Differentiable measures and the Malliavin calculus

506 Pages·2010·2.427 MB·English

by  Vladimir I. Bogachev

Mathematical
Surveys
and
Monographs
Volume 164
Differentiable Measures
and the Malliavin Calculus
Vladimir I. Bogachev
American Mathematical Society
Differentiable Measures Differentiable Measures
and the Malliavin Calculus and the Malliavin Calculus
Vladimir I. Bogachev

Mathematical
Surveys
and
Monographs
Volume 164
Differentiable Measures
and the Malliavin Calculus
Vladimir I. Bogachev
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Ralph L. Cohen, Chair Michael A. Singer
Eric M. Friedlander Benjamin Sudakov
Michael I. Weinstein
2010 Mathematics Subject Classification. Primary 28Cxx, 46Gxx, 58Bxx, 60Bxx, 60Hxx.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-164
Library of Congress Cataloging-in-Publication Data
Bogachev,V.I.(VladimirIgorevich),1961–
DifferentiablemeasuresandtheMalliavincalculus/VladimirI.Bogachev
p.cm. –(Mathematicalsurveysandmonographs;v. 164)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4993-4(alk.paper)
1. Measure theory. 2. Theory of distributions (Functional analysis) 3. Sobolev spaces.
4.Malliavincalculus. I.Title.
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Contents
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 IRn 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 IRn 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
v
vi Contents
4.5. 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
Contents vii
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

Preface
Among the most notable events in nonlinear functional analysis for the
last three decades one should mention the development of the theory of
differentiablemeasuresandtheMalliavincalculus. Thesetwocloselyrelated
theories can be regarded at the same time as infinite dimensional analogues
of such classical fields as geometric measure theory, the theory of Sobolev
spaces, and the theory of generalized functions.
The theory of differentiable measures was suggested by S.V. Fomin in
his address at the International Congress of Mathematicians in Moscow in
1966 as a candidate for an infinite dimensional substitute of the Sobolev–
Schwartz theory of distributions (see [440]–[443], [82], [83]). It was Fomin
who realized that in the infinite dimensional case instead of a pair of spaces
(test functions – generalized functions) it is natural to consider four spaces:
a certain space of functions S, its dual S0, a certain space of measures M,
and its dual M0. In the finite dimensional case, M is identified with S
by means of Lebesgue measure which enables us to represent measures via
their densities. The absence of translationally invariant nonzero measures
destroys this identification in the infinite dimensional case. Fomin found a
convenient definition of a smooth measure on an infinite dimensional space
corresponding to a measure with a smooth density on IRn. By means of
this definition one introduces the space M of smooth measures. The Fourier
transform then acts between S and M and between S0 and M0. In this
way a theory of pseudo-differential operators can be developed: the initial
objective of Fomin was a theory of infinite dimensional partial differential
equations. However, as it often happens with fruitful ideas, the theory of
differentiable measures overgrew the initial framework. This theory has be-
come an efficient tool in a wide variety of the most diverse applications
such as stochastic analysis, quantum field theory, and nonlinear analysis.
At present it is a rapidly developing field abundant with many challeng-
ing problems of great importance for better understanding of the nature of
infinite dimensional phenomena.
It is worth mentioning that before the pioneering works of Fomin re-
lated ideas appeared in several papers by T. Pitcher on the distributions
of random processes in functional spaces. T. Pitcher investigated an even
more general situation, namely he studied the differentiability of a family
ix