Brownian motion

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 10
Vectors, functions, and measures:......Page 12
Stopping times:......Page 13
Sets and processes associated with Brownian motion:......Page 14
Motivation......Page 15
1.1.1 Definition of Brownian motion......Page 21
1.1.2 Paul Lévy’s construction of Brownian motion......Page 23
1.1.3 Simple invariance properties of Brownian motion......Page 26
1.2 Continuity properties of Brownian motion......Page 28
1.3 Nondifferentiability of Brownian motion......Page 32
1.4 The Cameron–Martin theorem......Page 38
Exercises......Page 44
Notes and comments......Page 47
2.1 The Markov property and Blumenthal’s 0-1 law......Page 50
2.2 The strong Markov property and the reflection principle......Page 54
2.2.1 The reflection principle......Page 58
2.2.2 The area of planar Brownian motion......Page 59
2.3 Markov processes derived from Brownian motion......Page 62
2.4 The martingale property of Brownian motion......Page 67
Exercises......Page 73
Notes and comments......Page 77
3.1 Harmonic functions and the Dirichlet problem......Page 79
3.2 Recurrence and transience of Brownian motion......Page 85
3.3 Occupation measures and Green’s functions......Page 90
3.4 The harmonic measure......Page 98
Exercises......Page 105
Notes and comments......Page 108
4.1.1 The Minkowski dimension......Page 110
4.1.2 The Hausdorff dimension......Page 112
4.1.3 Upper bounds on the Hausdorff dimension......Page 115
4.2 The mass distribution principle......Page 119
4.3 The energy method......Page 122
4.4 Frostman’s lemma and capacity......Page 125
Exercises......Page 129
Notes and comments......Page 130
5.1 The law of the iterated logarithm......Page 132
5.2 Points of increase for random walk and Brownian motion......Page 137
5.3.1 The Dubins’ embedding theorem......Page 141
5.3.2 The Azéma–Yor embedding theorem......Page 143
5.3.3 The Donsker invariance principle......Page 145
5.4 The arcsine laws for random walk and Brownian motion......Page 149
5.5 Pitman’s 2M - B theorem......Page 154
Exercises......Page 160
Notes and comments......Page 163
6.1 The local time at zero......Page 167
6.2 A random walk approach to the local time process......Page 179
6.3 The Ray–Knight theorem......Page 184
6.4 Brownian local time as a Hausdorff measure......Page 192
Exercises......Page 200
Notes and comments......Page 201
7.1.1 Construction of the stochastic integral......Page 204
7.1.2 Itô’s formula......Page 209
7.2 Conformal invariance and winding numbers......Page 215
7.3 Tanaka’s formula and Brownian local time......Page 223
7.4 Feynman–Kac formulas and applications......Page 227
Exercises......Page 234
Notes and comments......Page 236
8.1 The Dirichlet problem revisited......Page 238
8.2 The equilibrium measure......Page 241
8.3 Polar sets and capacities......Page 248
8.4 Wiener’s test of regularity......Page 262
Exercises......Page 265
Notes and comments......Page 267
9.1.1 Existence of intersections......Page 269
9.1.2 Stochastic co-dimension and percolation limit sets......Page 272
9.1.3 Hausdorff dimension of intersections......Page 274
9.2 Intersection equivalence of Brownian motion and percolation limit sets......Page 277
9.3 Multiple points of Brownian paths......Page 286
9.4 Kaufman’s dimension doubling theorem......Page 293
Exercises......Page 299
Notes and comments......Page 301
10.1 The fast times of Brownian motion......Page 304
10.2 Packing dimension and limsup fractals......Page 312
10.3 Slow times of Brownian motion......Page 321
10.4 Cone points of planar Brownian motion......Page 326
Exercises......Page 336
Notes and comments......Page 338
11.1.1 The questions......Page 341
11.1.2 Reformulation in terms of Brownian hulls......Page 343
11.1.3 An alternative characterisation of Brownian hulls......Page 344
11.2.1 Heuristic description......Page 345
11.2.2 Loewner’s equation......Page 346
11.2.3 The loop-erased random walk......Page 347
11.2.5 Critical percolation and SLE(6)......Page 350
11.3 Special properties of SLE(6)......Page 353
11.4.1 A radial computation......Page 354
11.4.2 Consequences......Page 356
11.4.3 From exponents to dimensions......Page 357
12.1 Convergence of distributions......Page 360
12.2 Gaussian random variables......Page 363
Notes and comments......Page 358
12.3 Martingales in discrete time......Page 365
12.4 Trees and flows on trees......Page 372
Hints and solutions for selected exercises......Page 375
Selected open problems......Page 397
Bibliography......Page 400
Index......Page 414