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

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  Chapter One

 

SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS

1.1. DIRICHLET FORMS AND EXTENDED DIRICHLET SPACES

The concepts of Dirichlet form and Dirichlet space were introduced in 1959 by A. Beurling and J. Deny [8] and the concept of the extended Dirichlet space was given in 1974 by M. L. Silverstein [138]. They all assumed that the underlying state space E is a locally compact separable metric space. Concrete examples of Dirichlet forms (bilinear form, weak solution formulations) have appeared frequently in the theory of partial differential equations and Riemannian geometry. However, the theory of Dirichlet forms goes far beyond these.

In this section, we work with a σ-finite measure space (E, B(E), m) without any topological assumption ...

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