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.

该研究项目位于概率、分析和微分几何等几个数学领域的交叉点。 PI 将致力于研究与退化几何设置中的扩散过程及其热核相关的问题，合并上述三个领域的工具。其中一些问题与各个应用领域相关，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融入概率课程教学、指导学生和组织会议中。 PI 还将继续努力鼓励女性和其他在数学领域代表性不足的群体。该项目重点研究受先验非完整约束的随机扩散过程，该过程可以完美地融入亚黎曼几何框架。所提出的主题中反复出现的主题是基础几何结构与相关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。第一个是研究亚黎曼模型空间上的随机过程和显式热核，这为理解一般情况提供了很好的例子和强烈的直觉。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限制行为，这分别反映了测地线和里奇曲率界的基础几何信息。第三个主题是研究亚黎曼流形中有界域的热含量，揭示了域的几何形状，例如边界的表面积和总平均曲率。该奖项反映了 NSF 的法定使命，并被认为是值得的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来提供支持。

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

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

Quaternionic Brownian Windings

四元布朗绕组

• DOI: 10.1007/s10959-020-01034-9
• 发表时间: 2020
• 期刊: Journal of Theoretical Probability
• 影响因子: 0.8
• 作者: Baudoin, Fabrice;Demni, Nizar;Wang, Jing
• 通讯作者: Wang, Jing