DescriptionBrownian Motion is a fundamental stochastic process. It serves as the scaling limit of random walks, which are ubiquitous, and thus it describes the universal behaviour of all random walks. The development of the theory of Brownian Motion is fairly modern by mathematical standards. Here is a story recalled by Shizuo Kakutani, one of its pioneers: After Pearl Harbor, Kakutani took the last ship from the US to Japan. He was afraid the ship would be sunk, so wrote his latest theorems on pieces of paper that he sealed in bottles and threw overboard. When he arrived in Japan, Kakutani met with Ito and Yosida, and they decided to choose a topic and collaborate on it. After some debate, they chose Brownian Motion. They ended up working on it separately in a transformative way - Ito creating stochastic calculus, Kakutani developing the connection of Brownian motion and harmonic functions, and Yosida developing Semigroup theory. This project will explore geometric properties of one and two dimensional Brownian Motion. The emphasis will be on fine structural properties and connections to Analysis. Topics of interest include: One dimensional Brownian Motion:
Two dimensional Brownian Motion:
PrerequisitesProbability II and Analysis III are essential. Continuous and Discrete Probability is a desirable co-requisite. ResourcesStudents will be guided to books, lecture notes and papers. The following textbooks are a useful introduction and provide background.
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email: Mustazee Rahman