Topics in stochastic analysis

随机分析主题

• 批准号: 1511328
• 负责人: Fabrice Baudoin
• 金额: $ 30万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2016-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1511328&HistoricalAwards=false
• 关键词: Topics; stochastic; analysis

Probability theory is the mathematical theory concerned with the analysis of random phenomena. Many of such phenomena may be modeled by continuous time stochastic processes. The first stochastic process that has been extensively studied is the celebrated Brownian motion, named in honor of the botanist Robert Brown, who observed and described in 1828 the random movement of particles suspended in a liquid or gas. The same process was later used in 1900 by the mathematician Louis Bachelier to model stock prices on financial markets. Finally, in 1905, Albert Einstein brought this process to the attention of physicists by presenting it as a way to indirectly confirm the existence of atoms and molecules. The Brownian motion is an example of diffusion process. Like their ancestor the Brownian motion, diffusion processes appear in many different areas of sciences and economy and their theoretical mathematical study has far reaching consequences in understanding and making predictions about the phenomena they model. In this project, the PI will study several problems in the theory of diffusion processes. In particular, questions about the deep interaction between the diffusion process and the geometry of the ambient space will be addressed and rates of convergence to equilibrium will be studied.Mathematically speaking, the present project focuses on different aspects of the theory of diffusion processes and diffusion semigroups. The PI will investigate applications to sub-Riemannian geometry where diffusion methods turn out to be very fruitful to study generalized Ricci curvature lower bounds. The PI will also address several questions about hypocoercive diffusions. Hypocoercivity is a concept recently introduced by Cedric Villani to obtain quantitative estimates for the convergence to equilibrium of some highly degenerate hypoelliptic semigroup. Hypocoercive estimates are in general very difficult to prove and the PI will systematically study new methods which parallel the Bakry-Emery approach to hypercontractivity. Some problems in the rough paths theory of Terry Lyons will also be studied, in particular related to the properties of stochastic differential equations driven by Gaussian processes. These projects will involve the training of graduate students and junior mathematicians. The results will be disseminated through publications in professional journals, lectures and on the blog of the PI.

概率理论是与随机现象分析有关的数学理论。许多此类现象可以通过连续的随机过程来建模。 经过广泛研究的第一个随机过程是著名的布朗尼运动，以纪念植物学家罗伯特·布朗（Robert Brown）的名字命名，他在1828年观察并描述了悬浮在液体或气体中的颗粒的随机运动。数学家路易斯·巴切尔（Louis Bachelier）在1900年后来使用了同样的过程，为金融市场上的股价建模。最后，在1905年，阿尔伯特·爱因斯坦（Albert Einstein）将这一过程引起了物理学家的注意，以此作为间接确认原子和分子存在的一种方式。布朗运动是扩散过程的一个例子。像他们的祖先布朗运动一样，扩散过程出现在许多不同的科学和经济领域，其理论数学研究在理解和预测它们建模的现象方面产生了很大的影响。在该项目中，PI将研究扩散过程理论中的几个问题。特别是，将解决有关扩散过程与环境空间几何形状之间的深层相互作用的问题，并研究融合均与平衡的速率。从天上角度来看，目前的项目着重于扩散过程理论和扩散半群的不同方面。 PI将调查对亚裂口几何形状的应用，在研究广义的RICCI曲率下限中，扩散方法非常富有成果。 PI还将解决有关低碳扩散的几个问题。低调是Cedric Villani最近引入的一个概念，以获得与某些高度退化的低纤维化半群的融合的定量估计值。通常很难证明估计值非常困难，PI将系统地研究新方法，这些方法与Bakry-Emery的超扣除率相似。 也将研究特里·里昂斯（Terry Lyons）的粗糙路径理论中的一些问题，特别是与高斯过程驱动的随机微分方程的特性有关。这些项目将涉及研究生和初级数学家的培训。结果将通过专业期刊，讲座和PI博客的出版物进行传播。