Intertwining ideas for some problems in probability

一些概率问题的相互交织的想法

• 批准号: 2246766
• 负责人: Pierre Patie
• 金额: $ 30万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-15 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246766&HistoricalAwards=false
• 关键词: Intertwining; ideas; some; problems; probability

Stochastic models play a vital role in understanding complex phenomena that occur in the natural, social, and engineering sciences. Obtaining comprehensive and accurate information about these models is crucial for gaining a systematic understanding of the modeled system and for designing effective problem-solving strategies. The objective of this project is to provide fresh perspectives in the study of recently proposed models in various areas of mathematical physics and finance. The underlying concept is to establish a connection between a simple stochastic dynamic, which is easily analyzable and well-understood, and a family of complex stochastic models. This connection allows for the transfer of fundamental properties from the reference model to the entire family. To achieve this, the project aims to deepen the understanding of classification schemes can connect models of disparate phenomena. In fact, all stochastic models within a class are linked solely by a set of points known as the spectrum, which remains independent of the dynamics' structure. In addition to theoretical development, this project also incorporates a computational component: its goal is to develop precise and efficient numerical schemes for simulating such complex models. Both undergraduate and graduate students will participate, in an inclusive learning environment where students can contribute and gain valuable experience. The awardee will also organize conferences, facilitating opportunities for scholars and researchers to collaborate.By combining theoretical insights with practical computational methods, this project strives to advance the understanding and applicability of general Markov semigroups on Hilbert spaces. It encompasses three primary objectives, which can be described as follows: first, to deevelop a novel methodology to characterize different isospectral orbits, including unitary, intertwining, interweaving, and weak similarity orbits, of Markov semigroups on Hilbert spaces. This methodology aims to provide a comprehensive understanding of the various orbits exhibited by these semigroups. Second to utilize the aforementioned classification schemes to identify analytical, ergodic, and mixing properties that can be transferred from the reference semigroup to its corresponding orbit. Particular emphasis will be placed on studying Markov processes residing in subsets of Euclidean space and Weyl chambers, with the goal of conducting an in-depth analysis of dynamical determinantal point processes. Third, to use the classification schemes developed in this project to design precise and exact algorithms for simulating these dynamics.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.

随机模型在理解自然、社会和工程科学中发生的复杂现象方面发挥着至关重要的作用。获得关于这些模型的全面而准确的信息对于系统地理解建模系统和设计有效的问题解决策略至关重要。该项目的目标是为数学物理和金融各个领域最近提出的模型的研究提供新的视角。其基本概念是建立一个简单的随机动态，这是很容易分析和理解，和一个家庭的复杂的随机模型之间的连接。这种连接允许将基本属性从参照模型传递到整个族。为了实现这一目标，该项目旨在加深对分类方案的理解，这些方案可以将不同现象的模型联系起来。事实上，一个类中的所有随机模型仅由一组称为谱的点连接，而谱与动态结构无关。除了理论发展，该项目还包括一个计算部分：其目标是开发精确和有效的数值方案来模拟这种复杂的模型。本科生和研究生都将参加，在一个包容性的学习环境中，学生可以做出贡献，并获得宝贵的经验。获奖者还将组织会议，为学者和研究人员提供合作的机会。通过将理论见解与实际计算方法相结合，该项目致力于提高对Hilbert空间上一般Markov半群的理解和适用性。它包括三个主要目标，可以描述如下：第一，发展一种新的方法来表征不同的等谱轨道，包括酉，交织，交织，弱相似轨道，马尔可夫半群在Hilbert空间。这种方法的目的是提供一个全面的了解，这些半群所表现出的各种轨道。第二，利用上述分类方案，以确定分析，遍历，和混合性质，可以从参考半群转移到其相应的轨道。特别强调将放在研究马尔可夫过程居住在欧几里得空间和Weyl室的子集，进行深入分析的动态决定点过程的目标。第三，使用本项目开发的分类方案来设计用于模拟这些动力学的精确算法。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。