Description

Brownian 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:

• Levy's modulus of continuity
• The zero set of Brownian Motion and Hausdorff dimension
• Brownian local time
• The Ray-Knight Theorem
• Pitman's 2M-X Theorem

Two dimensional Brownian Motion:

• Recurrence and space filling curves
• Conformal invariance
• Harmonic functions
• Potential theory

Mode of operation and evidence of leanring

The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.

Prerequisites

Probability II and Analysis III are essential. Advanced Probability is a desirable co-requisite.

Resources

Students will be guided to books, lecture notes and papers. The following textbooks are a useful introduction and provide background.

• P. Morters and Y. Peres Brownian Motion.
• E. M. Stein and R. Shakarchi Real Analysis.

Get in touch by email if you have questions.