Random Walks on Spheres End of Term Report Summer 2013

We are examining a classical Kakutani result on the relationship between Brownian motion, a form of random movement, and harmonic functions, which are solutions to Laplace equation. We will use this result to numerically solve Laplace’s equation in certain regions with various boundary conditions via the Random Walks on Spheres method. Looking at several regions, we will discuss the distribution of the point of first encounter with the boundary. We will also approximate the solutions using Gaussian quadrature, a method which allows us to have incomplete data concerning the boundary conditions. Lastly we consider the accuracy of our numerical solutions