Problems at the Interface of Stochastics and Analysis

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Many phenomena that arise in statistical physics, engineering and biology are modeled by stochastic processes that are constrained to live within a domain. This proposal aims to further develop the theory of such processes, with three concrete application areas in mind. The first area concerns random networks that arise in biology, manufacturing, and other service systems when they operate near capacity. The performance of these networks can often be described by diffusions that are constrained to have nonnegative components. A second area is in mathematical finance, where the advent of electronic exchanges driven purely by the flow of orders has revolutionized the method by which prices are formed. The price process in a model of strategic agents who place buy and sell limit orders can be better understood by studying a class of constrained processes. The third area involves the study of scaling limits of random matrices, which arise in many areas, including physics and engineering. The gaps between the eigenvalues of some classes of high-dimensional random matrices, when properly scaled, can be shown to be approximated by constrained multi-dimensional diffusions with singular drift. The proposal seeks to develop a unified theory for the construction and study of these processes, and to examine their implications for the described applications. Another theme of the proposal involves the study of planar obliquely reflected diffusions. Planar stochastic processes have been the focus of active research over the last two decades. Finally, the proposal also has a substantial educational component that includes training of post-doctoral fellows, graduate students and undergraduate students, as well as new course development. It also entails a broader effort that coordinates several graduate students in outreach activities aimed at communicating mathematics to a broader audience.This is an interdisciplinary proposal that focuses on several problems in probability that require substantial use of analytical techniques. The first theme concerns various aspects of obliquely reflected diffusions, including the construction and properties of obliquely reflected Brownian motions in bounded planar domains, and also excursion reflected Brownian motion which arises in the boundary theory of Markov processes. It also involves the development of a common framework for the analysis of diffusions with both reflection and singular drift, large deviations of semimartingale reflected Brownian motions and a free boundary problem related to a two-dimensional reflected Brownian motion. These are motivated by applications in queuing networks, biology, and mathematical finance. The tools that will be used intersect with several areas of mathematics including analysis (in particular, complex analysis, conformal mappings, harmonic analysis, functional analysis, partial differential equations and free-boundary problems). There are also implications of this work for mathematical physics, specifically the study of repulsive particle models associated with random matrices.