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Brownian Motion
Peter MÄorters and Yuval Peres
Contents
Foreword 7
List of frequently used notation 9
Chapter 0. Motivation 13
Chapter 1. De¯nition and ¯rst properties of Brownian motion 21
1. Paul L¶evy's construction of Brownian motion 21
2. Continuity properties of Brownian motion 27
3. Nondi®erentiability of Brownian motion 31
4. The Cameron-Martin theorem 37
Exercises 38
Notes and Comments 41
Chapter 2. Brownian motion as a strong Markov process 43
1. The Markov property and Blumenthal's 0-1 Law 43
2. The strong Markov property and the re°ection principle 46
3. Markov processes derived from Brownian motion 53
4. The martingale property of Brownian motion 57
Exercises 64
Notes and Comments 68
Chapter 3. Harmonic functions, transience and recurrence 69
1. Harmonic functions and the Dirichlet problem 69
2. Recurrence and transience of Brownian motion 75
3. Occupation measures and Green's functions 80
4. The harmonic measure 87
Exercises 94
Notes and Comments 96
Chapter 4. Hausdor® dimension: Techniques and applications 97
1. Minkowski and Hausdor® dimension 97
2. The mass distribution principle 106
3. The energy method 108
4. Frostman's lemma and capacity 111
Exercises 117
Notes and Comments 119
Chapter 5. Brownian motion and random walk 121
1. The law of the iterated logarithm 121
2. Points of increase for random walk and Brownian motion 126
3. The Skorokhod embedding problem 129
4. The Donsker invariance principle 134
5. The arcsine laws 137
Exercises 142
Notes and Comments 144
Chapter 6. Brownian local time 147
1. The local time at zero 147
2. A random walk approach to the local time process 158
3. The Ray-Knight theorem 163
4. Brownian local time as a Hausdor® measure 171
Exercises 179
Notes and Comments 181
Chapter 7. Stochastic integrals and applications 183
1. Stochastic integrals with respect to Brownian motion 183
2. Conformal invariance and winding numbers 194
3. Tanaka's formula and Brownian local time 202
4. Feynman-Kac formulas and applications 206
Exercises 213
Notes and Comments 215
Chapter 8. Potential theory of Brownian motion 217
1. The Dirichlet problem revisited 217
2. The equilibrium measure 220
3. Polar sets and capacities 226
4. Wiener's test of regularity 233
Exercises 236
Notes and Comments 238
Chapter 9. Intersections and self-intersections of Brownian paths 239
1. Intersection of paths: existence and Hausdor® dimension 239
2. Intersection equivalence of Brownian motion and percolation limit sets 247
3. Multiple points of Brownian paths 256
4. Kaufman's dimension doubling theorem 264
Exercises 270
Notes and Comments 272
Chapter 10. Exceptional sets for Brownian motion 275
1. The fast times of Brownian motion 275
2. Packing dimension and limsup fractals 283
3. Slow times of Brownian motion 292
4. Cone points of planar Brownian motion 296
Exercises 306
Notes and Comments 309
Appendix I: Hints and solutions for selected exercises 311
Appendix II: Background and prerequisites 331
1. Convergence of distributions on metric spaces 331
2. Gaussian random variables 334
3. Martingales in discrete time 337
4. The max-°ow min-cut theorem 342
Index 345
Bibliography 349