**Mathematical Surveys and Monographs**

Volume: 164;
2010;
488 pp;
Hardcover

MSC: Primary 28; 46; 58; 60;

Print ISBN: 978-0-8218-4993-4

Product Code: SURV/164

List Price: $113.00

Individual Member Price: $90.40

**Electronic ISBN: 978-1-4704-1391-0
Product Code: SURV/164.E**

List Price: $113.00

Individual Member Price: $90.40

#### Supplemental Materials

# Differentiable Measures and the Malliavin Calculus

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*Vladimir I. Bogachev*

This book provides the reader with the principal concepts and results
related to differential properties of measures on infinite dimensional
spaces. In the finite dimensional case such properties are described in
terms of densities of measures with respect to Lebesgue measure. In the
infinite dimensional case new phenomena arise. For the first time a
detailed account is given of the theory of differentiable measures,
initiated by S. V. Fomin in the 1960s; since then the method has found
many various important applications. Differentiable properties are
described for diverse concrete classes of measures arising in
applications, for example, Gaussian, convex, stable,
Gibbsian, and for distributions of random processes. Sobolev classes for
measures on finite and infinite dimensional spaces are discussed in
detail. Finally, we present the main ideas and results of the Malliavin
calculus—a powerful method to study smoothness properties of the
distributions of nonlinear functionals on infinite dimensional spaces with
measures.

The target readership includes mathematicians and physicists whose
research is related to measures on infinite dimensional spaces,
distributions of random processes, and differential equations in infinite
dimensional spaces. The book includes an extensive bibliography on the
subject.

#### Readership

Graduate students and research mathematicians interested in measure theory and random processes.

#### Table of Contents

# Table of Contents

## Differentiable Measures and the Malliavin Calculus

- Contents v6 free
- Preface ix10 free
- Chapter 1. Background material 118 free
- Chapter 2. Sobolev spaces on $\mathbb{R}^n$ 3956
- Chapter 3. Differentiable measures on linear spaces 6986
- Chapter 4. Some classes of differentiable measures 105122
- Chapter 5. Subspaces of differentiability of measures 133150
- Chapter 6. Integration by parts and logarithmic derivatives 157174
- Chapter 7. Logarithmic gradients 189206
- Chapter 8. Sobolev classes on infinite dimensional spaces 227244
- Chapter 9. The Malliavin calculus 279296
- Chapter 10. Infinite dimensional transformations 329346
- Chapter 11. Measures on manifolds 369386
- Chapter 12. Applications 401418
- References 427444
- Subject Index 483500