Brownian Motion

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ISBN: 9780521760188 | 0521760186
Cover: Hardcover
Copyright: 3/25/2010

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This eagerly awaited textbook offers a broad and deep exposition of Brownian motion. Extensively class tested, it leads the reader from the basics to the latest research in the area.

Preface	p. viii
Frequently used notation	p. x
Motivation	p. 1
Brownian motion as a random function	p. 7
Paul Lévy's construction of Brownian motion	p. 7
Continuity properties of Brownian motion	p. 14
Nondifferentiability of Brownian motion	p. 18
The Cameron-Martin theorem	p. 24
Exercises	p. 30
Notes and comments	p. 33
Brownian motion as a strong Markov process	p. 36
The Markov property and Blumenthal's 0-1 law	p. 36
The strong Markov property and the reflection principle	p. 40
Markov processes derived from Brownian motion	p. 48
The martingale property of Brownian motion	p. 53
Exercises	p. 59
Notes and comments	p. 63
Harmonic functions, transience and recurrence	p. 65
Harmonic functions and the Dirichlet problem	p. 65
Recurrence and transience of Brownian motion	p. 71
Occupation measures and Green's functions	p. 76
The harmonic measure	p. 84
Exercises	p. 91
Notes and comments	p. 94
Hausdorff dimension: Techniques and applications	p. 96
Minkowski and Hausdorff dimension	p. 96
The mass distribution principle	p. 105
The energy method	p. 108
Frostman's lemma and capacity	p. 111
Exercises	p. 115
Notes and comments	p. 116
Brownian motion and random walk	p. 118
The law of the iterated logarithm	p. 118
Points of increase for random walk and Brownian motion	p. 123
Skorokhod embedding and Donsker's invariance principle	p. 127
The arcsine laws for random walk and Brownian motion	p. 135
Pitman's 2M - B theorem	p. 140
Exercises	p. 146
Notes and comments	p. 149
Brownian local time	p. 153
The local time at zero	p. 153
A random walk approach to the local time process	p. 165
The Ray-Knight theorem	p. 170
Brownian local time as a Hausdorff measure	p. 178
Exercises	p. 186
Notes and comments	p. 187
Stochastic integrals and applications	p. 190
Stochastic integrals with respect to Brownian motion	p. 190
Conformal invariance and winding numbers	p. 201
Tanaka's formula and Brownian local time	p. 209
Feynman-Kac formulas and applications	p. 213
Exercises	p. 220
Notes and comments	p. 222
Potential theory of Brownian motion	p. 224
The Dirichlet problem revisited	p. 224
The equilibrium measure	p. 227
Polar sets and capacities	p. 234
Wiener's test of regularity	p. 248
Exercises	p. 251
Notes and comments	p. 253
Intersections and self-intersections of Brownian paths	p. 255
Intersection of paths: Existence and Hausdorff dimension	p. 255
Intersection equivalence of Brownian motion and percolation limit sets	p. 263
Multiple points of Brownian paths	p. 272
Kaufman's dimension doubling theorem	p. 279
Exercises	p. 285
Notes and comments	p. 287
Exceptional sets for Brownian motion	p. 290
The fast times of Brownian motion	p. 290
Packing dimension and limsup fractals	p. 298
Slow times of Brownian motion	p. 307
Cone points of planar Brownian motion	p. 312
Exercises	p. 322
Notes and comments	p. 324
Further developments
Stochastic Loewner evolution and planar Brownian motion	p. 327
Some subsets of planar Brownian paths	p. 327
Paths of stochastic Loewner evolution	p. 331
Special properties of SLE(6)	p. 339
Exponents of stochastic Loewner evolution	p. 340
Notes and comments	p. 344
Background and prerequisites	p. 346
Convergence of distributions	p. 346
Gaussian random variables	p. 349
Martingales in discrete time	p. 351
Trees and flows on trees	p. 358
Hints and solutions for selected exercises	p. 361
Selected open problems	p. 383
Bibliography	p. 386
Index	p. 400
Table of Contents provided by Ingram. All Rights Reserved.

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Brownian Motion

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Table of Contents

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