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

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  Chapter Six

 

REFLECTED DIRICHLET SPACES

Reflected Dirichlet space was introduced for a regular Dirichlet form by M. L. Silverstein in 1974 in [138, 139] and further investigated by the first-named author in [16]. Reflected Dirichlet space plays an important role for the boundary theory of symmetric Markov processes. While it is possible to introduce reflected Dirichlet form for a quasi-regular Dirichlet form directly, we choose to do it first in the regular Dirichlet form setting as this allows us to define the reflected Dirichlet space via terminal random variables and harmonic functions of finite energy. This approach sheds more insight into the probabilistic meaning and the structure of the reflected Dirichlet form. As illuminated by the ...

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