Stochastic Processes and Diffusion Equations

Abstract

In these lecture notes, we explore the mathematical preliminaries and foundational concepts that connect stochastic processes with partial differential equations. We begin by investigating Brownian motion, which serves as a model for random fluctuations and is deeply connected to the heat equation. This connection forms the basis for understanding diffusion phenomena, where the probability distribution of Brownian motion evolves according to the heat equation over time. To extend this classical result to more general stochastic systems, we introduce the It\^o calculus, a powerful framework that allows us to analyze processes driven by both deterministic drift and stochastic fluctuations. This mathematical tool is essential for understanding the dynamics of more complex diffusion processes, where randomness is no longer purely Brownian, but also depends on the underlying system's state. Building on these concepts, we turn to the study of diffusion processes, which generalize Brownian motion by incorporating the Fokker-Planck equation. This equation describes how the probability density of a diffusion process evolves over time and serves as an extension of the heat equation to more complex stochastic systems. By using It\^o calculus, we can rigorously study how these processes behave and connect the microscopic randomness of individual particles to their macroscopic description via partial differential equations.