Collaborative Research: Stochastic Functional Systems: Analysis, Algorithms and Applications

The time evolution of many physical, biological, and engineering systems is described by functional differential equations, where the future state of the system is not only determined by its present state, but also by the state of the system at some prior time(s). Examples can be found in the study of epidemic and ecological models, multi-agent models in financial systems, neural network models, and other areas in statistics, data science, and engineering. Among the various modeling approaches in existence, stochastic functional differential equations (SFDE) and McKean-Vlasov stochastic functional differential equations (MVSFDE) play a crucial role in modeling complex systems across science and engineering. Despite extensive research, many questions about these systems remain unresolved due to their challenging past-dependent nature. At the same time, a growing interest in functional stochastic approximation algorithms (FSAA) has emerged from new problems in optimization, data science, and machine learning. This project aims to systematically investigate these systems to establish their critical properties, broaden current applications, and discover new applications in science, machine learning, and engineering. In addition, this project will provide research opportunities for graduate students, engage high school students through math tournaments, and work towards creating a network of academia, students, and industry representatives to enhance career opportunities for students and increase public awareness of the role of mathematics in real-world applications. This project aims to (i) explore long-term properties, such as ergodicity and stability, of SFDE; (ii) formulate a new approach for MVSFDE to systematically examine their fundamental properties and long-term behaviors; and (iii) propose a framework for FSAA dealing with discontinuous operators, establish convergence conditions and rates, and provide implementation methods. The project will apply these theories to address specific problems in ecology, infectious diseases, control engineering, networked systems, neutral network models, game theory, and cell biology, as well as emerging problems in statistics, data science, and engineering. To achieve these goals, the research will integrate Dupire's functional Itô's formula, inventive concepts of generalized coupling, and will bridge stochastic calculus and non-smooth analysis in infinite-dimensional spaces, in addition to employing other advanced techniques. 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.