Cornell Math - MATH 778, Spring 2004

MATH 778: Stochastic Processes - Random Walk on the Integer Lattice (Spring 2004)

Instructor: Gregory Lawler

Meeting Time & Room

In 1964, Frank Spitzer published the classic book on random walks, Principles of Random Walk. We will read this book (and in places update it). This will be the start of a long range project of mine to produce a working person's guide to random walk. We will restrict ourselves to walks on the integer lattice (both discrete and continuous time). There is plenty to consider in this simple setting!

We will also consider questions about strong convergence of random walk to Brownian motion including the KMT approximation, Beurling estimates, and their use in approximating rates of convergence on very rough domains (in two dimensions).

As pointed out by Spitzer, random walk on the integer lattice is essentially the same subject as discrete potential theory and this course could be of interest to people in differential equations and combinatorics. Convergence rates of random walks to Brownian motion are very closely related to rates of convergence of finite difference approximations to elliptic equations.

Prerequisites will be the equivalent of Math 671, in particular familiarity with discrete martingales and some knowledge of Brownian motion.