Lecture 8: From Random Walks to Diffusion

In this lecture we provide two examples of problems where reasonable models are given by the deterministic continuum limit of the discrete random walk through which the problem is naturally defined. Once suitably defined, this continuum limit usually takes the form of a partial differential equation (hence deterministic). One interesting use of the continuum limit we will see in upcoming lectures is to understand the behavior of random walks with boundary conditions. To properly incorporate boundary conditions into the calculation of the N -step point mass function of a discrete random walk usually introduces complicated combinatorial calculations. On the other hand, it may be much simpler to solve a differential equation subject to certain boundary conditions in order to answer some questions about the behavior of the random process. However, the applicability of this type of approximation has limits as well. We will explore when these approximations are valid and useful.