. Under topic (C), large deviations behavior of particle systems of reaction-diffusion type will be studied. Of particular interest are annihilating Brownian particles approximating reaction diffusion equations with a polynomial reaction term in which the interaction becomes singular in the limit. Finally topic (D) is concerned with locally interacting particle systems on discrete lattices with a suitable temporal and spatial scaling. Such systems arise, for example, as crystal growth models in chemistry. The goal is to develop stochastic control methods for studying large deviation properties of such systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.