Preface ........................................................ ix
Chapter 1 Background material ................................. 1
1.1 Functional analysis ........................................ 1
1.2 Measures on topological spaces ............................. 6
1.3 Conditional measures ...................................... 19
1.4 Gaussian measures ......................................... 23
1.5 Stochastic integrals ...................................... 29
1.6 Comments and exercises .................................... 35
Chapter 2 Sobolev spaces on  n .............................. 39
2.1 The Sobolev classes Wp,k .................................. 39
2.2 Embedding theorems for Sobolev classes .................... 45
2.3 The classes BV ............................................ 51
2.4 Approximate differentiability and Jacobians ............... 52
2.5 Restrictions and extensions ............................... 56
2.6 Weighted Sobolev classes .................................. 58
2.7 Fractional Sobolev classes ................................ 65
2.8 Comments and exercises .................................... 66
Chapter 3 Differentiable measures on linear spaces ............ 69
3.1 Directional differentiability ............................. 69
3.2 Properties of continuous measures ......................... 73
3.3 Properties of differentiable measures ..................... 76
3.4 Differentiable measures on n ............................. 82
3.5 Characterization by conditional measures .................. 88
3.6 Skorohod differentiability ................................ 91
3.7 Higher order differentiability ........................... 100
3.8 Convergence of differentiable measures ................... 101
3.9 Comments and exercises ................................... 103
Chapter 4 Some classes of differentiable measures ............ 105
4.1 Product measures ......................................... 105
4.2 Gaussian and stable measures ............................. 107
4.3 Convex measures .......................................... 112
4.4 Distributions of random processes ........................ 122
4.1 Gibbs measures and mixtures of measures .................. 127
4.6 Comments and exercises ................................... 131
Chapter 5 Subspaces of differentiability of measures ......... 133
5.1 Geometry of subspaces of differentiability ............... 133
5.2 Examples ................................................. 136
5.3 Disposition of subspaces of differentiability ............ 141
5.4 Differentiability along subspaces ........................ 149
5.5 Comments and exercises ................................... 155
Chapter 6 Integration by parts and logarithmic
derivatives ....................................... 157
6.1 Integration by parts formulae ............................ 157
6.2 Integrability of logarithmic derivatives ................. 161
6.3 Differentiability of logarithmic derivatives ............. 168
6.4 Quasi-invariance and differentiability ................... 170
6.5 Convex functions ......................................... 173
6.6 Derivatives along vector fields .......................... 180
6.7 Local logarithmic derivatives ............................ 183
6.8 Comments and exercises ................................... 187
Chapter 7 Logarithmic gradients .............................. 189
7.1 Rigged Hilbert spaces .................................... 189
7.2 Definition of logarithmic gradient ....................... 190
7.3 Connections with vector measures ......................... 194
7.4 Existence of logarithmic gradients ....................... 198
7.5 Measures with given logarithmic gradients ................ 204
7.6 Uniqueness problems ...................................... 209
7.7 Symmetries of measures and logarithmic gradients ......... 217
7.8 Mappings and equations connected with logarithmic
gradients ................................................ 222
7.9 Comments and exercises ................................... 223
Chapter 8 Sobolev classes on infinite dimensional spaces ..... 227
8.1 The classes Wp,r ......................................... 227
8.2 The classes Dp,r ......................................... 230
8.3 Generalized derivatives and the classes Gp,r ............. 233
8.4 The semigroup approach ................................... 235
8.5 The Gaussian case ........................................ 239
8.6 The interpolation approach ............................... 242
8.7 Connections between different definitions ................ 246
8.8 The logarithmic Sobolev inequality ....................... 250
8.9 Compactness in Sobolev classes ........................... 253
8.10 Divergence ............................................... 254
8.11 An approach via stochastic integrals ..................... 257
8.12 Some identities of the Malliavin calculus ................ 263
8.13 Sobolev capacities ....................................... 265
8.14 Comments and exercises ................................... 274
Chapter 9 The Malliavin calculus ............................. 279
9.1 General scheme ........................................... 279
9.2 Absolute continuity of images of measures ................ 282
9.3 Smoothness of induced measures ........................... 288
9.4 Infinite dimensional oscillatory integrals ............... 297
9.5 Surface measures ......................................... 299
9.6 Convergence of nonlinear images of measures .............. 307
9.7 Supports of induced measures ............................. 319
9.8 Comments and exercises ................................... 323
Chapter 10 Infinite dimensional transformations ............... 329
10.1 Linear transformations of Gaussian measures .............. 329
10.2 Nonlinear transformations of Gaussian measures ........... 334
10.3 Transformations of smooth measures ....................... 338
10.4 Absolutely continuous flows .............................. 340
10.5 Negligible sets .......................................... 342
10.6 Infinite dimensional Rademacher's theorem ................ 350
10.7 Triangular and optimal transformations ................... 358
10.8 Comments and exercises ................................... 365
Chapter 11 Measures on manifolds .............................. 369
11.1 Measurable manifolds and Malliavin's method .............. 370
11.2 Differentiable families of measures ...................... 379
11.3 Current and loop groups .................................. 390
11.4 Poisson spaces ........................................... 393
11.5 Diffeomorphism groups .................................... 394
11.6 Comments and exercises ................................... 398
Chapter 12 Applications ....................................... 401
12.1 A probabilistic approach to hypoellipticity ............. 401
12.2 Equations for measures .................................. 407
12.3 Logarithmic gradients and symmetric diffusions .......... 414
12.4 Dirichlet forms and differentiable measures ............. 416
12.5 The uniqueness problem for invariant measures ........... 420
12.6 Existence of Gibbs measures ............................. 422
12.7 Comments and exercises .................................. 424
References .................................................... 427
Subject Index ................................................. 483
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