Brownian Motion and Martingales in Analysis (Wadsworth & Brooks/Cole Mathematics Series)

Cover......Page 1 Title Page......Page 3 Copyright Page......Page 4 Preface......Page 5 Contents......Page 9 1.1 Definition and Construction......Page 13 1.2 The Markov Property......Page 19 1.3 The Right Continuous Filtration, Blumenthal's 0-1 Law......Page 23 1.4 Stopping Times......Page 29 1.5 The Strong Markov Property......Page 33 1.6 Martingale Properties of Brownian Motion......Page 37 1.7 Hitting Probabilities, Recurrence, and Transience......Page 39 1.8 The Potential Kernels......Page 42 1.9 Brownian Motion in a Half Space......Page 44 1.10 Exit Distributions for the Sphere......Page 48 1.11 Occupation Times for the Sphere......Page 51 Notes on Chapter 1......Page 55 2.1 Integration w.r.t. Brownian Motion......Page 56 2.2 Integration w.r.t. Discrete Martingales......Page 60 2.3 The Basic Ingredients for Our Stochastic Integral......Page 62 2.4 The Variance and Covariance of Continuous Local Martingales......Page 64 2.5 Integration w.r.t. Continuous Local Martingales......Page 67 2.6 The Kunita-Watanabe Inequality......Page 71 2.7 Stochastic Differentials, the Associative Law......Page 74 2.8 Change of Variables, Ito's Formula......Page 76 2.9 Extension to Functions of Several Semimartingales......Page 79 2.10 Applications of Ito's Formula......Page 82 2.11 Change of Time, Levy's Theorem......Page 87 2.12 Conformal Invariance in d > 2, Kelvin's Transformations......Page 90 2.13 Change of Measure, Girsanov's Formula......Page 94 2.14 Martingales Adapted to Brownian Filtrations......Page 97 Notes on Chapter 2......Page 101 A Word about the Notes......Page 102 3.1 Warm-Up: Conditioned Random Walks......Page 103 3.2 Brownian Motion Conditioned to Exit H = Rd-1 x (0, oo) at 0......Page 106 3.3 Other Conditioned Processes in H......Page 109 3.4 Inversion in d > 3, B, Conditioned to Converge to 0 as t -* oo......Page 112 3.5 A Zero-One Law for Conditioned Processes......Page 114 4.1 Probabilistic Analogues of the Theorems of Privalov and Spencer......Page 117 4.2 Probability Is Less Stringent than Analysis......Page 120 4.3 Equivalence of Brownian and Nontangential Convergence in d=2......Page 125 4.4 Burkholder and Gundy's Counterexample (d = 3)......Page 128 4.5 With a Little Help from Analysis, Probability Works in d > 3: Brossard's Proof of Calderon's Theorem......Page 131 5.1 Conformal Invariance, Applications to Brownian Motion......Page 135 5.2 Nontangential Convergence in D......Page 138 5.3 Boundary Limits of Functions in the Nevanlinna Class N......Page 140 5.4 Two Special Properties of Boundary Limits of Analytic Functions......Page 144 5.5 Winding of Brownian Motion in C - {0} (Spitzer's Theorem)......Page 146 5.6 Tangling of Brownian Motion in C - { -1, 11 (Picard's Theorem)......Page 151 6.1 Definition of HP, an Important Example......Page 156 6.2 First Definition of .11", Differences Between p > 1 and p = 1......Page 158 6.3 A Second Definition of #P......Page 164 6.4 Equivalence of H" to a Subspace of &P......Page 167 6.5 Boundary Limits and Representation of Functions in HP......Page 170 6.6 Martingale Transforms......Page 174 6.7 Janson's Characterization of \mathcal{U}^1......Page 178 6.8 Inequalities for Conjugate Harmonic Functions......Page 182 6.9 Conjugate Functions of Indicators and Singular Measures......Page 192 7.1 The Duality Theorem for .,#'......Page 196 7.2 A Second Proof of (Jiu)* = .V.#0......Page 200 7.3 Equivalence of BMO to a Subspace of M. &&......Page 204 7.4 The Duality Theorem for H 1, Fefferman-Stein Decomposition......Page 211 7.5 Examples of Martingales in -4.,#......Page 217 7.6 The John-Nirenberg Inequality......Page 220 7.7 The Garnett-Jones Theorem......Page 223 7.8 A Disappointing Look at (.,ff?* When p 1......Page 318 A.7 Uniform Integrability and Convergence in L'......Page 319 A.8 Optional Stopping Theorems......Page 321 References......Page 325 Index of Notation......Page 337 Subject Index......Page 339 In the last forty years, it has been shown that Brownian motion can be used to prove many results in classical analysis. This book provides a self-contained survey of this area beginning with the definition of Brownian motion and then develops the theory needed for the applications.