One of my probability teachers said, “Every probabilist must know something about Brownian motion (BM)”, and I believe she was right.
Heuristically, a BM is continuous version of a random walk. Two of the most popular texts on graduate level probability (Durrett’s Probability: Theory and Examples and Billingsley’s Probability and Measure) both conclude the course with introductions to Brownian motion that make this idea precise.
There is a huge literature on stochastic calculus that features BM as an ingredient (but generally not the focus) and BM is included in many books on stochastic processes at varying levels of rigor. But where does the graduate student go to learn about BM for its own sake, from a modern viewpoint? That is the task of Brownian Motion by Peter Mörters & Yuval Peres.
The book starts off with the basics: existence, construction, scaling properties, continuity, non-differentiability of paths, strong Markov property, etc. The results are explained clearly and readers who have completed a measure theoretic probability course will have no trouble following all the details. The book then develops tools such as Hausdorff dimension, local times, stochastic integrals needed for more advanced topics. The book concludes with several topics of current research such as intersections BMs, fast and slow times of BM and an appendix on stochastic Loewner evolution written by Oded Schramm and Wendelin Werner.
Note that even though there is a chapter on stochastic integration and applications, the applications are within mathematics — there are no applications to financial option pricing. To me, this is a definite plus of the book, as it focuses on the amazing amount of detail known about BM, not only what typical behavior is, but what behavior that is non-typical looks like.
Each chapter (except the appendix on SLE) concludes with about 10 to 15 interesting exercises, of which roughly half have solutions provided at the end of the book. These solutions range from hints or pointers to the literature to clear fully worked out solutions. This is a huge bonus for the student trying to learn this material outside of a classroom. Notes and comments at the end of each chapter describe historical context, connections to other areas of probability, and further reading. A very interesting section on several unsolved problems concludes the book.
Throughout there is good use of pictures to illustrate concepts, and the authors do their best to orient the reader via frequent guideposts. For example, when discussing Minkowski dimension, the authors note that it does not have a countable stability property, and that there are two ways out of this problem. They briefly describe the approaches and state that they will pursue the first one immediately, and the other later in the book, thus preparing the reader for what follows.
To balance this review, I now have to point out some extremely minor flaws. There are a few typos (some errata can already be found on Mörters’ web page), a few symbols that are undefined (the infinity norm is used without being defined, but the Euclidean metric is defined), and an exercise that is indicated to have a solution in the appendix, but does not. None of them detract from the clarity of the book.
In summary, this is an excellent book from which to learn all about Brownian motion.
Peter Rabinovitch is a Systems Architect at Research in Motion, and a Ph.D. student in probability.
Preface; Frequently used notation; Motivation; 1. Brownian motion as a random function; 2. Brownian motion as a strong Markov process; 3. Harmonic functions, transience and recurrence; 4. Hausdorff dimension: techniques and applications; 5. Brownian motion and random walk; 6. Brownian local time; 7. Stochastic integrals and applications; 8. Potential theory of Brownian motion; 9. Intersections and self-intersections of Brownian paths; 10. Exceptional sets for Brownian motion; Appendix A. Further developments: 11. Stochastic Loewner evolution and its applications to planar Brownian motion; Appendix B. Background and prerequisites; Hints and solutions for selected exercises; References; Index.