Stochastic Partial Differential Equations, Gauge Theories, and Scaling Limits

随机偏微分方程、规范理论和标度极限

• 批准号: 1954091
• 负责人: Hao Shen
• 金额: $ 19.85万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954091&HistoricalAwards=false
• 关键词: Stochastic; Partial; Differential; Equations; Gauge

A core problem of probability is to extract important laws and information from complex and large random systems. Such random systems may be stocks with fluctuating prices in financial markets, cancer cells generated by random mutations in biological tissues, a large number of sub-atomic quantum interaction particles, or how diseases randomly spread among people. This project focuses on two types of models: motion of a large number of elementary particles under the interaction of random quantum fields, and the growth of polymers under random interference environments. Due to the generality of mathematical models, the research results of these two types of models will also be of great help in studying other systems, such as systems in materials science, finance, and biology. The main mathematical tool to be used is called stochastic partial differential equations.More specifically, the theory of stochastic partial differential equations will be used to study stochastic quantization of gauge theories, which are fundamental models in elementary particle physics. The PI will prove well-posedness of such singular equations, show that the solutions have infinite dimensional gauge symmetries, and expand the scope of regularity structure theory. The PI will also study stochastic partial differential equation scaling limits of discrete probabilistic models from statistical physics, such as directed polymers in an environment of interacting particle systems. This will allow to demonstrate universality of these stochastic partial differential equations. The main methods and techniques are the new theory of stochastic partial differential equations such as regularity structures, and their discrete versions. The study brings together tools from different areas, such as stochastic analysis, partial differential equations, quantum field theory, and statistical mechanics.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.

概率的一个核心问题是从复杂、大型的随机系统中提取重要的规律和信息。这样的随机系统可能是金融市场价格波动的股票、生物组织随机突变产生的癌细胞、大量的亚原子量子相互作用粒子、或者疾病如何在人与人之间随机传播。该项目重点研究两类模型：随机量子场相互作用下大量基本粒子的运动，以及随机干扰环境下聚合物的生长。由于数学模型的通用性，这两类模型的研究成果对于研究其他系统，例如材料科学、金融、生物学中的系统也会有很大的帮助。使用的主要数学工具称为随机偏微分方程。更具体地说，随机偏微分方程理论将用于研究规范理论的随机量子化，规范理论是基本粒子物理学的基本模型。 PI将证明此类奇异方程的适定性，证明解具有无限维规范对称性，并扩展正则结构理论的范围。 PI 还将研究统计物理学中离散概率模型的随机偏微分方程标度限制，例如相互作用粒子系统环境中的定向聚合物。这将证明这些随机偏微分方程的普遍性。主要方法和技术是正则结构等随机偏微分方程的新理论及其离散形式。该研究汇集了来自不同领域的工具，例如随机分析、偏微分方程、量子场论和统计力学。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。

项目成果

A Stochastic Analysis Approach to Lattice Yang–Mills at Strong Coupling

强耦合下格子 Yang–Mills 的随机分析方法

• DOI: 10.1007/s00220-022-04609-1
• 发表时间: 2022-04-27
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Hao Shen;Rongchan Zhu;Xiangchan Zhu
• 通讯作者: Xiangchan Zhu

A stochastic PDE approach to large N problems in quantum field theory: A survey

量子场论中大 N 问题的随机偏微分方程方法：一项调查

• DOI: 10.1063/5.0089851
• 发表时间: 2022-08-01
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Hao Shen

Scaling limit of a directed polymer among a Poisson field of independent walks

独立游走泊松场中定向聚合物的缩放极限

• DOI: 10.1016/j.jfa.2021.109066
• 发表时间: 2021-09
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Shen, Hao;Song, Jian;Sun, Rongfeng;Xu, Lihu
• 通讯作者: Xu, Lihu

Stochastic Ricci Flow on Compact Surfaces

致密表面上的随机 Ricci 流

• DOI: 10.1093/imrn/rnab015
• 发表时间: 2019-04-24
• 期刊: arXiv: Probability
• 作者: Julien Dub'edat;Hao Shen
• 通讯作者: Hao Shen

Large N Limit of the O(N) Linear Sigma Model in 3D

3D 中 O(N) 线性西格玛模型的大 N 极限

• DOI: 10.1007/s00220-022-04414-w
• 发表时间: 2021-02-04
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Hao Shen;Rongchan Zhu;Xiangchan Zhu
• 通讯作者: Xiangchan Zhu