Descripción del título
Dirichlet forms and analysi...
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 "carré 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 (Itô-Ramer-Skorokhod integrals), etc.). This book focuses on the abstract structure of Dirichlet forms and Malliavin calculus rather than their applications. However, the authors give a lot of exercises and references and they may help the reader to study other topics which are not discussed in this book. Zentralblatt Math, Reviewer: S.Kusuoka (Hongo)
Monografía
monografia Rebiun36127591 https://catalogo.rebiun.org/rebiun/record/Rebiun36127591 m o d | cr -n--------- 910816s1991 gw ob 001 0 eng d (OCoLC)1002061161 3-11-085838-X 10.1515/9783110858389 doi UPVA 998797962403706 CBUC 991013161249506708 MiAaPQ eng rda pn MiAaPQ MiAaPQ eng gw DE MAT000000 bisacsh 519.2 20 SK 600 rvk Bouleau, Nicolas Dirichlet forms and analysis on Wiener space Nicolas Bouleau, Francis Hirsch Reprint 2010 Berlin New York W. de Gruyter 1991 Berlin New York Berlin New York W. de Gruyter 1 online resource (336 p.) 1 online resource (336 p.) Text txt computer c online resource cr De Gruyter studies in mathematics 14 Description based upon print version of record Includes bibliographical references (p. [309]-320) and index Frontmatter -- I General Dirichlet forms -- II Dirichlet forms on vector spaces -- III Analysis on Wiener space -- IV Stochastic differential equations -- V The algebra of Dirichlet structures -- VI An extension of Girsanov's theorem -- VII Quasi-everywhere convergence -- Notes -- Bibliography -- Index -- Backmatter 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 "carré 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 (Itô-Ramer-Skorokhod integrals), etc.). This book focuses on the abstract structure of Dirichlet forms and Malliavin calculus rather than their applications. However, the authors give a lot of exercises and references and they may help the reader to study other topics which are not discussed in this book. Zentralblatt Math, Reviewer: S.Kusuoka (Hongo) Issued also in print English Dirichlet forms Malliavin calculus Hirsch, F. Francis) De Gruyter studies in mathematics (CKB)991042745756700 3-11-012919-1 De Gruyter Studies in Mathematics