Degenerate Diffusions and Related Heat Kernel Estimates

简并扩散和相关的热核估计

• 批准号: 1855523
• 负责人: Jing Wang
• 金额: $ 12万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-01 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855523&HistoricalAwards=false
• 关键词: Degenerate; Diffusions; Related; Heat; Kernel

This research project lies at the intersection of several areas of mathematics: probability, analysis and differential geometry. The PI will work on problems related to diffusion processes and their heat kernels in degenerate geometric settings, merging tools of the three mentioned areas. Some of the problems are pertinent to various applied fields, including mathematical finance, control of dynamic systems, machine learning, and others. The PI will integrate her research expertise into teaching probability courses, mentoring students, and organizing conferences. The PI will also continue her effort in encouraging women and other underrepresented groups in math. The project focuses on random diffusion processes that are subject to a priori non-holonomic constraints which can perfectly fit into the framework of sub-Riemannian geometry. A recurring theme of proposed topics is the interaction between the underlying geometric structure and the limiting behaviors of the associated Markov processes. The project is primarily concerned with three topics. The first is to study stochastic processes and the explicit heat kernel on sub-Riemannian model spaces, which provides great examples and strong intuition for the understanding of general cases. The second topic concerns the limiting behaviors -in both small and large time scale -of a general degenerate diffusion process, which reflect the underlying geometric information of geodesic and Ricci curvature bound respectively. The third topic is to study the heat content of bounded domains in a sub-Riemannian manifold, which reveals the geometry of the domain such as surface area and total mean curvature of the boundary.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.

这个研究项目是几个数学领域的交集：概率、分析和微分几何。PI将在简并几何环境下研究与扩散过程及其热核有关的问题，合并上述三个领域的工具。其中一些问题涉及不同的应用领域，包括数学金融、动态系统控制、机器学习等。PI将把她的研究专业知识整合到教授概率课程、指导学生和组织会议中。国际数学联合会还将继续努力鼓励女性和其他在数学方面代表性不足的群体。该项目专注于受先验非完整约束的随机扩散过程，该约束完全符合次黎曼几何的框架。提出的主题的一个反复出现的主题是潜在的几何结构和相关的马尔可夫过程的限制行为之间的相互作用。该项目主要涉及三个主题。第一个是研究次黎曼模型空间上的随机过程和显式热核，为理解一般情形提供了很好的例子和很强的直观性。第二个主题涉及一般简并扩散过程在小时间尺度和大时间尺度上的极限行为，它们分别反映了测地线和Ricci曲率界的基本几何信息。第三个主题是研究次黎曼流形中有界域的热含量，它揭示了域的几何形状，如表面积和边界的总平均曲率。这一奖项反映了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。

项目成果

Parabolic Anderson model on Heisenberg groups: The Itô setting

海森堡群的抛物线安德森模型：Ità 设置

• DOI: 10.1016/j.jfa.2023.109920
• 发表时间: 2023
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Baudoin, Fabrice;Ouyang, Cheng;Tindel, Samy;Wang, Jing
• 通讯作者: Wang, Jing

Bounds for exit times of Brownian motion and the first Dirichlet eigenvalue for the Laplacian

布朗运动的退出时间和拉普拉斯的第一个狄利克雷特征值的界限

• DOI: 10.1090/tran/8903
• 发表时间: 2023
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Bañuelos, Rodrigo;Mariano, Phanuel;Wang, Jing
• 通讯作者: Wang, Jing

Quaternionic stochastic areas

四元数随机区域

• DOI: 10.1016/j.spa.2020.09.002
• 发表时间: 2021
• 期刊: Stochastic Processes and their Applications
• 影响因子: 1.4
• 作者: Baudoin, Fabrice;Demni, Nizar;Wang, Jing
• 通讯作者: Wang, Jing

Improved Upper Bounds for the Hot Spots Constant of Lipschitz Domains

改进的 Lipschitz 域热点常数的上限

• DOI: 10.1007/s11118-022-10001-4
• 发表时间: 2022
• 期刊: Potential Analysis
• 影响因子: 1.1
• 作者: Mariano, Phanuel;Panzo, Hugo;Wang, Jing
• 通讯作者: Wang, Jing

Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions

酉布朗运动的块行列式的渐近缠绕和相关扩散

• DOI: 10.1214/21-ejp600
• 发表时间: 2021
• 期刊: Electronic journal of probability
• 影响因子: 1.4
• 作者: Baudoin, Fabrice;Wang, Jing
• 通讯作者: Wang, Jing