Stochastic Models with Random Times: Long-Time Behavior and Large Population Limit

大型随机系统和其他相互作用的过程在科学发现和决策中越来越突出。各个学科的科学家都可以前所未有地访问大量数据，这些数据取决于具有复杂相互作用的结构。决策者还需要优化社会福利，这涉及到大量充满随机性的人口。传统的低维自然模型不再适合这些科学问题，并作为决策的基础。该项目将通过开发数学和计算工具来分析复杂的相互作用系统，从而解决这些问题，这些系统将对公共卫生、经济和科学产生深远的影响。该项目将侧重于开发数学理论和算法方面的几项创新，这将为不同学科中出现的基础科学和政策问题提供信息。研究结果将在不同的科学和社会群体中广泛传播。本计画将提供研究生训练机会，研究随机过程的长时间行为与大族群极限。该项目将涉及三个具体专题。第一个主题将关注涉及命中时间的随机模型，以了解平均场极限的长时间行为，并设计一个最佳策略来控制大型复杂系统。主要的工具，调查员计划开发将从概率论和偏微分方程。第二个主题将涉及加速梯度方法从鞍点的非凸高维目标函数逃逸。第三个主题将研究一些概率排序模型的灵敏度时，观察的数量是大的。在第二和第三个主题中，研究人员计划开发概率论和组合学的工具。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

Large stochastic systems and other interacting processes are increasingly conspicuous in scientific discoveries and decision making. Scientists in a variety of disciplines have unprecedented access to massive data which hinge on structures with complex interactions. Decision makers also need to optimize social welfare involving a large population full of randomness. Traditional models of low dimensional nature are no longer adequate for these scientific problems and as a basis for decision making. The project will address these problems by developing mathematical and computational tools for analyzing complex interacting systems that will have far-reaching public health, economic and scientific implications. The project will focus on developing several innovations in mathematical theory and algorithms, which will inform basic science and policy questions arising in diverse disciplines. The results will be disseminated broadly across diverse scientific and social communities. The project will provide training opportunities for graduate students.The project will investigate long-time behavior and large population limit of stochastic processes. The project will address three specific topics. The first topic will concern stochastic models involving hitting times to understand the long-time behavior on the mean-field limit and design an optimal strategy to control the large complex system. The main tools that the investigator plans to develop will be from probability theory and partial differential equations. The second topic will involve accelerating gradient methods for escaping from saddle points of a non-convex high-dimensional objective function. The third topic will investigate the sensitivity of some probabilistic ranking models when the number of observations is large. In both the second and third topics, the investigator plans to develop tools from probability theory and combinatorics.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.