Intertwining ideas for some problems in probability

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和准确的信息对于系统地理解所建模的系统和设计有效的问题解决策略至关重要。这个项目的目的是为最近提出的数学物理和金融领域的模型的研究提供新的视角。其基本概念是在易于分析和理解的简单随机动态与一族复杂的随机模型之间建立联系。此连接允许将基本属性从参照模型传递到整个族。为了实现这一点，该项目旨在加深对分类方案可以连接不同现象的模型的理解。事实上，一类中的所有随机模型都只由一组称为谱的点连接在一起，这与动力学的结构无关。除了理论上的发展，这个项目还包括一个计算部分：它的目标是开发精确和高效的数值格式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。本项目将理论见解与实际计算方法相结合，努力促进对希尔伯特空间上一般马尔可夫半群的理解和应用。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刻画Hilbert空间上马氏半群的不同等谱轨道，包括酉轨道、交织轨道、交织轨道和弱相似轨道。这种方法旨在提供对这些半群所表现出的各种轨道的全面了解。第二，利用前述分类方案来识别可以从参考半群转移到其对应轨道的分析、遍历和混合性质。将特别强调研究欧几里德空间和Weyl腔的子集中的马尔可夫过程，目的是对动态行列式点过程进行深入分析。第三，使用本项目中开发的分类方案来设计精确和精确的算法来模拟这些动态。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.