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中的反射布朗运动和相应的热方程的Neumann边界条件。主要重点是热方程和反射布朗运动解的存在性和唯一性，以及热方程解在移动边界附近的奇异性。该项目的第二部分是关于奇异随机微分方程的随机流。这种类型的流具有有趣的组合性质，而在对应于具有光滑系数的方程的流中不存在。“热点”猜想指出，在一个绝缘体的最热点位于其边界上。虽然这在一般情况下不是真的，但人们普遍认为该猜想在凸域中成立。PI目前正在研究对称凸域。最后，从势理论的角度研究了稳定过程和相关过程。随机分析是世纪概率论与分析之间最重要的发展之一。它现在提供了研究自然随机的真实的生命现象的基本性质的基础。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。由于传统的连续过程模型并不总是适用于金融数学和其他应用科学，稳定过程是该项目的主题之一。奇异流的研究直接受到一位PI（Burdzy）与经济学家合作的启发，发表在一份领先的期刊“计量经济学”上。在含时域中对反射布朗运动的研究使人想起与融冰有关的“斯特凡问题”。类似的物理模型对于模拟真实的生活环境变化的科学家来说是非常感兴趣的。

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.