Mathematical Sciences: Brownian Motion and Related Processes

Bass教授和burzy教授提出在以下领域研究一些问题：布朗运动和调和函数的边界行为、反映布朗运动、分形上的扩散、二维随机游走的占用时间、平面布朗运动的例外点、平面布朗运动的交点局部时间、迭代布朗运动、Fleming- Viot过程的数值结果。概率论中研究的布朗运动是一种数学模型，当人们观察奶油如何混入咖啡或烟雾如何在房间空气中扩散时，人们会看到什么。布朗运动的研究在物理、化学、工程和量子力学中都有应用，这并不奇怪；然而，它在过滤噪声信号、研究人口增长、股票市场行为以及其他许多应用方面也非常有用。Bass教授和burzy教授建议研究如何利用布朗运动和相关物体来解决数学物理中的问题。这些问题中的许多可以简化为某些偏微分方程的解，而这些偏微分方程又可以通过使用布朗运动和其他扩散来解决。

Bass Professors Bass and Burdzy propose to study some problems in the areas: boundary behavior of Brownian motion and harmonic functions, reflecting Brownian motion, diffusions on fractals, occupation times for random walks in two dimensions, exceptional points of planar Brownian motion, intersection local times for planar Brownian motion, iterated Brownian motion, and numerical results for Fleming--Viot processes. Brownian motion as studied in probability theory is a mathematical model of what one sees when one watches how cream gets mixed into coffee or how smoke diffuses through the air of a room. It is not surprising that the study of Brownian motion has applications to physics, chemistry, engineering, and quantum mechanics; however, it is also extremely useful for filtering noisy signals, studying the growth of populations, the behavior of the stock market, and a whole host of other applications. Professors Bass and Burdzy propose to study how Brownian motion and related objects can be used in solving problems that come up in mathematical physics. Many of these problems can be reduced to the solution of certain partial differential equations, which in turn can be resolved through the use of Brownian motion and other diffusions.