Brownian Motion
Peter Mörters and Yuval Peres
with an appendix by Oded Schramm and Wendelin Werner

Preface page viii
Frequently used notation x
Motivation 1
1 Brownian motion as a random function 7
1.1 Paul Lévy’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 − B 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