The heat equation can be derived by averaging over a very large number of
particles. Traditionally, the resulting PDE is studied as a deterministic
equation, an approach that has brought many significant results and a deep
understanding of the equation and its solutions. By studying the heat
equation by considering the individual random particles, however, one
gains further intuition into the problem. While this is now standard for
many researchers, this approach is generally not presented at the
undergraduate level. In this book, Lawler introduces the heat equation and
the closely related notion of harmonic functions from a probabilistic
perspective.

The theme of the first two chapters of the book is the relationship
between random walks and the heat equation. The first chapter discusses
the discrete case, random walk and the heat equation on the integer
lattice; and the second chapter discusses the continuous case, Brownian
motion and the usual heat equation. Relationships are shown between the
two. For example, solving the heat equation in the discrete setting
becomes a problem of diagonalization of symmetric matrices, which becomes
a problem in Fourier series in the continuous case. Random walk and
Brownian motion are introduced and developed from first principles. The
latter two chapters discuss different topics: martingales
and fractal dimension, with the chapters tied together by one example, a
random Cantor set.

The idea of this book is to merge probabilistic and deterministic
approaches to heat flow. It is also intended as a bridge from
undergraduate analysis to graduate and research perspectives. The book is
suitable for advanced undergraduates, particularly those considering
graduate work in mathematics or related areas.

Readership

Undergraduate students interested in probability and connections between probability and classical analysis.