Diffusion Processes and Stochastic Analysis

pi研究了随机分析中的几个问题。第一个问题涉及时变域的反射布朗运动和相应的具有诺伊曼边界条件的热方程。重点讨论了热方程解和反射布朗运动解的存在性和唯一性，以及热方程解在运动边界附近的奇异性。项目的第二部分关注与奇异随机微分方程相关的随机流。这种类型的流具有有趣的组合特性，这在与光滑系数方程相对应的流中是不存在的。“热点”猜想指出，在一个绝缘物体中，最热的点位于它的边界上。虽然这在一般情况下是不正确的，但人们普遍认为这个猜想在凸域是成立的。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20世纪概率分析领域最重要的发展之一。它现在为研究现实生活中自然随机现象的基本性质提供了基础。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。稳定过程作为课题之一，由于传统的连续模型并不总是足够的，在金融数学和其他应用科学中得到越来越多的应用。对单一流动的研究直接受到了其中一位pi （burzy）与经济学家合作的启发，该合作发表在权威期刊《计量经济学》（Econometrics）上。在时间依赖域中对反射布朗运动的研究使人想起与冰融化有关的“斯蒂芬问题”。类似的物理模型对模拟现实生活环境变化的科学家非常感兴趣。

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.