##
**Brownian motion. With an appendix by Oded Schramm and Wendelin Werner.**
*(English)*
Zbl 1243.60002

Cambridge Series in Statistical and Probabilistic Mathematics 30. Cambridge: Cambridge University Press (ISBN 978-0-521-76018-8/hbk). xii, 403 p. (2010).

Since its first theoretical description by Einstein in 1905, Brownian motion (BM) has become one of the central objects of interest in mathematics of the 20th century. More than a hundred years later, the astonishing recent developments for complex-valued BM subsumed under the label “Stochastic Loewner Evolution” (SLE) give rise to the expectation that its victory march is far from completed and that more and more profound insight in its properties remains of primordial importance. The monograph under review by Peter Mörters and Yuval Peres provides a largely self-contained introduction to BM with a particular interest in its path properties in different space dimensions, reaching from the very basics to many results at the edge of our knowledge.

The book starts with Lévy’s construction of scalar BM and several classical path properties, followed by a study of the strong Markov nature of BM and its consequences before changing the perspective on BM towards potential theory. It gives an introduction of the notion of harmonic functions and its basic properties and poses the classical Dirichlet problem with regard to the stochastic interpretation of its solution including the hitting times of BM. The special focus of this book on path properties can be seen in Chapter 4, where the authors cover a topic rarely included in monographs about BM, treating it as a random fractal with typical properties such as several notions of fractal dimension. The highlight of this chapter includes the proof of the spectacular theorem that the zero level set and the record set (times when BM coincides with zero or its running maximum) of BM have exactly Hausdorff dimension \(1/2\). The following section is concerned with the derivation of the asymptotic law of the iterated logarithm for the upper limit, first for BM and then for (embedded) simple random walks, laying the basics of Donsker’s theory. In Chapter 6 the reader gets acquainted with the Brownian local time and it closes the narrative circle by connecting the theory of Brownian local times with the random fractal perspective of Chapter 4. In Chapter 7 the part of rather general results on BM finds its end; before entering much less standard results it gathers an excursion to stochastic analysis, the Feynman-Kac formulae and fine hitting probabilities such as for cones on the complex plane.

In the four sections of Chapter 8 the authors develop the key ingredients for potential theory of transient BM with the aim to characterize regular points of the domain for the Dirichlet problem in terms of notions related to BM. In Chapter 9 the authors carry out fine analysis of the path intersections of several independent \(d\)-dimensional BM and later of self-intersections of the Brownian path in different dimensions. These results are refined considerably in Chapter 10 by generalizing the previously obtained path properties found for typical (deterministic) times to (random) exceptional times and points in space. The appendix had been initially designed by Oded Schramm and after his tragic and unforeseen passing away; it was serendipitously completed by Wendelin Werner, who gives an inspiring, less formal invitation to SLE and critical branching.

If you agree with the thesis that the evolution of probability theory and more generally of mathematics, since its very beginnings, has led to a co-evolution in the style of writing mathematics towards an ever more comprehensive, intuitive, but rigorous, self-contained, collaborative and interactive exposition, in order to give better and better access to deeper and deeper results, this book is without any doubt a splendid example. The material of this book contains an essential selection of topics that could be also found in books on stochastic analysis, but mostly restricted to the framework of BM. Hence many far reaching concepts like continuous martingales, Markov processes, Itô calculus, potential theory can be explained in its probably easiest context. This concentration on the core ideas of the arguments enables the authors to attain a quick and direct access to many deep results available in the sequel. Apart form the excellent selection of more classical results in stochastic analysis, Mörters and Peres place particular emphasis on the concepts of fractal geometry, which turns out to be extraordinarily fruitful for the understanding of the fine structure of BM paths. Special attention receives in this context the notion of Hausdorff dimension, initiated in Chapter 4 and continued in Chapters 8, 9 and 10. This part fills an important gap in the literature, since elementary expositions of this subject in a probabilistic context had rarely been accessible so far apart from the respective research articles. In addition, it is impressive to see the powerful connection to other fields previously introduced in this monograph at work in the later chapters. One example is the widely exploited link between the Hausdorff dimension and different kinds of capacities.

Almost all results of this book are motivated in a very vivid and insightful way with many precious remarks that help the reader to build his or her proper intuition for the problem (and related problems) under study. Some carefully chosen pictures support the understanding very much. Attached to each chapter the interested reader finds a lot of helpful exercises that contain complementary material which is occasionally cited in the text. For about half of the exercises solution hints or even detailed solutions are provided at the end of the book. After the exercises of each chapter there is always a small, but precious section, where the history and the research context with the relevant references of the previously presented results and proofs are unfolded. This separation of content and context improves the reading of the book remarkably since the mathematical text is presented as an integrated story in its own right.

