Doug Coulter is a fourth-year maths and physics student at the Australian National University. He plans to do honours in maths in 2019.
Harmonic Functions and Brownian Motion
This project aims to study the Laplace equation and harmonic functions using Brownian motion and probability theory. Then we can derive some standard facts of analysis, such as Harnack’s inequality and the boundary behavior of harmonic functions, using Brownian motion and probability theory instead of conventional analysis. After this, the aim is to give probabilistic proofs for properties of Green’s functions, the Riemann mapping theorem, the boundary behaviour of conformal mappings, and look at Loewner’s differential equation.