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

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.