Topics in stochastic analysis

概率论是有关随机现象分析的数学理论。许多这样的现象可以通过连续时间随机过程来建模。 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·布朗的荣誉命名，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·巴舍利耶（Louis Bachelier）用同样的方法模拟了金融市场的股票价格。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。像他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，他们的理论数学研究在理解和预测他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的几何形状之间的深层相互作用的问题将得到解决，收敛到平衡的速度将研究。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将研究应用到次黎曼几何中，扩散方法在研究广义Ricci曲率下界方面非常富有成效。PI还将解决几个关于低矫顽扩散的问题。亚椭圆性是Cedric Villani最近引入的一个概念，它给出了一类高度退化亚椭圆半群收敛于平衡点的定量估计。一般来说，低强制性估计很难证明，PI将系统地研究与Bakry-Emery方法平行的超收缩性新方法。 一些问题的粗糙路径理论的特里里昂也将进行研究，特别是有关的性质随机微分方程驱动的高斯过程。这些项目将涉及培训研究生和初级数学家。研究结果将通过在专业期刊、讲座和PI博客上发表的出版物进行传播。

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.