Diffusion Processes and Stochastic Analysis

PI研究随机分析中的几个问题。第一个问题涉及含时区域上的反射布朗运动和相应的具有Neumann边界条件的热方程。主要讨论了热方程解的存在唯一性和反射布朗运动解的存在唯一性，以及运动边界附近热方程解的奇性。该项目的第二部分涉及与奇异随机微分方程相关的随机流。这类流具有与光滑系数方程对应的流中所没有的有趣的组合性质。“热点”猜想指出，绝缘体中最热的点在其边界上。虽然这在总体上不是正确的，但人们普遍认为该猜想在凸域中成立。目前，PI正在研究对称凸域。最后，从位势理论的角度研究了稳定过程及其相关过程。随机分析是二十世纪概率分析边缘最重要的发展之一。它现在为研究现实生活现象的基本性质提供了基础，这些现象本质上是随机的。该理论最近最引人注目的成功之一是所谓的金融数学。这一理论为证券交易提供了坚实的理论基础--其创始人最近获得了诺贝尔经济学奖的认可。稳定过程是该项目的主题之一，由于传统的连续模型并不总是足够的，因此在金融数学和其他应用科学中越来越多地被应用。对奇异流的研究直接受到了一位PI(Burdzy)与经济学家的合作的启发，该研究发表在领先的期刊《计量经济学》上。对依赖于时间的区域中反射布朗运动的研究让人想起与融化冰有关的“斯特凡问题”。对现实生活中的环境变化进行模拟的科学家对类似的物理模型非常感兴趣。

The PIs study several problems in stochastic analysis. The first problem involves reflected Brownian motion in time-dependent domains and the corresponding heat equation with the Neumann boundary conditions. The main emphasis is on the existence and uniqueness of the solutions to the heat equation and the reflected Brownian motion, and on the singularities of the heat equation solutions close to the moving boundary. The second part of the project is concerned with stochastic flows related to singular stochastic differential equations. Flows of this type have interesting combinatorial properties not present in flows corresponding to equations with smooth coefficients. The "hot spot" conjecture states that hottest point in an insulated body lies on its boundary. While this is not true in general, it is a widespread belief that the conjecture holds in convex domains. The PIs are currently studying symmetric convex domains. Finally, stable processes and related processes are studied from the point of view of potential theory. Stochastic analysis was one of the most important developments on the borderline of probability and analysis in the twentieth century. It now provides the basis of studying fundamental properties of real life phenomena which are random by nature. One of the most spectacular recent successes of the theory is the so-called financial mathematics. This theory provides a solid theoretical basis for trading securities - its founding fathers were recently recognized by a Nobel Prize in Economics. Stable processes, one of the topics of the project, are more and more often applied in financial mathematics and other applied sciences because the traditional continuous models are not always adequate. The study of a singular flow was directly inspired by a collaboration of one of the PIs (Burdzy) with economists, published in a leading journal "Econometrics." The study of reflected Brownian motion in time dependent domains is reminiscent of the "Stefan problem" concerned with melting ice. Similar physical models are of great interest to scientists who model real life environmental changes.