Brownian motion

This book on Brownian motion takes a concrete approach, emphasizing sample path properties. It developed out of Lectures by Yuval Peres delivered at Berkeley in 1998. Since its publication, it has been used in courses in many universities, including Stanford, Princeton, MIT, Toronto and Tel Aviv. As noted in the review by Rene Schilling at Mathscinet, the first three introductory chapters are followed by the presentation of some of the main tools of the book: fractal dimensions, covering techniques, capacities and Frostman’s lemma. The idea is that many properties of the Brownian paths are best understood within the framework of random fractals, trees and (fractal) geometric measure theory. Although the ideas are not new—they date back to McKean’s papers in the late 1950s, and Hawkes’ and Kahane’s contributions in the 1960s and 1970s—it is surprisingly difficult to find an easily accessible presentation. Hausdorff measure and Hausdorff dimensions are central ingredients for the remaining 200 pages of the book.
Chapter 5 returns to the proper topic of the tract: the Brownian paths, this time seen from a random walk perspective. We get a gentle introduction to (path) properties which have `random-walk character’, culminating in Donsker’s invariance principle and some fluctuation identities. The proof of Donsker’s theorem is based on Skorokhod’s embedding theorem, which can be seen as the missing link for the duality between Brownian motion and classical random walks. The exposition of Brownian local time (Chapter 6) is best characterized as Paul Lévy’s way to his local time leading to an elegant existence proof for Brownian local times by random walk approximations. Other topics of this chapter are an introduction to Ray-Knight theorems and an interpretation of local time as the Hausdorff measure of the time spent in a point x. Stochastic calculus is not the main aim of the book and therefore its presentation is limited to the bare necessities needed later on. The readers get a brief introduction to Itô’s stochastic integral and the change-of-variable formula (Itô’s Lemma) which is generalized to non-smooth functions, i.e., Tanaka’s formula. Continuing the theme from Chapter 3 and as a preparation for Chapter 8, the authors use stochastic integration to give a probabilistic proof of the Feynman-Kac formula. The highlight of this chapter, however, is the proof of the conformal invariance of Brownian motion and of harmonic measure. This may well be seen as an hors d’oeuvre for Chapter 11, treating stochastic Loewner evolution.
While most topics up to this point are in one way or another well known, Chapters 9 and 10 contain topics of the authors’ own research. Chapter 9 focusses on self-intersections of Brownian paths, in particular, the existence of multiple points and the size—again measured in terms of Hausdorff measures—of the set of multiple points. The rather surprising fact (due to Dvoretzky, Erdős and Kakutani and to Le Gall) that there is some point x in the plane such that planar Brownian motion returns to it uncountably many times, is one of the highlights of the book. The closing chapter of the book is again devoted to exceptional sets of Brownian motion. Using methods from fractal geometry, fast points, slow points and cone points of planar Brownian motion, in particular their (packing) dimension, are studied.
In an appendix, a short non-technical survey of stochastic Loewner evolution is given; the focus is on how certain Loewner exponents are related to dimension numbers of some classes of exceptional points of (self-avoiding) planar Brownian motion. Originally, the authors asked Oded Schramm to write this chapter; after his tragic death Wendelin Werner completed the task.

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Related open problems

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Consider two independent Brownian paths in the plane, run for unit time. Are all connected components of their intersection single points?


Consider the complementary components of a planar Brownian path, run for unit time. Call two components adjacent if removing one point from the Brownian path connects them. Is any complementary component reachable from the unbounded complementary component via finitely many steps, each step moving from a component to an adjacent one?

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