Problems at the Interface of Stochastics and Analysis

统计物理、工程和生物学中出现的许多现象都是由随机过程建模的，这些随机过程被限制在一个领域内。这项提议旨在进一步发展这种过程的理论，并考虑到三个具体的应用领域。第一个领域涉及生物、制造业和其他服务系统在接近产能时出现的随机网络。这些网络的性能通常可以用被约束为具有非负分量的扩散来描述。第二个领域是数学金融，在这个领域，纯粹由订单流动驱动的电子交易所的出现，彻底改变了价格的形成方法。通过研究一类受约束的过程，可以更好地理解下达限购限价订单的战略代理人模型中的价格过程。第三个领域涉及对随机矩阵的比例极限的研究，这在包括物理和工程在内的许多领域中都存在。某些类型的高维随机矩阵的特征值之间的间隔，当适当缩放时，可以用具有奇异漂移的约束多维扩散来近似。该提案力求为这些进程的建设和研究发展一个统一的理论，并审查其对所述应用的影响。该提案的另一个主题涉及平面斜反射扩散的研究。在过去的二十年里，平面随机过程一直是研究的热点。最后，该提案还包括大量的教育内容，包括博士后研究员、研究生和本科生的培训以及新课程的开发。它还需要更广泛的努力，在旨在向更广泛的受众交流数学的外展活动中协调几名研究生。这是一项跨学科的建议，侧重于几个需要大量使用分析技术的概率问题。第一个主题涉及斜反射扩散的各个方面，包括有界平面区域中斜反射布朗运动的结构和性质，以及马尔可夫过程边界理论中出现的游程反射布朗运动。它还涉及建立一个共同的框架，用于分析同时具有反射和奇异漂移的扩散、半鞅反射布朗运动的大偏差以及与二维反射布朗运动有关的自由边界问题。这些都是由排队网络、生物学和数学金融中的应用程序推动的。将使用的工具与数学的几个领域交叉，包括分析(特别是复数分析、保角映射、调和分析、泛函分析、偏微分方程式和自由边界问题)。这项工作对数学物理也有影响，特别是与随机矩阵相关的排斥粒子模型的研究。

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.