Solving random walk recursions in the plane using separation of variables and Fourier analysis

Abstract

The aim of this article is first of all to show how to solve a two-dimensional random walk problem on an arbitrary rectangle with certain boundary conditions, using separation of variables and discrete Fourier analysis. Second, my intention is to demonstrate how ordinary Fourier analysis will solve the same problem if residue calculations are applied. Finally, I pay special attention to a resulting formula, connecting discrete and ordinary Fourier coefficients.