Differentiable measures and the Malliavin calculus. (English) Zbl 0929.58015
The article is a detailed survey of the topic formulated in its title that includes both the contemporary state of the subject in the large and the author’s results. Taking into account the length of the paper and 616 items in References it may be considered as a monograph giving a systematic exposition of this theory.
The paper consists of 12 chapters. The first one explains the background of the theory. Chapters 2-6 are devoted to the properties of measures on infinite-dimensional spaces analogous to those in \(\mathbb{R}^n\) such as existence and smoothness of the densities of measures with respect to the Lebesgue measure (recall that nothing like this is possible in infinite dimensions). Chapter 7 deals with Sobolev classes in infinite-dimensional spaces. Chapters 6 and 7 contain the author’s results on vector logarithmic derivatives and Sobolev spaces over smooth measures.
In Chapters 8-10 the nonlinear transformations of smooth measures are under consideration. In particular, a central problem of the Malliavin calculus such as the regularity of measures induced by smooth functionals are described here. Also the measurable manifolds (measurable spaces with a certain differentiable structure) are discussed. Chapters 11 and 12 contain some applications (to Malliavin calculus, Dirichlets forms, diffusions, etc.).
The paper consists of 12 chapters. The first one explains the background of the theory. Chapters 2-6 are devoted to the properties of measures on infinite-dimensional spaces analogous to those in \(\mathbb{R}^n\) such as existence and smoothness of the densities of measures with respect to the Lebesgue measure (recall that nothing like this is possible in infinite dimensions). Chapter 7 deals with Sobolev classes in infinite-dimensional spaces. Chapters 6 and 7 contain the author’s results on vector logarithmic derivatives and Sobolev spaces over smooth measures.
In Chapters 8-10 the nonlinear transformations of smooth measures are under consideration. In particular, a central problem of the Malliavin calculus such as the regularity of measures induced by smooth functionals are described here. Also the measurable manifolds (measurable spaces with a certain differentiable structure) are discussed. Chapters 11 and 12 contain some applications (to Malliavin calculus, Dirichlets forms, diffusions, etc.).
Reviewer: Yu.E.Gliklikh (Voronezh)
MSC:
| 58J65 | Diffusion processes and stochastic analysis on manifolds |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 28A99 | Classical measure theory |
| 60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |
| 58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |
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