Solution Theories and Scaling Limit Problems in Stochastic Partial Differential Equations

随机偏微分方程中的解理论和标度极限问题

• 批准号: 1712684
• 负责人: Hao Shen
• 金额: $ 14.53万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712684&HistoricalAwards=false
• 关键词: Solution; Theories; Scaling; Limit; Problems

In modern physics and other areas of science many problems have random components and are modeled by equations or systems of equations that are probabilistic. An important part of understanding the physics is knowing whether, or under what conditions, these equations have solutions. The principal investigator will further develop methods to study these types of equations. He will also organize conferences and develop courses on this topic. Stochastic partial differential equations (SPDEs) arise from extremely important models in areas such as statistical physics, quantum field theory and fluid mechanics. Solving these equations, including proving existence and uniqueness of their solutions, is exceedingly difficult. This is often due to the presence of very singular random forcing, as well as nonlinearities. Various solution theories were established based on different approaches, the most powerful of which is the theory of regularity structures introduced by Hairer around 2013 and further developed by the PI and a few other authors since then. This research will apply the theory, combined with ideas from other areas such as quantum field theory to investigate more SPDE problems. The PI will provide solutions to new important examples of SPDEs, including equations with gauge symmetry and the sine-Gordon equation near criticality. The PI also plans to prove scaling limit results for these singular SPDEs. In particular the PI will study convergence of discrete systems, such as the Glauber dynamics of ferromagnetic systems and directed polymers in random media, to the solutions to these SPDEs in various scaling regimes. The PI will also organize conferences and develop courses to disseminate this research.

在现代物理学和其他科学领域中，许多问题都具有随机成分，并通过概率方程或方程组进行建模。理解物理学的一个重要部分是知道这些方程是否有解，或者在什么条件下有解。首席研究员将进一步开发研究这些类型方程的方法。他还将组织有关该主题的会议并开发课程。 随机偏微分方程 (SPDE) 源自统计物理学、量子场论和流体力学等领域极其重要的模型。求解这些方程，包括证明其解的存在性和唯一性，是极其困难的。这通常是由于存在非常奇异的随机强迫以及非线性。基于不同的方法建立了各种解理论，其中最强大的是 Hairer 在 2013 年左右提出的正则结构理论，此后由 PI 和其他几位作者进一步发展。本研究将应用该理论，结合量子场论等其他领域的思想来研究更多的 SPDE 问题。 PI 将为 SPDE 的新的重要示例提供解决方案，包括规范对称性方程和接近临界点的正弦戈登方程。 PI 还计划证明这些奇异 SPDE 的缩放极限结果。特别是，PI 将研究离散系统的收敛性，例如铁磁系统的格劳伯动力学和随机介质中的定向聚合物，以及各种缩放机制中这些 SPDE 的解决方案。 PI 还将组织会议并开发课程来传播这项研究。

项目成果

Stochastic telegraph equation limit for the stochastic six vertex model

随机六顶点模型的随机电报方程极限

• DOI: 10.1090/proc/14415
• 发表时间: 2019-06
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Shen, Hao;Tsai, Li
• 通讯作者: Tsai, Li

Stochastic PDE Limit of the Six Vertex Model

六顶点模型的随机偏微分方程极限

• DOI: 10.1007/s00220-019-03678-z
• 发表时间: 2020-05
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Corwin, Ivan;Ghosal, Promit;Shen, Hao;Tsai, Li
• 通讯作者: Tsai, Li

Open ASEP in the Weakly Asymmetric Regime

在弱不对称机制中打开 ASEP

• DOI: 10.1002/cpa.21744
• 发表时间: 2016-10-17
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: Ivan Corwin;Hao Shen
• 通讯作者: Hao Shen

Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem

非高斯场 SPDE 的矩界及其在 Wong-Zakai 问题中的应用

• DOI: 10.1214/17-ejp84
• 发表时间: 2016-05-18
• 期刊: arXiv: Probability
• 作者: A. Ch;ra;ra;Hao Shen
• 通讯作者: 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