Topics in stochastic analysis

概率论是分析随机现象的数学理论。许多这样的现象可以用连续时间随机过程来模拟。第一个被广泛研究的随机过程是著名的布朗运动，以植物学家罗伯特·布朗的名字命名，他在1828年观察并描述了悬浮在液体或气体中的粒子的随机运动。1900年，数学家路易斯·巴切利耶（Louis Bachelier）用同样的方法在金融市场上建立了股票价格模型。最后，在1905年，阿尔伯特·爱因斯坦把这个过程作为一种间接证实原子和分子存在的方法，引起了物理学家的注意。布朗运动是扩散过程的一个例子。就像它们的祖先布朗运动一样，扩散过程出现在科学和经济的许多不同领域，它们的理论数学研究在理解和预测它们所模拟的现象方面具有深远的影响。在这个项目中，PI将研究扩散过程理论中的几个问题。特别是，关于扩散过程和周围空间几何之间的深层相互作用的问题将得到解决，并将研究收敛到平衡的速率。从数学上讲，本项目侧重于扩散过程和扩散半群理论的不同方面。PI将研究在亚黎曼几何中的应用，其中扩散方法对于研究广义里奇曲率下界是非常有效的。PI还将讨论关于低强制扩散的几个问题。次矫顽力是Cedric Villani最近为了得到一些高度简并的次椭圆半群收敛到平衡的定量估计而引入的一个概念。一般来说，低强制估计很难证明，PI将系统地研究与Bakry-Emery方法平行的超收缩性新方法。本文还将研究特里·莱昂斯粗糙路径理论中的一些问题，特别是关于高斯过程驱动的随机微分方程的性质。这些项目将包括培养研究生和初级数学家。研究结果将通过专业期刊、讲座和PI博客的出版物进行传播。

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.