Mathematical Sciences: Brownian Motion and Related Processes

巴斯教授和伯齐建议研究以下领域的一些问题：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，二维随机游走的占据时间，平面布朗运动的异常点，平面布朗运动的交点局部时间，迭代布朗运动以及弗莱明-维奥特过程的数值结果。 概率论中研究的布朗运动是一种数学模型，描述了当人们观察奶油如何混合到咖啡中或烟雾如何在房间空气中扩散时所看到的情况。布朗运动的研究应用于物理、化学、工程和量子力学，这并不奇怪。然而，它对于过滤噪声信号、研究人口增长、股票市场行为以及许多其他应用也非常有用。 巴斯和伯齐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。其中许多问题可以简化为某些偏微分方程的求解，而这些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.