## Brownian MotionThis eagerly awaited textbook covers everything the graduate student in probability wants to know about Brownian motion, as well as the latest research in the area. Starting with the construction of Brownian motion, the book then proceeds to sample path properties like continuity and nowhere differentiability. Notions of fractal dimension are introduced early and are used throughout the book to describe fine properties of Brownian paths. The relation of Brownian motion and random walk is explored from several viewpoints, including a development of the theory of Brownian local times from random walk embeddings. Stochastic integration is introduced as a tool and an accessible treatment of the potential theory of Brownian motion clears the path for an extensive treatment of intersections of Brownian paths. An investigation of exceptional points on the Brownian path and an appendix on SLE processes, by Oded Schramm and Wendelin Werner, lead directly to recent research themes. |

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### Contents

7 | |

Brownian motion as a strong Markov process | 36 |

Harmonic functions transience and recurrence | 65 |

Techniques and applications | 96 |

Brownian motion and random walk | 118 |

Brownian local time | 153 |

Stochastic integrals and applications | 190 |

Potential theory of Brownian motion | 224 |

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### Common terms and phrases

ball Borel set Borel-Cantelli lemma boundary Brownian motion B(t Brownian motion started Brownian paths Brownian scaling compact set completes the proof conformal map constant construction convergence cubes deﬁne Deﬁnition denote density differentiable dimp Dirichlet problem domain dyadic Exercise exists exit finite ﬁrst fixed follows Fubini's theorem Gaussian given harmonic function harmonic measure Hausdorff dimension Hausdorff measure Hence hitting implies independent inequality inf{t inﬁnite intersection interval invariance principle kernel Lemma Let B(t limsup linear Brownian motion lower bound Markov process Markov property martingale Note obtain packing dimension percolation limit sets planar Brownian motion point of increase probability measure proof of Theorem prove random variables Recall reﬂection Remark result right hand side sequence simple random walk standard Brownian motion stopping strong Markov property Suppose surely upper bound vector zero