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Random Walk and the Heat Equation

Gregory F. Lawler
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mathematical Library 55
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

This book was derived from lectures prepared for a Research Experiences for Undergraduates (REU) program at the University of Chicago. As with many books in the AMS Student Mathematical Library series, it takes up a subject that undergraduates are unlikely to encounter in the usual mathematics curriculum. Also like many books in the series, this one challenges students with concepts, techniques and arguments they are likely to find difficult and does so at a rapid pace.

The subject (at least at first) is a probabilistic treatment of the diffusion of heat. Most commonly this phenomenon is modeled via the deterministic heat equation that is derived using macroscopic physical principles. One can also study the diffusion of heat at the micro level by considering the movement of many individual randomly-moving particles. That’s the approach the author takes. It leads naturally to a probabilistic treatment that starts with random walk on a lattice in discrete time and proceeds to Brownian motion in continuous time and space. The interplay between deterministic and random, discrete and continuous is subtle. It provides some intriguing mathematical insights and can also spur better intuition of the underlying physics.

The author begins with simple random walk and develops the probabilistic background necessary to establish the heat equation on an integer lattice. Then he introduces discrete harmonic functions. Along the way we see a nice application of linear algebra: we solve the discrete heat equation by diagonalizing a symmetric matrix.

In the next section we take the limit of discrete random walk to get Brownian motion. Actually, it takes a fair bit of work to show the limit exists. The author introduces the continuous version of harmonic functions and gets to the standard deterministic version of the heat equation. Then the matrix diagonalization encountered in the discrete problem turns into a question about Fourier series.

The third part of the book moves away from the heat equation to martingales, the optional sampling theorem, betting and models of fair games. A fourth section is an introduction to fractal dimension that concludes with a calculation of the Hausdorff dimension of the random Cantor set introduced in a previous chapter.

This book is aimed at strong mathematics undergraduates who are considering graduate work. Readers are expected to have had a course in undergraduate analysis (and it had better have been a good one) and some linear algebra. The author uses the heat equation as a hook to draw attention, but his real interest is clearly probability and analysis. Strong students are likely to enjoy the approach, but even they are likely to struggle with the contents and the pace. Each section has several exercises; few of them are routine. The book would provide a good basis for an independent project or as ancillary reading for an analysis or probability course.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.