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中起着至关重要的作用。获得有关这些模型的全面和准确的信息对于获得对建模系统的系统理解和设计有效解决问题的策略至关重要。该项目的目的是在对数学物理和金融各个领域的最近提出的模型的研究中提供新的观点。潜在的概念是建立一个简单的随机动态（易于分析且易于理解）与复杂随机模型的家族之间的联系。该连接允许将基本属性从参考模型转移到整个家庭。为了实现这一目标，该项目旨在加深对分类方案的理解，可以连接不同现象的模型。实际上，一类中的所有随机模型仅由一组称为频谱的点链接，该点仍然独立于动力学结构。除了理论开发外，该项目还结合了一个计算组件：其目标是开发精确有效的数值方案，以模拟此类复杂模型。本科生和研究生都将参加包容性学习环境，学生可以贡献并获得宝贵的经验。获奖者还将组织会议，促进学者和研究人员合作的机会。通过将理论见解与实用的计算方法相结合，该项目致力于促进Markov Semigroup对Hilbert Space的理解和适用性。它包含三个主要目标，可以描述如下：首先，挖掘一种新的方法，以表征马尔伯特（Hilbert）在希尔伯特（Hilbert）空间上的马尔可夫半群岛（Markov semigroup）的不同等光轨道，包括单一，交织，交织，交织和弱相似轨道。该方法旨在提供对这些半群展示的各种轨道的全面理解。其次利用上述分类方案来识别可以从参考半群转移到其相应轨道的分析，千古和混合特性。将特别强调研究位于欧几里得空间和Weyl室子集中的马尔可夫过程，目的是对动态确定点过程进行深入分析。第三，使用该项目中开发的分类方案来设计精确和确切的算法来模拟这些动态。该奖项反映了NSF的法定任务，并使用基金会的智力优点和更广泛的影响评估标准，认为值得通过评估来获得支持。