Introduction to the theory of (non-symmetric) Dirichlet forms.

*(English)*Zbl 0826.31001
Universitext. Berlin: Springer-Verlag. viii, 209 p. (1992).

Since the book was published in 1992 (and a first draft was sent to the reviewer by the authors much earlier), a lot of reviews had meanwhile been published, including one by M. Fukushima, the doyen in the theory of Dirichlet forms, in Metrika. For this reason the review here will be rather short.

The objective of the book is to give an introduction to the theory of non-symmetric Dirichlet forms in general state spaces. After giving some background material on functional analysis and discussing a lot of examples, the authors give a detailed discussion of the potential theory of Dirichlet forms, especially they handle exceptional sets and stress the importance of the notion of quasi-continuity. In chapter IV they construct the Markov process associated with a Dirichlet form. The main point is that the authors work with quasi-regular forms, not only with regular ones. Finally they handle concrete processes in the frame of Dirichlet forms.

The book is well written and may serve as a textbook for an introductory course on Dirichlet forms.

The objective of the book is to give an introduction to the theory of non-symmetric Dirichlet forms in general state spaces. After giving some background material on functional analysis and discussing a lot of examples, the authors give a detailed discussion of the potential theory of Dirichlet forms, especially they handle exceptional sets and stress the importance of the notion of quasi-continuity. In chapter IV they construct the Markov process associated with a Dirichlet form. The main point is that the authors work with quasi-regular forms, not only with regular ones. Finally they handle concrete processes in the frame of Dirichlet forms.

The book is well written and may serve as a textbook for an introductory course on Dirichlet forms.

Reviewer: N.Jacob (München)

##### MSC:

31-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to potential theory |

31C25 | Dirichlet forms |

60J45 | Probabilistic potential theory |