Abstract: Publisher SummaryThis chapter discusses the basic theory of Dirichlet forms. The Dirichlet form on an L2-space is defined as a Markovian closed symmetric form. The link connecting the theory of Dirichlet forms with Markov processes is in that the Markovian nature of a closed symmetric form is equivalent to the Markovian properties of the associated semigroup and resolvent on L2. A Markovian semigroup is transient if and only if the domain of the associated Dirichlet form can be extended to a function space (an extended Dirichlet space) relative to a weighted L2 -space by means of the completion with respect to the 0-order form. Thus, the Dirichlet space in the original sense of Beurling and Deny is a special extended Dirichlet space in the chapter. In addition, translation invariant Dirichlet forms are studied in examples.
Publication Year: 1980
Publication Date: 1980-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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