## Conformally Invariant Processes in the PlaneTheoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are in some sense conformally invariant. This belief has allowed physicists to predict many quantities for these critical systems. The nature of these scaling limits has recently been described precisely by using one well-known tool, Brownian motion, and a new construction, the Schramm-Loewner evolution (SLE). This book is an introduction to the conformally invariant processes that appear as scaling limits. The following topics are covered: stochastic integration; complex Brownian motion and measures derived from Brownian motion; conformal mappings and univalent functions; the Loewner differential equation and Loewner chains; the Schramm-Loewner evolution (SLE), which is a Loewner chain with a Brownian motion input; and applications to intersection exponents for Brownian motion. The prerequisites are first-year graduate courses in real analysis, complex analysis, and probability. The book is suitable for graduate students and research mathematicians interested in random processes and their applications in theoretical physics. |

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### Contents

1 | |

Chapter 1 Stochastic calculus | 11 |

Chapter 2 Complex Brownian motion | 43 |

Chapter 3 Conformal mappings | 57 |

Chapter 4 Loewner differential equation | 91 |

Chapter 5 Brownian measures on paths | 119 |

Chapter 6 SchrammLoewner evolution | 147 |

Chapter 7 More results about SLE | 177 |

Chapter 9 Restriction measures | 205 |

Appendix A Hausdorff dimension | 217 |

Appendix B Hypergeometric functions | 229 |

Appendix C Reflecting Brownian motion | 233 |

237 | |

240 | |

242 | |

Chapter 8 Brownian intersection exponent | 187 |

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### Common terms and phrases

analytic function assume boundary bounded Brownian motion starting chordal SLE complex Brownian motion conformal invariance conformal maps conformal transformation connected component consider continuous function converges Corollary defined derived dimh dist(z distribution driving function eacist estimate excursion measure exponent follows gambler's ruin gives half-plane harmonic function Hausdorff dimension heap(A Hence hulls Im(z implies inf{t integral Itō's formula Jordan domain Lemma let g limit Loewner chain Loewner equation loop loop-erased random walk Markov property martingale Möbius transformation Note O O O one-dimensional Brownian motion one-to-one parametrization particular PROOF PROPOSITION rad(A random walk Re(z REMARK satisfies sequence simple curve simply connected simply connected domain solution standard Brownian motion strong Markov property suffices to prove Theorem whole-plane write z e H