Conformally invariant processes in the plane.

*(English)*Zbl 1074.60002
Mathematical Surveys and Monographs 114. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3677-3/hbk). xi, 242 p. (2005).

This nice book celebrates the fruitful marriage of Brownian motion and complex analysis. Old conjectures on percolation theory or fractal dimension of exceptional sets on planar Brownian motion have now found an elegant and rigorous proof. The main tool is the Schramm-Loewner evolution theory whose study with related applications can be considered as one of the major advances in probability theory and complex analysis in the recent years.

Basic facts on Brownian motion, stochastic calculus and complex analysis are recalled in the first three chapters. The Loewner differential equations are presented in Chapter 4. Measures on spaces of curves and specially Brownian measures are the topic of Chapter 5. The main chapter is Chapter 6 which introduces the Schramm-Loewner evolution (SLE) which is the Loewner differential equation driven by a Brownian motion. More results about SLE are given in Chapter 7. Chapter 8 is devoted to the computation of planar Brownian intersection exponents from which the Hausdorff dimension of exceptional sets of the Brownian path can be determined. The final Chapter 9 shows the relationship between SLE and some scale-invariant measures called restriction measures.

Reading this beautiful book is sometimes demanding. But it will let the reader know a significant new area in probability theory with many by-products in statistical mechanics. Most of the displayed material comes from recent joint work of the author with Oded Schramm and Wendelin Werner.

Basic facts on Brownian motion, stochastic calculus and complex analysis are recalled in the first three chapters. The Loewner differential equations are presented in Chapter 4. Measures on spaces of curves and specially Brownian measures are the topic of Chapter 5. The main chapter is Chapter 6 which introduces the Schramm-Loewner evolution (SLE) which is the Loewner differential equation driven by a Brownian motion. More results about SLE are given in Chapter 7. Chapter 8 is devoted to the computation of planar Brownian intersection exponents from which the Hausdorff dimension of exceptional sets of the Brownian path can be determined. The final Chapter 9 shows the relationship between SLE and some scale-invariant measures called restriction measures.

Reading this beautiful book is sometimes demanding. But it will let the reader know a significant new area in probability theory with many by-products in statistical mechanics. Most of the displayed material comes from recent joint work of the author with Oded Schramm and Wendelin Werner.

Reviewer: Dominique Lepingle (OrlĂ©ans)

##### MSC:

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

30C35 | General theory of conformal mappings |

31A15 | Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

60J65 | Brownian motion |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

82B27 | Critical phenomena in equilibrium statistical mechanics |