Thesis - Differential Equations and Brownian Motion

In this project, I explored the interplay between partial differential equations and stochastic calculus, focusing on the behavior of Brownian motion and its connection to heat flow. I first established the existence and continuity of Brownian motion, observing its intriguing property of being continuous but nowhere differentiable. This framework provides a probabilistic interpretation of heat flow, aligning with the approach used by Albert Einstein in his 1905 derivation of the heat equation. Building on this, I demonstrated that the heat equation itself can be used to prove harmonic martingale properties on bounded domains, offering an alternative to the traditional stochastic calculus proofs. Additionally, leveraging this connection, I showed recurrence of Brownian motion in one and two dimensions. The project highlights how stochastic processes and classical PDEs can inform and reinforce each other, providing both theoretical insights and alternative methods for analysis.