ISBN: 3-540-66455-6
TITLE: Transformation of Measure on Wiener Space
AUTHOR: stnel, A.Sleyman; Zakai, Moshe
TOC:

Introduction 1
1. Some Background Material and Preliminary Results 5
1.1 The Radon-Nikodym Theorem 5
1.2 Uniform Integrability 10
1.3 Two Measures Associated with a Point Transformation 13
1.4 Kakutani's Dichotomy Theorem 16
1.5 The Structure of Non-negative Continuous Martingales 17
Notes and References 19
2. Transformation of Measure Induced by Adapted Shifts 21
2.1 The Girsanov Theorem 21
2.2 Integrability Conditions on Lambda_t 25
2.3 Transformation of Measure Induced on C_0 ([0, 1]) by Direct Shifts 28
2.4 Transformations of Measure Induced on C_0 ([0, 1]) by Indirect Shifts 30
2.5 The Innovation Theorem 37
2.6 A Dimension-free Extension of Results of the Previous Sections 41
2.7 The Representability of Measures by Shifts 47
Notes and References 51
3. Transformation of Measure Induced by General Shifts 53
3.1 Introduction 53
3.2 The Change of Variables Formula for a Small Perturbation of the Identity 56
3.3 H-Regularity of Random Variables 71
3.4 Some Preliminaries 80
3.5 The Change of Variables Formula. 86
3.6 A Comparison Between the Formulas Associated with Adapted and Non Adapted Cases 94
Notes and References 97
4. The Sard Inequality 99
4.1 Introduction and Preliminaries 99
4.2 The Measurability of the Forward Image 102
4.3 The Sard Inequality I 103
4.4 The Sard Inequality II 109
4.5 Some Applications to Absolute Continuity 112
Notes and References 113
5. Transformation of Measure Under Anticipative Flows 115
5.1 Introduction and Finite Dimensional Flows 115
5.2 Cylindrical Flows 120
5.3 Infinite Dimensional Flows 127
5.4 A Singular Flow on the Classical Wiener Space 140
Notes and References 156
6. Monotone Shifts 157
6.1 Introduction 157
6.2 Monotone Shifts 158
6.3 Absolute Continuity of Monotone Shifts-I 162
6.4 Absolute Continuity of Monotone Shifts-II 171
6.5 Shifts of Hammerstein Type 176
Notes and References 180
7. Generalized Radon-Nikodym Derivatives 181
7.1 Introduction 181
7.2 The G Class of Wiener Functionals and its Composition with Shifts 181
7.2.1 The G -class of Wiener Functionals 181
7.2.2 The Extendibility of G Functionals 184
7.3 A Generalized R-N Derivative for G Functionals 185
7.4 The Conditioning of G Functionals with Respect to Certain Sub-sigma-fields 189
7.5 Composition of the Rademacher Class of Wiener Functionals with Shifts 190
7.6 The Composition Rules 195
7.6.1 The Cylindrical Case 195
7.6.2 Extensions of the Composition Rules197
Notes and References 205
8. Random Rotations207
8.1 Introduction 207
8.2 Random Rotations 208
8.3 A Partial Converse to Theorem 8.2.1 216
8.4 The Invertibility of Tw=w+R(w)h and that of R 218
8.5 Stochastic Calculus of Rotations 220
8.5.1 A New Derivation and Calculation of E[delta eta B] 223
8.5.2 Case of Deterministic R 225
8.6 Transformations of Measure Induced by Euclidean Motions of the Wiener Path 226
Notes and References 231
9. The Degree Theorem on Wiener Space 233
9.1 Introduction 233
9.2 Measure Theoretic Degree 234
9.3 Applications to Absolute Continuity 240
9.4 Relations with Leray-Schauder Degree 246
Notes and References 254
A. Some Inequalities 255
A.1 Gronwall and Young Inequalities 255
A.1.1 Gronwall Inequality 255
A.1.2 Young Inequality 255
A.2 Some Inequalities for det2(I_H+A) 256
B. An Introduction to Malliavin Calculus 259
B.1 Introduction to Abstract Wiener Space 259
B.2 An Introduction to Analysis on Wiener Space 261
B.3 Construction of Sobolev Derivatives 263
B.4 The Divergence 266
B.5 Ornstein-Uhlenbeck Operator and Meyer Inequalities 269
B.6 Some Useful Lemmas 272
B.7 Local Versus Global Differentiability of Wiener Functionals 279
B.8 Exponential Integrability of Wiener Functionals and Poincar
Inequality 281
Notes and References 288
References 289
Index 295
Notations 297
END
