Dirichlet Forms and Analysis on Wiener Space
Dirichlet Forms and Analysis on Wiener Space
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DOI：
10.1515/9783110858389
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发表时间：
1991-10
期刊：
影响因子：
--
通讯作者：
N. Bouleau;F. Hirsch
中科院分区：
文献类型：
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作者：
N. Bouleau;F. Hirsch
这本书的主题是利用Dirichlet形式和Malliavin演算分析Wiener空间。关于这个主题已经有几篇文献了，但这本书有一些不同的观点。作者首先回顾了Dirichlet形式的理论，但它们只观察到泛函解析性质、位势理论性质和代数性质。它们没有像通常的书(如《福岛S》一书)所讨论的那样，提到与马尔可夫过程或随机微积分的关系。即使是关于解析性质，他们也没有提到Beuring-Deny公式，而是非常仔细地讨论了Meyer和Bakry引入的Carre du Champ算子。虽然他们讨论了Carre-Du Champ算子在一般情况下何时存在，但他们给出的条件很难证明，所以他们后来在Wiener空间中证明了Ornstein-Uhlenbeck算子的情况。(应该注意的是，在这种情况下，人们可以很容易地利用Shigekawa S的H-导数证明Carre Du Champ算子的存在性。)在Malliavin演算部分，主要讨论了Wiener泛函概率律的绝对连续性。Dirichlet形式只对应于一阶导数，因此在这个框架中考虑高阶导数并不容易。这就是为什么他们只讨论Malliavin微积分的第一步。另一方面，他们成功地处理了一些微妙问题(具有Lipschitz连续系数的随机微分方程解的概率律的绝对连续性，随机积分(Ito-Ramer-Skorokhod积分)的区域等)。这本书侧重于狄利克雷特形式和Malliavin演算的抽象结构，而不是它们的应用。然而，作者提供了大量的练习和参考资料，他们可能会帮助读者学习本书中没有讨论的其他主题。中央布拉特数学，审稿人：S.Kusuoka(Hongo)
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 carre du champ operators introduced by Meyer and Bakry very carefully. Although they discuss when this carre 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 carre 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 (Ito-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) "