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portada Dirichlet Forms and Analysis on Wiener Space
Type
Physical Book
Publisher
Author
Year
1991
Language
English
Pages
325
Format
Hardcover
ISBN
3110129191
ISBN13
9783110129199
Edition No.
2010
Categories

Dirichlet Forms and Analysis on Wiener Space

Nicolas Bouleau; Francis Hirsch (Author) · De Gruyter · Hardcover

Dirichlet Forms and Analysis on Wiener Space - Nicolas Bouleau; Francis Hirsch

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Synopsis "Dirichlet Forms and Analysis on Wiener Space "

The subject of this book is analysis on Wiener space by means of Dirichlet forms and Malliavin calculus. There are already several literature on this topic, but this book has some different viewpoints. First the authors review the theory of Dirichlet forms, but they observe only functional analytic, potential theoretical and algebraic properties. They do not mention the relation with Markov processes or stochastic calculus as discussed in usual books (e.g. Fukushima's book). Even on analytic properties, instead of mentioning the Beuring-Deny formula, they discuss "carréeacute; du champ" operators introduced by Meyer and Bakry very carefully. Although they discuss when this “carré du champ” operator exists in general situation, the conditions they gave are rather hard to verify, and so they verify them in the case of Ornstein-Uhlenbeck operator in Wiener space later. (It should be noticed that one can easily show the existence of “carré du champ” operator in this case by using Shigekawa’s H-derivative.) In the part on Malliavin calculus, the authors mainly discuss the absolute continuity of the probability law of Wiener functionals. The Dirichlet form corresponds to the first derivative only, and so it is not easy to consider higher order derivatives in this framework. This is the reason why they discuss only the first step of Malliavin calculus. On the other hand, they succeeded to deal with some delicate problems (the absolute continuity of the probability law of the solution to stochastic differential equations with Lipschitz continuous coefficients, the domain of stochastic integrals (Itôocirc;-Ramer-Skorokhod integrals), etc.). This book focuses

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