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Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35) by Zhenqing Chen, Masatoshi Fukushima

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  Chapter Two

 

BASIC PROPERTIES AND EXAMPLES OF DIRICHLET FORMS

2.1. TRANSIENCE, RECURRENCE, AND IRREDUCIBILITY

In this section, we introduce the concepts of the transience, recurrence, and irreducibility of the semigroup for general Markovian symmetric operators and present their characterizations by means of the associated Dirichlet form as well as the associated extended Dirichlet space. These notions are invariant under the time changes of the associated Markov process. We shall also examine them in the concrete examples of the next section.

As in Section 1.1, we let (E, B(E), m) be a σ-finite measure space and consider a strongly continuous contraction semigroup {Tt; t > 0} of symmetric Markovian operators and a Dirichlet form (ε, ) on ...

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