CAREER: Conditional Theory of Large-Scale Stochastic Systems

当代科学、工程和技术中的问题越来越需要对高度复杂的系统进行分析，这些系统既具有高维随机动力学或相互作用，又具有大量的观测数据。为了在这类系统中获得可靠的预测，必须综合利用大规模随机模型和观测数据。这项建议的目标是启动一项系统的研究，研究观测数据的条件如何影响大规模随机模型的性质，如相互作用的粒子系统、随机偏微分方程组和马尔可夫随机场。研究将集中于发展无限维马尔可夫过程和条件无限吉布斯测度的条件遍历理论的基础；研究诸如条件相变之类的概率现象；发展与测度论、统计力学和高维概率中的问题的联系；以及潜在的应用于设计和分析用于高维系统中过滤和预测的蒙特卡罗算法，其中经典方法是无效的。大规模预报问题出现在无数重要的应用中，例如天气预报、地球物理和海洋数据同化、图像分析、交通预报和网络预报。这些问题直接影响我们的日常生活，并出现在我们社会的关键领域，如国家安全、能源资源管理、气候预测和医学成像。这个项目的广泛目标是对复杂模型、随机性和处于任何预测问题核心的观测数据之间的相互作用形成系统的理解。通过专注于各种应用程序共同的基本结构，数学家可以为复杂问题提供独特的见解和新的方向，并为发展跨学科联系提供动力。与此同时，强大的数学科学劳动力对未来的技术创新和教育至关重要。该项目的一个组成部分是一系列教育、指导和推广活动，旨在提高学生对大学预科、本科生和研究生数学科学的兴趣和多样性，并培训下一代研究人员和教育工作者。

Contemporary problems in science, engineering and technology increasingly demand the analysis of highly complex systems that feature both high-dimensional random dynamics or interactions and a large amount of observed data. In order to obtain reliable predictions in such systems, it is essential to exploit large-scale stochastic models and observed data in an integrated fashion. The goal of this proposal is to initiate a systematic study of how conditioning on observed data affects the properties of large-scale stochastic models such as interacting particle systems, stochastic partial differential equations, and Markov random fields. Research will focus on developing the foundations of a conditional ergodic theory for infinite-dimensional Markov processes and of conditional infinite Gibbs measures; on the investigation of probabilistic phenomena such as conditional phase transitions; on developing connections with problems in measure theory, statistical mechanics, and high-dimensional probability; and on potential applications to the design and analysis of Monte Carlo algorithms for filtering and prediction in high-dimensional systems, where classical methods are known to fail.Large-scale forecasting problems arise in a myriad of important applications such as weather forecasting, geophysical and oceanographic data assimilation, image analysis, traffic forecasting, and prediction in networks. Such problems have a direct impact on our daily lives, and arise in crucial areas of our society such as national security, energy resource management, climate prediction, and medical imaging. The broad goal of this project is to develop a systematic understanding of the interplay between complex models, randomness, and observed data that lies at the heart of any forecasting problem. By focusing on the fundamental structures that are common to a diverse range of applications, mathematicians can provide unique insights and new directions to complex problems and provide an impetus for developing interdisciplinary connections. At the same time, a strong workforce in the mathematical sciences is of crucial importance to the future of technological innovation and education. An integral part of this project is formed by a range of educational, mentoring and outreach activities aimed at increasing student interest and diversity in the mathematical sciences across pre-college, undergraduate and graduate student levels, and at training the next generation of researchers and educators.