Preface ...................................................... viii
Frequently used notation ........................................ x
Motivation ...................................................... 1
1 Brownian motion as a random function ......................... 7
1.1 Paul Levy's construction of Brownian motion ............. 7
1.2 Continuity properties of Brownian motion ............... 14
1.3 Nondifferentiability of Brownian motion ................ 18
1.4 The Cameron-Martin theorem ............................. 24
Exercises ................................................... 30
Notes and comments .......................................... 33
2 Brownian motion as a strong Markov process .................. 36
2.1 The Markov property and Blumenthal's 0-1 law ........... 36
2.2 The strong Markov property and the reflection
principle .............................................. 40
2.3 Markov processes derived from Brownian motion .......... 48
2.4 The martingale property of Brownian motion ............. 53
Exercises ................................................... 59
Notes and comments .......................................... 63
3 Harmonic functions, transience and recurrence ............... 65
3.1 Harmonic functions and the Dirichlet problem ........... 65
3.2 Recurrence and transience of Brownian motion ........... 71
3.3 Occupation measures and Green's functions .............. 76
3.4 The harmonic measure ................................... 84
Exercises ................................................... 91
Notes and comments .......................................... 94
4 Hausdorff dimension: Techniques and applications ............ 96
4.1 Minkowski and Hausdorff dimension ...................... 96
4.2 The mass distribution principle ....................... 105
4.3 The energy method ..................................... 108
4.4 Frostman's lemma and capacity ......................... 111
Exercises .................................................. 115
Notes and comments ......................................... 116
5 Brownian motion and random walk ............................ 118
5.1 The law of the iterated logarithm ..................... 118
5.2 Points of increase for random walk and Brownian
motion ................................................ 123
5.3 Skorokhod embedding and Donsker's invariance
principle ............................................. 127
5.4 The arcsine laws for random walk and Brownian motion .. 135
5.5 Pitman's 2M - В theorem ............................... 140
Exercises .................................................. 146
Notes and comments ......................................... 149
6 Brownian local time ........................................ 153
6.1 The local time at zero ................................ 153
6.2 A random walk approach to the local time process ...... 165
6.3 The Ray-Knight theorem ................................ 170
6.4 Brownian local time as a Hausdorff measure ............ 178
Exercises .................................................. 186
Notes and comments ......................................... 187
7 Stochastic integrals and applications ...................... 190
7.1 Stochastic integrals with respect to Brownian motion .. 190
7.2 Conformal invariance and winding numbers .............. 201
7.3 Tanaka's formula and Brownian local time .............. 209
7.4 Feynman-Kac formulas and applications ................. 213
Exercises .................................................. 220
Notes and comments ......................................... 222
8 Potential theory of Brownian motion ........................ 224
8.1 The Dirichlet problem revisited ....................... 224
8.2 The equilibrium measure ............................... 227
8.3 Polar sets and capacities ............................. 234
8.4 Wiener's test of regularity ........................... 248
Exercises .................................................. 251
Notes and comments ......................................... 253
9 Intersections and self-intersections of Brownian paths ..... 255
9.1 Intersection of paths: Existence and Hausdorff
dimension ............................................. 255
9.2 Intersection equivalence of Brownian motion and
percolation limit sets ................................ 263
9.3 Multiple points of Brownian paths ..................... 272
9.4 Kaufman's dimension doubling theorem .................. 279
Exercises .................................................. 285
Notes and comments ......................................... 287
10 Exceptional sets for Brownian motion ....................... 290
10.1 The fast times of Brownian motion ..................... 290
10.2 Packing dimension and limsup fractals ................. 298
10.3 Slow times of Brownian motion ......................... 307
10.4 Cone points of planar Brownian motion ................. 312
Exercises .................................................. 322
Notes and comments ......................................... 324
Appendix A: Further developments
11 Stochastic Loewner evolution and planar Brownian motion .... 327
by Oded Schramm and Wendelin Werner
11.1 Some subsets of planar Brownian paths ................. 327
11.2 Paths of stochastic Loewner evolution ................. 331
11.3 Special properties of SLE(6) .......................... 339
11.4 Exponents of stochastic Loewner evolution ............. 340
Notes and comments ......................................... 344
Appendix B: Background and prerequisites ...................... 346
12.1 Convergence of distributions ......................... 346
12.2 Gaussian random variables ............................ 349
12.3 Martingales in discrete time ......................... 351
12.4 Trees and flows on trees ............................. 358
Hints and solutions for selected exercises ................. 361
Selected open problems ..................................... 383
Bibliography .................................................. 386
Index ......................................................... 400
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