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用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，以纪念植物学家罗伯特·布朗，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。后来，数学家路易斯·巴谢里耶在1900年使用了同样的方法来对金融市场的股票价格进行建模。最后，在1905年，阿尔伯特·爱因斯坦将这一过程作为间接证实原子和分子存在的一种方式，引起了物理学家的注意。布朗运动是扩散过程的一个例子。像它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，它们的理论数学研究对于理解和预测它们所模拟的现象具有深远的影响。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度相互作用的问题将被解决，并且收敛到平衡的速率将被研究。从数学上讲，本项目集中在扩散过程和扩散半群理论的不同方面。PI将研究在次黎曼几何中的应用，在这些几何中，扩散方法在研究广义Ricci曲率下界方面被证明是非常有效的。PI还将解决几个关于亚胁性扩散的问题。亚矫顽力是Cedric Villani最近提出的一个概念，目的是得到一些高度退化的亚椭圆半群收敛到平衡点的定量估计。一般来说，次强制性估计很难证明，PI将系统地研究新的方法，这些方法与Bakry-Emery方法平行于超收缩。文中还将研究Terry Lyons粗糙路径理论中的一些问题，特别是与由高斯过程驱动的随机微分方程的性质有关的问题。这些项目将涉及对研究生和初级数学家的培训。结果将通过专业期刊、讲座和国际和平协会的博客上的出版物传播。

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.