- Published 2011

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 on E and establish the correspondence of the above-mentioned notions to the semigroups of symmetric Markovian linear operators. The present arguments are a little longer than the usual ones under the topological assumption found in [39] and [73, §1.4] but they are quite elementary in nature. Only at the end of this section, we shall assume that E is a Hausdorff topological space and consider the semigroups and Dirichlet forms generated by symmetric Markovian transition kernels on E. Let (E,B(E)) be a measurable space and m a σ -finite measure on it. Let Bm(E) be the completion of B(E) with respect to m. Numerical functions f , g on E are said to be m-equivalent (f = g [m] in notation) if m({x ∈ E : f (x) = g(x)}) = 0. For p ≥ 1 and a numerical function f ∈ Bm(E), we put

@inproceedings{SEMIGROUPS2011ChapterOS,
title={Chapter One SYMMETRIC MARKOVIAN SEMIGROUPS AND DIRICHLET FORMS},
author={SYMMETRIC MARKOVIAN SEMIGROUPS},
year={2011}
}