Problems at the Interface of Stochastics and Analysis

随机学与分析的交叉问题

• 批准号: 1407504
• 负责人: Kavita Ramanan
• 金额: $ 30.67万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1407504&HistoricalAwards=false
• 关键词: Problems; Interface; Stochastics; Analysis

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.

在统计物理学、工程学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域内。 该提案旨在进一步发展这种过程的理论，考虑到三个具体的应用领域。 第一个领域涉及生物学、制造业和其他服务系统中出现的随机网络，当它们接近满负荷运行时。 这些网络的性能通常可以通过限制为具有非负分量的扩散来描述。 第二个领域是数学金融，纯粹由订单流驱动的电子交易所的出现彻底改变了价格形成的方法。 通过研究一类约束过程，可以更好地理解在一个模型中的战略代理人谁购买和出售限价订单的价格过程。 第三个领域涉及随机矩阵的标度极限的研究，它出现在许多领域，包括物理和工程。 某些类别的高维随机矩阵的特征值之间的差距，适当缩放时，可以被证明是近似的约束多维扩散奇异漂移。 该提案旨在为这些过程的构建和研究开发一个统一的理论，并研究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。 平面随机过程是近二十年来研究的热点。 最后，该提案还有一个实质性的教育部分，包括培训博士后研究员、研究生和本科生，以及新课程的开发。 它还需要一个更广泛的努力，协调几个研究生在推广活动，旨在沟通数学更广泛的观众，这是一个跨学科的建议，侧重于几个问题的概率，需要大量使用分析技术。 第一个主题涉及斜反射扩散的各个方面，包括有界平面区域上斜反射布朗运动的构造和性质，以及在马尔可夫过程的边界理论中出现的偏移反射布朗运动。 它还涉及到一个共同的框架的发展与反射和奇异漂移，大偏差的半鞅反射布朗运动和自由边界问题的二维反射布朗运动的分析扩散。 这些是由排队网络，生物学和数学金融中的应用所激发的。 将使用的工具与数学的几个领域，包括分析（特别是复分析，保形映射，调和分析，泛函分析，偏微分方程和自由边界问题）相交。 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。

项目成果

A conditional limit theorem for high-dimensional ℓᵖ-spheres

高维α球的条件极限定理

• DOI: 10.1017/jpr.2018.71
• 发表时间: 2018
• 期刊: Journal of Applied Probability
• 影响因子: 1
• 作者: Kim, Steven S.;Ramanan, Kavita
• 通讯作者: Ramanan, Kavita