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Differentiable Measures and the Malliavin Calculus Mathematical Surveys and Monographs Volume 164 Differentiable Measures and the Malliavin Calculus Vladimir I. Bogachev American Mathematical Society http://dx.doi.org/10.1090/surv/164 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 MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 28Cxx, 46Gxx, 58Bxx, 60Bxx, 60Hxx. For additional information and updates on this book, visit www.ams.org/bookpages/surv-164 Library of Congress Cataloging-in-Publication Data Bogachev, V. I. (Vladimir Igorevich), 1961– Differentiable measures and the Malliavin calculus / Vladimir I. Bogachev p. cm. – (Mathematical surveys and monographs ; v. 164) Includes bibliographical references and index. ISBN 978-0-8218-4993-4 (alk. paper) 1. Measure theory. 2. Theory of distributions (Functional analysis) 3. Sobolev spaces. 4. Malliavin calculus. I. Title. QA312.B638 2010 515.42–dc22 2010005829 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 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 W p,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 W p,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 differentiable measures and the Malliavin calculus. These two closely related 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 M 0. 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 M 0. 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 x Preface −1 of measures µt = µ◦Tt generated by a family of transformations Tt of a fixed measure µ. Fomin’s differentiability corresponds to the case where the measures µt are the shifts µth of a measure µ on a linear space. However, the theory constructed for this more special case is much richer. Let us also mention similar ideas of L. Gross [322] developed in the analysis on Wiener space. In the mid 1970s P. Malliavin [751] suggested an elegant method for proving the smoothness of the transition probabilities of finite dimensional diffusions. The essence of the method is to consider the transition proba- bilities Pt as images of the Wiener measure under some nonlinear transfor- mations (generated by stochastic differential equations) and then apply an integration by parts formula on the Wiener space leading to certain estimates ∞ of the generalized derivatives of Pt which ensure the membership in C .
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