Brownian Motion and Martingales in Analysis (The Wadsworth Mathematics Series)

Cover......Page 1
Title Page......Page 3
Copyright Page......Page 4
Preface......Page 5
Contents\0......Page 9
1.1 Definition and Construction\0......Page 13
1.2 The Markov Property\0......Page 19
1.3 The Right Continuous Filtration, Blumenthal\'s 0-1 Law\0......Page 23
1.4 Stopping Times\0......Page 29
1.5 The Strong Markov Property\0......Page 33
1.6 Martingale Properties of Brownian Motion\0......Page 37
1.7 Hitting Probabilities, Recurrence, and Transience\0......Page 39
1.8 The Potential Kernels\0......Page 42
1.9 Brownian Motion in a Half Space\0......Page 44
1.10 Exit Distributions for the Sphere\0......Page 48
1.11 Occupation Times for the Sphere\0......Page 51
Notes on Chapter 1\0......Page 55
2.1 Integration w.r.t. Brownian Motion\0......Page 56
2.2 Integration w.r.t. Discrete Martingales\0......Page 60
2.3 The Basic Ingredients for Our Stochastic Integral\0......Page 62
2.4 The Variance and Covariance of Continuous Local Martingales\0......Page 64
2.5 Integration w.r.t. Continuous Local Martingales\0......Page 67
2.6 The Kunita-Watanabe Inequality\0......Page 71
2.7 Stochastic Differentials, the Associative Law\0......Page 74
2.8 Change of Variables, Ito\'s Formula\0......Page 76
2.9 Extension to Functions of Several Semimartingales\0......Page 79
2.10 Applications of Ito\'s Formula\0......Page 82
2.11 Change of Time, Levy\'s Theorem\0......Page 87
2.12 Conformal Invariance in d > 2, Kelvin\'s Transformations\0......Page 90
2.13 Change of Measure, Girsanov\'s Formula\0......Page 94
2.14 Martingales Adapted to Brownian Filtrations\0......Page 97
Notes on Chapter 2\0......Page 101
A Word about the Notes\0......Page 102
3.1 Warm-Up: Conditioned Random Walks\0......Page 103
3.2 Brownian Motion Conditioned to Exit H = Rd-1 x (0, oo) at 0\0......Page 106
3.3 Other Conditioned Processes in H\0......Page 109
3.4 Inversion in d > 3, B, Conditioned to Converge to 0 as t -* oo\0......Page 112
3.5 A Zero-One Law for Conditioned Processes\0......Page 114
4.1 Probabilistic Analogues of the Theorems of Privalov and Spencer\0......Page 117
4.2 Probability Is Less Stringent than Analysis\0......Page 120
4.3 Equivalence of Brownian and Nontangential Convergence in d=2\0......Page 125
4.4 Burkholder and Gundy\'s Counterexample (d = 3)\0......Page 128
4.5 With a Little Help from Analysis, Probability Works in d > 3: Brossard\'s Proof of Calderon\'s Theorem\0......Page 131
5.1 Conformal Invariance, Applications to Brownian Motion\0......Page 135
5.2 Nontangential Convergence in D\0......Page 138
5.3 Boundary Limits of Functions in the Nevanlinna Class N\0......Page 140
5.4 Two Special Properties of Boundary Limits of Analytic Functions\0......Page 144
5.5 Winding of Brownian Motion in C - {0} (Spitzer\'s Theorem)\0......Page 146
5.6 Tangling of Brownian Motion in C - { -1, 11 (Picard\'s Theorem)\0......Page 151
6.1 Definition of HP, an Important Example\0......Page 156
6.2 First Definition of .11\", Differences Between p > 1 and p = 1\0......Page 158
6.3 A Second Definition of #P\0......Page 164
6.4 Equivalence of H\" to a Subspace of &P\0......Page 167
6.5 Boundary Limits and Representation of Functions in HP\0......Page 170
6.6 Martingale Transforms\0......Page 174
6.7 Janson\'s Characterization of \\mathcal{U}^1\0......Page 178
6.8 Inequalities for Conjugate Harmonic Functions\0......Page 182
6.9 Conjugate Functions of Indicators and Singular Measures\0......Page 192
7.1 The Duality Theorem for .,#\'\0......Page 196
7.2 A Second Proof of (Jiu)* = .V.#0\0......Page 200
7.3 Equivalence of BMO to a Subspace of M. &&\0......Page 204
7.4 The Duality Theorem for H 1, Fefferman-Stein Decomposition\0......Page 211
7.5 Examples of Martingales in -4.,#\0......Page 217
7.6 The John-Nirenberg Inequality\0......Page 220
7.7 The Garnett-Jones Theorem\0......Page 223
7.8 A Disappointing Look at (.,ff?* When p < 1\0......Page 227
A Parabolic Equations\0......Page 231
8.1 The Heat Equation\0......Page 232
8.2 The Inhomogeneous Equation\0......Page 235
8.3 The Feynman-Kac Formula\0......Page 241
8.4 The Cameron-Martin Transformation\0......Page 246
B  Elliptic Equations\0......Page 257
8.5 The Dirichlet Problem\0......Page 258
8.6 Poisson\'s Equation\0......Page 263
8.7 The Schrodinger Equation\0......Page 267
8.8 Eigenvalues of A + c\0......Page 275
9.1 PDE\'s That Can Be Solved by Running an SDE\0......Page 283
9.2 Existence of Solutions to SDE\'s with Continuous Coefficients\0......Page 286
9.3 Uniqueness of Solutions to SDE\'s with Lipschitz Coefficients\0......Page 290
9.4 Some Examples\0......Page 295
9.5 Solutions Weak and Strong, Uniqueness Questions\0......Page 298
9.6 Markov and Feller Properties\0......Page 300
9.7 Conditions for Smoothness\0......Page 302
Notes on Chapter 9\0......Page 305
A.1 Some Differences in the Language\0......Page 306
A.2 Independence and Laws of Large Numbers\0......Page 308
A.3 Conditional Expectation\0......Page 312
A.4 Martingales\0......Page 314
A.5 Gambling Systems and the Martingale Convergence Theorem\0......Page 315
A.6 Doob\'s Inequality, Convergence in L\", p > 1\0......Page 318
A.7 Uniform Integrability and Convergence in L\'\0......Page 319
A.8 Optional Stopping Theorems\0......Page 321
References\0......Page 325
Index of Notation\0......Page 337
Subject Index\0......Page 339