With this monograph, Mörters and Peres in collaboration with many excellent colleagues enrich the already broad literature on the subject with a gently readable, innovative and stimulating account of a great deal of what is known to date about the path properties of Brownian Motion.

The book starts with Lévy’s construction of scalar BM and several classical path properties, followed by a study of the strong Markov nature of BM and its consequences before changing the perspective on BM towards potential theory. It gives an introduction of the notion of harmonic functions and its basic properties and poses the classical Dirichlet problem with regard to the stochastic interpretation of its solution including the hitting times of BM. The special focus of this book on path properties can be seen in Chapter 4, where the authors cover a topic rarely included in monographs about BM, treating it as a random fractal with typical properties such as several notions of fractal dimension. The highlight of this chapter includes the proof of the spectacular theorem that the zero level set and the record set (times when BM coincides with zero or its running maximum) of BM have exactly Hausdorff dimension \(1/2\). The following section is concerned with the derivation of the asymptotic law of the iterated logarithm for the upper limit, first for BM and then for (embedded) simple random walks, laying the basics of Donsker’s theory. In Chapter 6 the reader gets acquainted with the Brownian local time and it closes the narrative circle by connecting the theory of Brownian local times with the random fractal perspective of Chapter 4. In Chapter 7 the part of rather general results on BM finds its end; before entering much less standard results it gathers an excursion to stochastic analysis, the Feynman-Kac formulae and fine hitting probabilities such as for cones on the complex plane.

In the four sections of Chapter 8 the authors develop the key ingredients for potential theory of transient BM with the aim to characterize regular points of the domain for the Dirichlet problem in terms of notions related to BM. In Chapter 9 the authors carry out fine analysis of the path intersections of several independent \(d\)-dimensional BM and later of self-intersections of the Brownian path in different dimensions. These results are refined considerably in Chapter 10 by generalizing the previously obtained path properties found for typical (deterministic) times to (random) exceptional times and points in space. The appendix had been initially designed by Oded Schramm and after his tragic and unforeseen passing away; it was serendipitously completed by Wendelin Werner, who gives an inspiring, less formal invitation to SLE and critical branching.

If you agree with the thesis that the evolution of probability theory and more generally of mathematics, since its very beginnings, has led to a co-evolution in the style of writing mathematics towards an ever more comprehensive, intuitive, but rigorous, self-contained, collaborative and interactive exposition, in order to give better and better access to deeper and deeper results, this book is without any doubt a splendid example. The material of this book contains an essential selection of topics that could be also found in books on stochastic analysis, but mostly restricted to the framework of BM. Hence many far reaching concepts like continuous martingales, Markov processes, Itô calculus, potential theory can be explained in its probably easiest context. This concentration on the core ideas of the arguments enables the authors to attain a quick and direct access to many deep results available in the sequel. Apart form the excellent selection of more classical results in stochastic analysis, Mörters and Peres place particular emphasis on the concepts of fractal geometry, which turns out to be extraordinarily fruitful for the understanding of the fine structure of BM paths. Special attention receives in this context the notion of Hausdorff dimension, initiated in Chapter 4 and continued in Chapters 8, 9 and 10. This part fills an important gap in the literature, since elementary expositions of this subject in a probabilistic context had rarely been accessible so far apart from the respective research articles. In addition, it is impressive to see the powerful connection to other fields previously introduced in this monograph at work in the later chapters. One example is the widely exploited link between the Hausdorff dimension and different kinds of capacities.

Almost all results of this book are motivated in a very vivid and insightful way with many precious remarks that help the reader to build his or her proper intuition for the problem (and related problems) under study. Some carefully chosen pictures support the understanding very much. Attached to each chapter the interested reader finds a lot of helpful exercises that contain complementary material which is occasionally cited in the text. For about half of the exercises solution hints or even detailed solutions are provided at the end of the book. After the exercises of each chapter there is always a small, but precious section, where the history and the research context with the relevant references of the previously presented results and proofs are unfolded. This separation of content and context improves the reading of the book remarkably since the mathematical text is presented as an integrated story in its own right.

With this monograph, Mörters and Peres in collaboration with many excellent colleagues enrich the already broad literature on the subject with a gently readable, innovative and stimulating account of a great deal of what is known to date about the path properties of Brownian Motion.

Reviewer: Michael Högele (Berlin)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60J65 | Brownian motion |

60G15 | Gaussian processes |

60G17 | Sample path properties |

60J45 | Probabilistic potential theory |

60J55 | Local time and additive functionals |