Mathematical Sciences: Brownian Motion and Related Processes

Bass教授和Burdzy教授建议研究如下领域的问题：反映布朗运动的布朗运动和调和函数的边界行为，分数维上的扩散，二维随机游动的占据时，平面布朗运动的例外点，平面布朗运动的相交局部时，迭代布朗运动，以及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，描述了当人们观察到奶油如何混合到咖啡中，或者烟雾如何在房间的空气中扩散时所看到的东西。布朗运动的研究在物理、化学、工程和量子力学方面的应用并不令人惊讶；然而，它在过滤噪声信号、研究人口增长、股票市场行为和许多其他应用方面也非常有用。巴斯和伯兹教授提议研究如何使用布朗运动和相关物体来解决数学物理中出现的问题。这些问题中的许多可以归结为某些偏微分方程组的解，而偏微分方程组又可以通过使用布朗运动和其他扩散来求解。

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.