Intertwining ideas for some problems in probability

随机模型在理解自然科学、社会科学和工程科学中的复杂现象方面起着至关重要的作用。获得关于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的问题解决策略至关重要。该项目的目的是为最近在数学、物理和金融等各个领域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，这是容易分析和容易理解的，和一系列复杂的随机模型之间的联系。这种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，分类方案可以连接不同现象的模型。事实上，一个类中的所有随机模型都是由一组被称为频谱的点连接起来的，这与动力学结构无关。除了理论发展之外，该项目还包括计算部分：其目标是开发精确而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与其中，在一个包容的学习环境中，学生可以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机会。通过将理论见解与实际计算方法相结合，本项目力求提高对希尔伯特空间上一般马尔可夫半群的理解和适用性。它包含三个主要目标，可以描述如下：首先，开发一种新的方法来表征希尔伯特空间上马尔可夫半群的不同等谱轨道，包括幺正轨道、交织轨道、交织轨道和弱相似轨道。这种方法的目的是提供对这些半群所显示的各种轨道的全面了解。其次，利用上述分类方案来识别可从参考半群转移到其相应轨道的解析性、遍历性和混合性。特别的重点将放在研究马尔可夫过程驻留在欧几里得空间和Weyl室的子集，以进行一个深入的动态决定点过程的分析的目标。第三，使用本项目开发的分类方案来设计精确的算法来模拟这些动态。该奖项反映了美国国家科学基金会的法定使命，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。

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.